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"proof-obligations/GAP_LEDGER.md", "proof-obligations/FINITE_TO_UNBOUNDED_FAMILY_AUDIT.json", "proof-obligations/FINITE_TO_UNBOUNDED_FAMILY_AUDIT.md", "complexity/COMPLEXITY_LEDGER.json", "complexity/COMPLEXITY_LEDGER.md", "trust-base/TRUST_BASE.json", "trust-base/SHRINK_PLAN.json", "trust-base/SHRINK_PLAN.md", "release/PUBLIC_REVIEW_BOUNDARY.json", "release/PUBLIC_REVIEW_BOUNDARY.md", "release/PUBLIC_REVIEW_HANDOFF.json", "release/PUBLIC_REVIEW_HANDOFF.md", "review/PUBLIC_REVIEW_CHECKLIST.json", "review/PUBLIC_REVIEW_CHECKLIST.md", "release/PUBLIC_THEOREM_EMISSION_DENIAL.json", "release/PUBLIC_THEOREM_EMISSION_DENIAL.md", "release/PUBLIC_THEOREM_EMISSION_GATE.json", "release/PUBLIC_THEOREM_EMISSION_GATE.md", "release/PUBLIC_THEOREM_EMISSION_NEGATIVE_TRANSITIONS.json", "release/PUBLIC_THEOREM_EMISSION_NEGATIVE_TRANSITIONS.md", "release/PUBLIC_THEOREM_EMISSION_PREFLIGHT.json", "release/PUBLIC_THEOREM_EMISSION_PREFLIGHT.md", "release/RELEASE_BLOCKER_CLEARANCE.json", "release/RELEASE_BLOCKER_CLEARANCE.md", "review/EXTERNAL_REVIEW_STATUS.md", "REVIEWER_MAP.md", "TRUST_BASE.md", "PUBLIC_REVIEW.json", "PUBLIC_REVIEW.md", "RELEASE_LADDER.md", "release/RELEASE_LADDER.json", "review/EXTERNAL_REVIEW_STATUS.json" ], "subordinateLegacySurfaceRoots": [ "checker-cycles/", "checker-mutations/", "checker-totality/", "complexity/", "independent-verifiers/", "kernel/", "oracle-audit/", "proof-obligations/", "release/", "report-bindings/", "reproducibility/", "review/", "trust-base/", "artifacts/multi-platform-ci/", "artifacts/regeneration/", "archive/legacy-v0/" ], "remainingFormalObligations": [ "Formal.ConcreteSAT", "Formal.ResidualBandMinimizer", "Formal.ZeroSlack", "Formal.PolynomialRuntimeAndCertificateBounds", "Formal.RootTheoremAndAxiomAudit" ], "remainingBlockers": [ "Formal.ConcreteSAT", "Formal.ResidualBandMinimizer", "Formal.ZeroSlack", "Formal.PolynomialRuntimeAndCertificateBounds", "Formal.RootTheoremAndAxiomAudit" ], "projectSpecificAxiomInventory": [ "PNP.CheckPCCPackexp", "PNP.GeneratePCCPack", "PNP.LockedNANDThreshold", "PNP.ResidualBandExactMinimization" ], "lockedNANDThresholdHostileReviewLemmaInventory": [ "DirectWireOutputLowerBound", "MacroDistinct", "TraceEquivalence", "ZeroOutputConvention", "FinalLockSeparation" ], "leanLockedNANDThresholdPremiseInventory": [ "baselineCandidate", "fullCandidate", "baselineConditions", "initialOutputsPreserved", "unsatisfiableFinalZero", "satisfiableFinalConditions" ], "leanLockedNANDThresholdMissingInstantiationInventory": [], "verificationCommands": [ "node pcc-formal-reconstruction-status0.mjs --json", "node pcc-formal-public-surface0.mjs --json", "npm run legacy:v0:check", "npm run pnp:verify -- --no-write", "node --test audits/lean-root-target0.test.mjs", "node --test audits/lean-concrete-machine0.test.mjs", "node --test audits/lean-concrete-tape-handoff0.test.mjs", "node --test audits/lean-concrete-tape-blank-equivalence0.test.mjs", "node --test audits/lean-concrete-pipeline-tape-geometry0.test.mjs", "node --test audits/lean-concrete-pipeline-input-framer0.test.mjs", "node --test audits/lean-concrete-pipeline-output-handoff0.test.mjs", "node --test audits/lean-concrete-pipeline-state-namespace0.test.mjs", "node --test audits/lean-concrete-pipeline-sequential-state-namespace0.test.mjs", "node --test audits/lean-concrete-pipeline-sequential-compiler0.test.mjs", "node --test audits/lean-concrete-pipeline-stage-bridges0.test.mjs", "node --test audits/lean-concrete-terminal-output-packer0.test.mjs", "node --test audits/lean-concrete-pipeline-terminal-bridge0.test.mjs", "node --test audits/lean-concrete-pipeline-paired-compiler0.test.mjs", "node --test audits/lean-concrete-pipeline-compiler0.test.mjs", "node --test audits/lean-concrete-pipeline-machine-simulation0.test.mjs", "node --test audits/lean-concrete-complexity0.test.mjs", "node --test audits/lean-concrete-pipeline-refinement0.test.mjs", "node --test audits/lean-concrete-cnf0.test.mjs", "node --test audits/lean-concrete-cook-levin-raw-tape-bridge0.test.mjs", "node --test audits/lean-concrete-cook-levin-formula-size0.test.mjs", "node --test audits/lean-concrete-cook-levin-formula-schedule0.test.mjs", "node --test audits/lean-concrete-cook-levin-formula-cursor0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-input-length0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-input-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-token-appender0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-first-token-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-complete-header0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-body-start-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-first-literal-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-first-clause-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-dynamic-token-cursor-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-first-clause-padding-run0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-clause-separator-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-clause-first-literal-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-clause-second-literal-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-clause-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-clause-padding-run0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-third-clause-separator-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-third-clause-first-literal-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-third-clause-second-literal-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-third-clause-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-third-clause-padding-run0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-fourth-clause-separator-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-fourth-clause-first-literal-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-fourth-clause-second-literal-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-fourth-clause-prefix0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-fourth-clause-padding-run0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-fifth-clause-padding-run0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-first-constraint-padding-run0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-separator-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-first-literal-sign-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-first-literal-first-unary-unit-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-first-literal-second-unary-unit-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-first-literal-third-unary-unit-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-first-literal-terminator-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-first-literal-successor-token-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-padding-or-unary-opportunity-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-second-padding-or-unary-opportunity-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-third-padding-or-unary-opportunity-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-fourth-padding-or-unary-opportunity-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-fifth-padding-or-terminator-opportunity-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-sixth-padding-or-opening-unary-opportunity-step0.test.mjs", "node --test audits/lean-concrete-cook-levin-builder-second-constraint-seventh-padding-or-unary-opportunity-step0.test.mjs", "node --test audits/lean-nand-semantics0.test.mjs", "node --test audits/lean-nand-enumerator0.test.mjs", "node --test audits/lean-nand-reference-minimum0.test.mjs", "node --test audits/lean-locked-nand-baseline0.test.mjs", "node --test audits/lean-locked-nand-threshold-boundary0.test.mjs", "node --test audits/lean-locked-nand-carrier-trace0.test.mjs", "node --test audits/lean-residual-routes0.test.mjs", "node --test audits/lean-residual-gain-chain0.test.mjs", "node --test audits/lean-residual-gain-stopping0.test.mjs", "node --test audits/lean-residual-terminal-full-bridge0.test.mjs", "lake build PNP", "lake env lean -DwarningAsError=true lean-audit/PNPBridgeAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteBitStringAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteMachineAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteTapeHandoffAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteTapeBlankEquivalenceAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteTapeBlankEquivalence.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcretePipelineTapeGeometryAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcretePipelineInputFramerAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcretePipelineOutputHandoffAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcretePipelineStateNamespaceAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcretePipelineSequentialStateNamespaceAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcretePipelineSequentialCompilerAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcretePipelineSequentialCompiler.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcretePipelineStageBridgesAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteTerminalOutputPackerAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcretePipelineTerminalBridgeAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcretePipelinePairedCompilerAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcretePipelinePairedCompiler.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcretePipelineCompilerAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcretePipelineCompiler.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcretePipelineMachineSimulationAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteComplexityAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcretePipelineRefinementAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcretePipelineRefinementRecursive.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteTargetAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCNFAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCNFWorkInputAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCNFVerifierAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCNFWorkAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinRawTapeBridgeAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinFormulaSizeAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinFormulaSize.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinFormulaScheduleAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinFormulaSchedule.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinFormulaCursorAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinFormulaCursor.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderInputLengthAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderInputLength.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderInputPrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderInputPrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderTokenAppenderAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderTokenAppender.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderFirstTokenPrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderFirstTokenPrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderUnaryPolynomialAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderCompleteHeaderAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderCompleteHeader.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderBodyStartPrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderBodyStartPrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderFirstLiteralPrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderFirstLiteralPrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderFirstClausePrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderFirstClausePrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderDynamicTokenCursorStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderDynamicTokenCursorStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderFirstClausePaddingRunAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderFirstClausePaddingRun.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondClauseSeparatorStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondClauseSeparatorStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondClauseFirstLiteralPrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondClauseFirstLiteralPrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondClauseSecondLiteralPrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondClauseSecondLiteralPrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondClausePrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondClausePrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondClausePaddingRunAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondClausePaddingRun.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderThirdClauseSeparatorStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderThirdClauseSeparatorStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderThirdClauseFirstLiteralPrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderThirdClauseFirstLiteralPrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderThirdClauseSecondLiteralPrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderThirdClauseSecondLiteralPrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderThirdClausePrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderThirdClausePrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderThirdClausePaddingRunAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderThirdClausePaddingRun.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderFourthClauseSeparatorStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderFourthClauseSeparatorStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderFourthClauseFirstLiteralPrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderFourthClauseFirstLiteralPrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderFourthClauseSecondLiteralPrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderFourthClauseSecondLiteralPrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderFourthClausePrefixAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderFourthClausePrefix.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderFourthClausePaddingRunAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderFourthClausePaddingRun.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderFifthClausePaddingRunAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderFifthClausePaddingRun.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderFirstConstraintPaddingRunAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderFirstConstraintPaddingRun.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintSeparatorStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintSeparatorStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintFirstLiteralSignStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintFirstLiteralSignStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintFirstLiteralFirstUnaryUnitStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintFirstLiteralSecondUnaryUnitStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintFirstLiteralThirdUnaryUnitStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintFirstLiteralTerminatorStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintFirstLiteralTerminatorStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintFirstLiteralSuccessorTokenStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintFirstLiteralSuccessorTokenStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintPaddingOrUnaryOpportunityStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintPaddingOrUnaryOpportunityStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintSecondPaddingOrUnaryOpportunityStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintThirdPaddingOrUnaryOpportunityStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintFourthPaddingOrUnaryOpportunityStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.lean", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCookLevinBuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStepAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCookLevinBuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCNFWorkCanonical.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteWorkCompilerEdges.lean", "lake env lean -DwarningAsError=true --run lean-regression/PNPConcreteCNFWorkCanonicalExtended.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCNFWorkExhaustive.lean", "lake env lean -DwarningAsError=true lean-audit/PNPNANDSemanticsAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPNANDEnumeratorAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPNANDTruthTableAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPNANDMinimumAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPNANDCompositionAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPNANDSlackAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPLockedNANDDirectAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPDirectWireBaselineAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPLockedNANDBaselineAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPLockedNANDLocalBaselineAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPLockedNANDThresholdBoundaryAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPLockedNANDCarrierTraceAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPLockedNANDCarrierTrace.lean", "lake env lean -DwarningAsError=true lean-audit/PNPLockedNANDGlobalCandidatesAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPLockedNANDGlobalCandidates.lean", "lake env lean -DwarningAsError=true lean-audit/PNPLockedNANDGlobalBaselineDistinctAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPLockedNANDGlobalBaselineDistinct.lean", "node --test audits/lean-locked-nand-global-baseline-distinct0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPLockedNANDGlobalUnsatisfiableFinalZeroAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPLockedNANDGlobalUnsatisfiableFinalZero.lean", "node --test audits/lean-locked-nand-global-unsatisfiable-final-zero0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPLockedNANDGlobalSemanticThresholdAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPLockedNANDGlobalSemanticThreshold.lean", "node --test audits/lean-locked-nand-global-semantic-threshold0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteLockedNANDSemanticReductionAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteLockedNANDSemanticReduction.lean", "node --test audits/lean-concrete-locked-nand-semantic-reduction0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteLockedNANDSourceParserAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteLockedNANDSourceParser.lean", "node --test audits/lean-concrete-locked-nand-source-parser0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteLockedNANDTargetEmitterAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteLockedNANDTargetEmitter.lean", "node --test audits/lean-concrete-locked-nand-target-emitter0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteLockedNANDPolynomialReductionAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteLockedNANDPolynomialReduction.lean", "node --test audits/lean-concrete-locked-nand-polynomial-reduction0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCNFToNANDAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCNFToNAND.lean", "node --test audits/lean-concrete-cnf-to-nand0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteCNFToNANDPolynomialReductionAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPConcreteCNFToNANDPolynomialReduction.lean", "node --test audits/lean-concrete-cnf-to-nand-polynomial-reduction0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPConcreteLockedNANDThresholdPublicationAxiomAudit.lean", "node --test audits/lean-concrete-locked-nand-threshold-publication0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualRoutesAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPResidualGainChainAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-audit/PNPLockedNANDResidualGainBoundAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualGainChain.lean", "lake env lean -DwarningAsError=true lean-audit/PNPResidualGainStoppingAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualGainStopping.lean", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalFullBridgeAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalFullBridge.lean", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalModeFirewallAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalModeFirewall.lean", "node --test audits/lean-residual-terminal-mode-firewall0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalProjectionMinimumAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalProjectionMinimum.lean", "node --test audits/lean-residual-terminal-projection-minimum0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalProjectionTransferAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalProjectionTransfer.lean", "node --test audits/lean-residual-terminal-projection-transfer0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalSaturationAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalSaturation.lean", "node --test audits/lean-residual-terminal-saturation0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalPhysicalSupportCompletionAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalPhysicalSupportCompletion.lean", "node --test audits/lean-residual-terminal-physical-support-completion0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalSupportExtractionAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalSupportExtraction.lean", "node --test audits/lean-residual-terminal-support-extraction0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalProperSupportAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalProperSupport.lean", "node --test audits/lean-residual-terminal-proper-support0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalSupportSquareClosureAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalSupportSquareClosure.lean", "node --test audits/lean-residual-terminal-support-square-closure0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalGovernedSupportCompletionAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalGovernedSupportCompletion.lean", "node --test audits/lean-residual-terminal-governed-support-completion0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalFrontierPushoutAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalFrontierPushout.lean", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalProjectionSquareAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalProjectionSquare.lean", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalSideTightMinimumAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalSideTightMinimum.lean", "node --test audits/lean-residual-terminal-side-tight-minimum0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalFourCornerCarrierAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalFourCornerCarrier.lean", "node --test audits/lean-residual-terminal-four-corner-carrier0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalFourCornerOptimumCompatibilityAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalFourCornerOptimumCompatibility.lean", "node --test audits/lean-residual-terminal-four-corner-optimum-compatibility0.test.mjs", "node --test audits/lean-residual-terminal-frontier-pushout0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalBN3RequestEnvelopeAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalBN3RequestEnvelope.lean", "node --test audits/lean-residual-terminal-bn3-request-envelope0.test.mjs", "lake env lean -DwarningAsError=true lean-regression/PNPBN3JointRealizabilityGap.lean", "node --test audits/bn3-joint-realizability-gap0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalBN4ActivationCancellationAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalBN4ActivationCancellation.lean", "node --test audits/lean-residual-terminal-bn4-activation-cancellation0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalBN5FullShadowLocalizationAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalBN5FullShadowLocalization.lean", "node --test audits/lean-residual-terminal-bn5-full-shadow-localization0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalPkgCSeparatingConsumersAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalPkgCSeparatingConsumers.lean", "node --test audits/lean-residual-terminal-pkgc-separating-consumers0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalPkgCTypedRestorationAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalPkgCTypedRestoration.lean", "node --test audits/lean-residual-terminal-pkgc-typed-restoration0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalPkgCSameKeyCancellationAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalPkgCSameKeyCancellation.lean", "node --test audits/lean-residual-terminal-pkgc-same-key-cancellation0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalPkgCAmbientBN4LedgerAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalPkgCAmbientBN4Ledger.lean", "node --test audits/lean-residual-terminal-pkgc-ambient-bn4-ledger0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalPkgCAmbientBN4ResidualReductionAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalPkgCAmbientBN4ResidualReduction.lean", "node --test audits/lean-residual-terminal-pkgc-ambient-bn4-residual-reduction0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalConsumerAntichainNormalFormAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalConsumerAntichainNormalForm.lean", "node --test audits/lean-residual-terminal-consumer-antichain-normal-form0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalConstantCutHypergraphRigidityAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalConstantCutHypergraphRigidity.lean", "node --test audits/lean-residual-terminal-constant-cut-hypergraph-rigidity0.test.mjs", "lake env lean -DwarningAsError=true lean-audit/PNPResidualTerminalBN6HypergraphPacketAxiomAudit.lean", "lake env lean -DwarningAsError=true lean-regression/PNPResidualTerminalBN6HypergraphPacket.lean", "node --test audits/lean-residual-terminal-bn6-hypergraph-packet0.test.mjs", "node scripts/export-lean-theorem-inventory.mjs --check", "node scripts/generate-formal-publication.mjs --check", "node --test audits/lean-theorem-inventory0.test.mjs audits/formal-publication0.test.mjs", "npm run report:check" ], "nonClaims": [ "The repository does not currently establish P = NP.", "Legacy JavaScript checker acceptance verifies assertion-bearing records under implemented predicates; it is not a formal proof of the named mathematical propositions.", "The BN3 joint-realizability gap still shows that arbitrary per-cut side-tight existence cannot imply a stable family. The successful computed BCEL nucleus has a candidate-derived finite repair with canonical request identities, exact minimal consumers, duplicate-free incidence, and one jointly side-tight basis selection function, but its all-subsets enumeration is exponential. The finite BN4 kernel consumes that repaired envelope without repairing arbitrary caller-supplied per-cut witnesses.", "The finite BN4 activation-exact cancellation kernel classifies exact integer positive and negative mass at each complete typed key over an explicit caller-supplied cell ledger. It does not derive the cells, semantic signatures, or transport types from four-corner bases; establish the full historical BN4 theorem; construct PkgC or BN6; complete global routes or selectors; establish ZeroSlack or polynomial PCCMin; put SAT in P; or prove P = NP.", "The finite BN5 full-shadow localization kernel uniformly handles arbitrary finite exact-coordinate unit and quotient-shadow ledgers. It validates negative-mass refinement, computes cut silence, and returns complete multiplicity coverage or a strict Hall deficit routed to local X1. The payloads and shadow universe are explicit inputs; complete matching is not connected back to a BN4 contradiction; full CritC/Q/E/L/X2/X3/X4 diagnosis, the full historical BN5 theorem, full PkgC and BN6, global routes, selectors, polynomial generation and runtime, ZeroSlack, PCCMin, SAT in P, and P = NP remain unproved.", "The finite PkgC separating-consumer classifier scans an arbitrary explicit minimal-consumer antichain for the first disjoint nonsingleton pair, canonically indexes its atoms, and returns exact-coordinate restoration coverage or a strict Hall deficit routed to local Q; absence is exactly the V54 singletonization premise. The restoration coordinate universe remains explicit, coverage is not connected back to a BN4 or BN5 contradiction, the local route is not embedded in the complete global outcome system, and full PkgC route silence, derivation from a terminal candidate, polynomial generation and runtime, ZeroSlack, PCCMin, SAT in P, and P = NP remain unproved.", "The finite PkgC typed-restoration realization accepts an explicit typed coordinate-preserving restoration operation, materializes one full candidate for each atom of the first separating pair, proves exact positional coordinates and multiplicity coverage, and excludes a strict Hall deficit for that equality-fibre graph. It does not construct the restoration operation from a terminal candidate, prove its full semantic adequacy, connect complete restoration to a BN4 or BN5 contradiction, embed local routes in the global outcome system, establish full PkgC, BN6, or Packet completeness, prove polynomial generation or runtime, ZeroSlack, PCCMin, SAT in P, or P = NP.", "The finite PkgC same-key cancellation kernel mechanically generates an exactly balanced opposite-sign BN4 ledger from an explicit typed restorer. The ambient-ledger milestone embeds those cells into an explicit BN4 ledger by exact permutation, preserves multiplicity, proves balanced-subledger removal leaves exactly the explicit remainder, and retains successful candidate-derived BN4 canonical-atom evidence. It does not derive the restorer or ambient ledger from a terminal candidate, prove semantic restoration adequacy or complete global route silence, establish full historical PkgC, BN6, or Packet completeness, prove polynomial generation or runtime, ZeroSlack, PCCMin, SAT in P, or P = NP.", "The finite PkgC ambient BN4 residual-reduction kernel proves that removing the exact balanced generated subledger preserves every executable per-key residual cell and the complete residual ledger over the ambient canonical key universe; an empty explicit remainder then yields an empty ambient residual ledger. The ambient ledger, typed restorer, exact embedding, and remainder remain explicit proof-bearing inputs. It does not derive those inputs, prove that a candidate-derived remainder is empty or route-producing, establish restoration semantic adequacy or complete global route silence, complete historical PkgC, BN6, or Packet, prove polynomial runtime, ZeroSlack, PCCMin, SAT in P, or P = NP.", "The arbitrary-finite V54 consumer-antichain normal form proves that two-sided activation is nonzero exactly when a minimal-consumer antichain has a disjoint pair and, under the exact PkgC singletonization premise, is literally the cut indicator of the singleton footprint. The theorem itself consumes an explicit antichain. The finite BN6 bridge now transports explicitly grouped instances into V53, but full PkgC construction and route silence, derivation and grouping from terminal candidates, global routes, selectors, polynomial runtime, ZeroSlack, PCCMin, SAT in P, and P = NP remain unproved.", "The arbitrary-finite V53 constant-cut hypergraph rigidity theorem proves the exact q=2, q=3, and q>=4 classification for sparse positive hypergraphs with one common nonempty proper-cut value. The finite BN6 bridge now constructs such a hypergraph from explicit grouped V54 cells and retains payload witnesses, but full PkgC construction, terminal-candidate derivation and grouping, full historical BN6 and Packet selector/realizer completeness, global routes, polynomial runtime, ZeroSlack, PCCMin, SAT in P, and P = NP remain unproved.", "The finite BN6 hypergraph-packet bridge transports an explicit already-grouped positive payload-bearing V54 survivor family into the exact V53 cut sum and returns the pair, mixed three-anchor, or full-span packet classification with source payload witnesses. It does not complete PkgC, derive or group survivors from a terminal candidate, establish full historical BN6 or Packet selector/realizer completeness, complete global routes, prove polynomial generation or runtime, ZeroSlack or PCCMin, SAT in P, or P = NP.", "The current Lean bridge is partial and does not contain the required concrete, assumption-audited root theorem.", "The pinned Lean library/root-status build is reconstruction data, not a proof of P = NP.", "Blank-delimited Tape.outputBits removes dependence on the unobservable represented-list boundary; Tape.handoffTarget itself is a pure canonical specification. PipelineOutputHandoff is a separate executable internal represented handoff, not terminal raw output normalization.", "TapeBlankEquivalence proves that finite tapes which differ only by materialized exterior blanks have identical raw execution and observable output. PipelineCompiler uses that relation to connect every ordinary empty, odd, or even raw startConfig to the same literal four-stage work pipeline.", "PipelineTape frames raw cells between distinct two-track markers and tolerates arbitrary exterior garbage; its boundary expansions are pure tape identities, not transition rules, a handoff machine, a compiler, or a runtime proof.", "PipelineInputFramer is one literal finite machine for every raw bitstring. It handles the empty word, complete two-bit work cells, and an odd final raw bit through explicit finite transitions; reaches an accepting represented frame with permitted exterior garbage at exact branch costs; and from ordinary startConfig accepts without timeout within 6 * m * m + 39 * m + 75 raw steps. All 70 public declarations have empty axiom closure. PipelineCompiler and PipelineRefinement now transport that endpoint through complete recursive raw refinement, but CNF-SAT in P, NP-completeness, and P = NP remain absent.", "PipelineOutputHandoff is one literal finite machine for an already represented internal tape. For logical output length n it reaches an accepting representation of Tape.handoffTarget in exactly 2 * n + 4 work steps and 12 * n + 24 compiled steps. PipelineTerminalBridge preserves the earlier ordinary-input trace in its extended rule table and composes it with verdict-indexed terminal-packer copies for every supplied exact target execution. This still supplies no target-termination theorem, complete external-input-size polynomial, or RawRefinement. CNF-SAT in P, NP-completeness, and P = NP remain unproved.", "PipelineMachineSimulation extracts from every at-most raw run with fuel F an exact successful prefix of length k at most F reaching the same endpoint, and each supplied exact n-step run costs exactly 3 * n work transitions. PipelineCompiler extracts that prefix internally on every raw bitstring, bounds target output by m + F + 1, composes all four executable stages in one literal raw machine, and proves exact verdict, no-timeout, and machineOutput equality at an explicit polynomial in external length m. PipelineRefinement now recursively composes those raw implementations; CNF-SAT in P, NP-completeness, and P = NP remain missing.", "PipelineStateNamespace remains the injective renaming and lookup-isolation prerequisite. PipelineTerminalBridge now proves that every successful earlier bridge step and exact trace is preserved without shadowing, then composes accepting and rejecting supplied traces through two disjoint packer copies. Target termination, an external-input-size polynomial, and pipeline RawRefinement remain missing.", "PipelineSequentialStateNamespace nests two complete component machines in disjoint outer state images and supplies literal accept/reject launches from the first component into the second simulator. PipelineSequentialCompiler composes both exact executions in that one finite table for every raw input, passes either first verdict onward, preserves the second verdict and ordinary output, retains stuck-first timeout, and proves the external polynomial R(m) = PipelineRaw(p)(m) + 6 + PipelineRaw(q)(m + p(m) + 1). PipelineRefinement recursively applies that compiler to function composition and decision precomposition, closing the concrete machine-link blocker. CNF-SAT in P, NP-completeness, and P = NP remain absent.", "PipelineStageBridges proves exact framer-to-simulator and accept/reject-to-handoff launches, collision-free first-match dispatch, exact cumulative work cost, six-for-one compiled raw cost, and accept/reject/timeout classification for supplied exact target traces. PipelineTerminalBridge transports those traces into the extended machine and composes the terminal suffix, with exact ordinary machineOutput and the local 18*n^2 + 36*n + 12 suffix bound. This module alone does not prove target termination or an external-input-size polynomial; downstream modules now supply both and recursive RawRefinement. CNF-SAT in P, NP-completeness, and P = NP remain absent.", "TerminalOutputPacker is a literal finite work machine that uniformly packs empty, one-bit, odd, even, all-zero, all-one, and mixed logical outputs in the presence of arbitrary exterior garbage. It proves exact work execution, a designated halt, ordinary blank-delimited raw output equality, exact six-for-one compilation, the local raw bound 18*n^2 + 36*n + 6, and one-step-short timeout, all with empty axiom closure. Downstream modules connect it to target termination, an external encoded-input-size polynomial, and recursive RawRefinement. CNF-SAT in P, NP-completeness, and P = NP remain absent.", "PipelineTerminalBridge is one literal extended finite work machine containing the earlier bridge rules and two disjoint terminal-packer copies. PipelineCompiler transports every raw input through that table, supplies target termination from PolynomialTimeMachine.haltsWithin, preserves exact verdict and ordinary machineOutput, and retains stuck-endpoint timeout. PipelineRefinement now supplies the recursive FunctionProgram and DecisionProgram RawRefinement constructors.", "PipelinePairedCompiler remains the sharper canonical-pair theorem and compiles the same four-stage work table. PipelineCompiler is its all-input successor: all 29 public declarations have empty axiom closure, and pipeline_boundedDecide_eq, pipeline_machineOutput_eq, pipeline_ne_timeout, and pipeline_accepts_iff quantify every raw BitString. The output polynomial is B(m) = m + p(m) + 1; the runtime polynomial adds 6*m*m + 39*m + 75, two six-step launches, 18*p(m), and output-bound substitutions into the handoff and terminal-packer polynomials. PipelineRefinement recursively compiles composite charged programs to one raw machine; there is still no CNF-SAT in P, NP-completeness, or P = NP claim.", "The concrete bitstring, natural-polynomial, and finite-rule machine kernel is consumed by a finite charged-pipeline P/NP/reduction interface, and every finite function/decision program tree now has a literal raw-machine refinement with a recursively generated polynomial bound. This closes only the concrete machine link.", "The concrete complexity interface proves P subset NP, reduction composition and transport, and the NP-complete-in-P implication relative to its exact pipeline semantics; it does not prove concrete SAT complete or in P.", "The integrated direct CNF-SAT finite-machine verifier proves exact accept/reject correctness, bounded no-timeout behavior, and PNP.Concrete.FinalUniversalDesign.cnfSATInNP; it proves CNF-SAT membership in NP only, not CNF-SAT in P, NP-completeness, or P = NP.", "CookLevinRawTapeBridge proves that the finite tableau semantics exactly represent ordinary two-sided raw Tape execution for both input-only and paired-certificate verifier modes, and derives CNFSAT problem.encodedFormula iff language problem.input. CookLevinFormulaSize supplies the external encoded-formula-size polynomial, CookLevinFormulaSchedule supplies an exact answer-independent padded slot schedule, CookLevinFormulaCursor supplies direct coordinate decoders plus exact fuelled traversal, and BuilderInputLength supplies the first literal raw builder stage: an input-preserving unary length tally. Formula emission, a complete raw finite formula builder, a concrete PolynomialReduction, CNFSAT NP-completeness, CNFSAT in P, and P = NP remain absent.", "CookLevinFormulaSize bounds the actual canonical unary-indexed CNF bitstring by an explicit fixed-verifier NatPolynomial evaluated only at external source-input length. Exact codec costs, both input modes, all concrete constraint families, program and clause counts, and clause width are covered. This is an output-size theorem only: it does not implement or time a raw finite formula builder or package a PolynomialReduction.", "CookLevinFormulaSchedule allocates exact rectangular constraint, clause, token, and raw-bit slots without reading the canonical program, formula, token encoding, or encoded formula as schedule inputs. Filtering populated slots reproduces those existing canonical objects, and the bit-slot count is exactly encodedFormulaSizePolynomial at external input length. This is a pure schedule specification: it is not a raw finite builder, a construction-runtime theorem, a RawRefinement, or a PolynomialReduction.", "CookLevinFormulaCursor decodes constraint, clause, token, and raw-bit coordinates without constructing the complete canonical program, formula, token stream, or encoded formula. Its nested options distinguish out-of-range, valid padding, and populated slots; exact prefix, full, one-step-short, terminal, and excess-fuel theorems reproduce the canonical schedule and encoded output. A token-level specification cursor now exposes exact in-range, done, and terminal single-step laws. This remains a Lean specification cursor, not a constant-time raw slot interpreter, raw finite builder, construction-runtime theorem, RawRefinement, or PolynomialReduction.", "CookLevin.BuilderInputLength is a fixed 19-rule work machine that preserves every source bit, appends exactly one unary tally cell per bit in fresh right-side workspace, and accepts after exactly 2*n*n + 4*n + 2 work steps. Its compiled raw run uses exactly 12*n*n + 24*n + 12 steps, malformed internal scan symbols and one-step-short fuel time out, and its start tape is definitionally the proved total-input-framer endpoint. This is one preparatory builder stage only: it does not emit a formula bit, interpret a formula cursor slot, compose the complete builder, establish RawRefinement or PolynomialReduction, prove CNFSAT NP-completeness or in P, or prove P = NP.", "CookLevin.BuilderInputPrefix places the total all-input framer, one total symbol-preserving launch, and BuilderInputLength in one collision-free finite work-machine table. Every raw bitstring reaches the exact preserved-input unary-tally endpoint within 18*n*n + 63*n + 93 compiled raw steps; a tally scan configuration headed by the unused zeroOne symbol times out for every exterior tape and fuel, and one-step-short fuel remains fail-closed. This is still only an input-preparation prefix: it emits no formula bit, interprets no formula cursor coordinate, supplies no complete builder, RawRefinement, PolynomialReduction, CNFSAT NP-completeness or in-P theorem, or P = NP theorem.", "CookLevin.BuilderTokenAppender remains the independently audited fixed 59-rule state-selected component. For every raw source word, arbitrary exterior-left garbage, canonical prior token list, and each of F, T, separator, and finish, it appends exactly the selected two-bit token and restores the input focus. Its distinguished start requests the first T header token, reaches a supplied tally endpoint within 24*n + 48 compiled raw steps, and those bits are proved equal to direct formula coordinates zero and one and to encodedFormula.take 2 for both Cook-Levin input modes.", "CookLevin.BuilderFirstTokenPrefix is one literal 184-rule finite work machine: all 116 input-prefix rules and all 59 token-appender rules occupy disjoint injective state images after a total nine-symbol bridge. Every raw bitstring reaches the preserved input/tally workspace containing exactly the first T header token within 18*n*n + 87*n + 147 compiled raw steps. The exact prefix endpoint before launch, malformed prefix/appender phases, and one-step-short fuel remain timeout. This emits only the first two fixed formula bits; the remaining width header, dynamic cursor interpreter, complete formula builder, builder RawRefinement, PolynomialReduction, CNFSAT NP-completeness or in-P theorem, and P = NP theorem remain absent.", "CookLevin.BuilderCompleteHeader compiles the verifier-fixed formula-width NatPolynomial into a literal unary evaluator, then composes the 184-rule raw-input/first-token prefix, five total nine-symbol bridges, a 16-rule unary-root controller, and two 59-rule appender copies in disjoint injective state images. Every raw bitstring reaches exactly FormulaWidth copies of T followed by F under an external NatPolynomial compiled-time bound, and those token pairs equal the complete canonical encodedFormula width header. The 74 evaluator and 84 composed-header public declarations use only propext and Quot.sound, never Classical.choice or a project axiom. This does not implement the dynamic formula cursor, emit the formula body, supply builder RawRefinement or PolynomialReduction, establish CNFSAT NP-completeness or in P, or prove P = NP.", "CookLevin.BuilderBodyStartPrefix composes the complete width-header machine, a structurally generated unary evaluator for the retained next-token coordinate FormulaVariableSlotBound + 2, two total nine-symbol bridges, and a 59-rule Sep appender in disjoint injective state images. Every raw bitstring reaches exactly FormulaWidth copies of T followed by F and Sep, retains the corresponding doubled bit cursor, and obeys the external compiled bound CompleteHeader.rawTimeBound + 72 + 6*Unary.workSteps(nextTokenSlotPolynomial) + 24*n + 12*width. Its 60 public declarations use only propext and Quot.sound, never Classical.choice or a project axiom. This is not a dynamic cursor or complete formula builder and supplies no builder RawRefinement, PolynomialReduction, CNFSAT NP-completeness or in-P result, or P = NP theorem.", "CookLevin.BuilderFirstLiteralPrefix composes the complete body-start machine, a structurally generated unary evaluator for the retained next-token coordinate FormulaVariableSlotBound + 4, three total nine-symbol bridges, and fixed T/F appender copies in four disjoint injective state images. Every raw bitstring reaches exactly FormulaWidth copies of T followed by F, Sep, T, and F; these bits are the canonical encoded-formula prefix through positive variable zero. The external compiled bound is BodyStartPrefix.rawTimeBound + 174 + 6*Unary.workSteps(nextTokenSlotPolynomial) + 48*n + 24*width. Its current 74 public declarations use only propext and Quot.sound, never Classical.choice or a project axiom; the added halt-separation interface supports the reviewed first-clause composition. This retained coordinate is not a dynamic cursor, and the milestone supplies no remaining body emission, complete builder, RawRefinement, PolynomialReduction, CNFSAT NP-completeness or in-P result, or P = NP theorem.", "CookLevin.BuilderFirstClausePrefix composes the complete first-literal machine, a structurally generated unary evaluator for the retained next-token coordinate FormulaVariableSlotBound + 12, and a fixed 535-rule eight-token tail in disjoint injective state images. Every raw bitstring reaches exactly FormulaWidth copies of T followed by F, Sep, T, F, T, T, F, T, T, T, F, and Finish; these bits are the canonical encoded-formula prefix through the complete positive clause on variables zero, one, and two, and the following direct slot is proved to be valid padding. The literal table has exactly 1138 plus the four compiled unary-evaluator rule counts, and its external compiled bound is FirstLiteralPrefix.rawTimeBound + 1158 + 6*Unary.workSteps(nextTokenSlotPolynomial) + 192*n + 96*width. Its 79 public declarations use only propext and Quot.sound, never Classical.choice or a project axiom. The combined 80-declaration audit also covers the predecessor halt-separation theorem. This retained coordinate alone is not a dynamic cursor, and the milestone supplies no remaining formula body, complete builder, RawRefinement, PolynomialReduction, CNFSAT NP-completeness or in-P result, or P = NP theorem.", "CookLevin.BuilderDynamicTokenCursorStep composes that complete first-clause machine with one total launch and one fixed 45-rule cursor-advance table. Every raw bitstring preserves the emitted clause, consumes the proved first in-range padding opportunity, and advances the retained unary token coordinate from FormulaVariableSlotBound + 12 to FormulaVariableSlotBound + 13 in exactly 2*cursorWord.length + 8 additional work steps and within FirstClausePrefix.rawTimeBound + 48 + 12*cursorWord.length compiled raw steps. Its 47 public declarations, including the two reviewed dead-state dispatch facts used by downstream collision-free composition, use only propext and Quot.sound, never Classical.choice or a project axiom. This is one literal padding transition, not a complete dynamic cursor loop, arbitrary raw slot decoder, remaining formula-body emitter, complete builder, RawRefinement, PolynomialReduction, CNFSAT NP-completeness or in-P result, or P = NP theorem.", "CookLevin.BuilderFirstClausePaddingRun composes the preceding one-slot cursor step with two structurally generated unary evaluators and one fixed 25-rule countdown controller. Every raw bitstring executes all D = (FormulaVariableSlotBound - 1) * (FormulaVariableSlotBound + 6) remaining padding opportunities without emitting a token, reaches FormulaVariableSlotBound + 1 + FormulaTokensPerClause, and proves that the next direct schedule outcome is Sep. Its literal table has 1244 plus six inherited/generated evaluator rule counts, and its compiled execution has an explicit external input-size polynomial bound. The 84-line combined audit covers all 83 public declarations plus one predecessor transport theorem: 37 have empty closure, 11 use only propext, and 36 use only propext and Quot.sound, never Classical.choice or a project axiom. This is the exact remaining first-clause padding block and second-clause boundary only, not a general dynamic formula cursor, remaining formula-body emitter, complete builder, RawRefinement, PolynomialReduction, CNFSAT NP-completeness or in-P result, or P = NP theorem.", "CookLevin.BuilderSecondClauseSeparatorStep composes the complete first-clause padding run with a selected 59-rule Sep appender, two total nine-symbol bridges, and the existing 45-rule cursor advance. Its literal table has 1366 plus six inherited/generated unary-evaluator rule counts. Every raw bitstring emits the canonical separator beginning clause two, advances the retained unary coordinate by one, and proves that the following direct token is F; its emitted bits equal encodedFormula.take (2 * (FormulaWidth + 13)). The external compiled bound is BuilderFirstClausePaddingRun.rawTimeBound + 246 + 24*n + 12*FormulaWidth + 12*cursorWord.length. The combined 56-declaration audit has 15 empty closures, 11 using only propext, and 30 using only propext and Quot.sound, never Classical.choice or a project axiom. Malformed appender tally/output, malformed cursor scratch, both unlaunched endpoints, and one-step-short fuel remain timeout. This is one fixed populated transition, not a general dynamic formula cursor, following-F or remaining-body emitter, complete builder, RawRefinement, PolynomialReduction, CNFSAT NP-completeness or in-P result, or P = NP theorem.", "CookLevin.BuilderSecondClauseFirstLiteralPrefix composes the complete second-clause separator prefix with two selected 59-rule F appenders, four total symbol-preserving bridges, and two copies of the existing 45-rule cursor advance. Its literal table has 1610 plus six inherited/generated unary-evaluator rule counts. Every raw bitstring emits the complete negative literal on variable zero in clause two, retains the following negative-sign coordinate on variable one, and proves exact equality with encodedFormula.take (2 * (FormulaWidth + 15)). The external compiled bound is BuilderSecondClauseSeparatorStep.rawTimeBound + 564 + 48*n + 24*FormulaWidth + 24*cursorWord.length. The combined 87-declaration audit has 25 empty closures, 18 using only propext, and 44 using only propext and Quot.sound, never Classical.choice or a project axiom. All four pre-launch endpoints, malformed appender tally/output in both copies, malformed cursor scratch in both copies, and one-step-short fuel remain timeout. This is one fixed negative literal, not a complete clause-two emitter, general dynamic formula cursor, remaining-body emitter, complete builder, RawRefinement, PolynomialReduction, CNFSAT NP-completeness or in-P result, or P = NP theorem.", "CookLevin.BuilderSecondClauseSecondLiteralPrefix composes the complete clause-two first-literal prefix with selected 59-rule F, T, and F appenders, six total symbol-preserving bridges, and three copies of the existing 45-rule cursor advance. Its literal table has 1976 plus six inherited/generated unary-evaluator rule counts. Every raw bitstring emits the complete negative literal on variable one in clause two, retains the following Finish coordinate, and proves exact equality with encodedFormula.take (2 * (FormulaWidth + 18)). The external compiled bound is BuilderSecondClauseFirstLiteralPrefix.rawTimeBound + 1026 + 72*n + 36*FormulaWidth + 36*cursorWord.length. The combined 115-declaration audit has 34 empty closures, 25 using only propext, and 56 using only propext and Quot.sound, never Classical.choice or a project axiom. All six pre-launch endpoints, malformed appender tally/output in all three copies, malformed cursor scratch in all three copies, and one-step-short fuel remain timeout. This is one fixed negative literal and does not emit the clause terminator, complete clause two, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondClausePrefix composes the complete clause-two second-literal prefix with one selected 59-rule Finish appender, two total symbol-preserving bridges, and the existing 45-rule cursor advance. Its literal table has 2098 plus six inherited/generated unary-evaluator rule counts. Every raw bitstring emits the complete second clause, advances to its first in-range padding coordinate, and proves exact equality with encodedFormula.take (2 * (FormulaWidth + 19)). The external compiled bound is BuilderSecondClauseSecondLiteralPrefix.rawTimeBound + 390 + 24*n + 12*FormulaWidth + 12*BuilderSecondClauseSeparatorStep.cursorWord.length. The combined 57-declaration audit has 15 empty closures, 10 using only propext, and 32 using only propext and Quot.sound, never Classical.choice or a project axiom. Both pre-launch endpoints, malformed appender tally/output, malformed cursor scratch, and one-step-short fuel remain timeout. This completes clause two only and does not traverse its padding, reach clause three, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondClausePaddingRun composes the complete second-clause prefix with two fixed unary-polynomial evaluators, the reused 25-rule PaddingCountdown machine, and three total symbol-preserving WorkChain bridges. Its literal table has 2150 plus the six inherited/generated predecessor unary-evaluator rule counts and the two new evaluator rule counts. For every raw bitstring it evaluates and traverses exactly C - 7 remaining padding coordinates, reaches the third-clause start coordinate V + 1 + 2*C, proves every traversed slot is padding and the retained target slot is Sep, and preserves the exact output encodedFormula.take (2 * (FormulaWidth + 19)). The external compiled bound is BuilderSecondClausePrefix.rawTimeBound + 18 + 6*countEvaluator.timePoly + 6*(D*(2*R + 8) + D^2) + 6*targetEvaluator.timePoly, with D = C - 7 and R = countRootPrefixLength + countControllerPrefixLength. The combined 68-declaration audit has 26 empty closures, 9 using only propext, and 33 using only propext and Quot.sound, never Classical.choice or a project axiom. Both malformed countdown phases, the unlaunched predecessor endpoint, and one-step-short total fuel remain timeout. This traverses clause-two padding and reaches clause three only as a retained schedule coordinate; it does not emit the third-clause Sep, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderThirdClauseSeparatorStep composes the complete second-clause padding run with the reused selected 59-rule Sep appender, two total symbol-preserving bridges, and the reused 45-rule cursor advance. Its literal table has 2272 plus eight inherited unary-evaluator rule counts. Every raw bitstring emits the separator beginning clause three, retains the following negative-sign coordinate, and proves exact equality with encodedFormula.take (2 * (FormulaWidth + 20)). Direct and specification cursors both prove that the following token is F. The external compiled bound is BuilderSecondClausePaddingRun.rawTimeBound + 330 + 24*n + 12*FormulaWidth + 12*cursorWord.length. The combined 56-declaration audit has 14 empty closures, 11 using only propext, and 31 using only propext and Quot.sound, never Classical.choice or a project axiom. Both pre-launch endpoints, malformed appender tally/output, malformed cursor scratch, and one-step-short fuel remain timeout. This emits only the third-clause separator and does not emit the following F, complete clause three, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderThirdClauseFirstLiteralPrefix composes the complete third-clause separator prefix with the reused 235-rule two-F appender/cursor suffix behind one total symbol-preserving bridge. Its literal table has 2516 plus eight inherited unary-evaluator rule counts. Every raw bitstring emits the complete negative literal on variable zero in clause three, retains the following negative-sign coordinate on variable two, and proves exact equality with encodedFormula.take (2 * (FormulaWidth + 22)). The external compiled bound is BuilderThirdClauseSeparatorStep.rawTimeBound + 732 + 48*n + 24*FormulaWidth + 24*cursorWord.length. The combined 87-declaration audit has 24 empty closures, 18 using only propext, and 45 using only propext and Quot.sound, never Classical.choice or a project axiom. All four pre-launch endpoints, malformed appender tally/output in both copies, malformed cursor scratch in both copies, and one-step-short fuel remain timeout. This is one fixed negative literal and does not emit the following F, complete clause three, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderThirdClauseSecondLiteralPrefix composes the complete third-clause first-literal prefix with one fixed 479-rule F/T/T/F appender/cursor suffix behind one total symbol-preserving bridge. Its literal table has 3004 plus eight inherited unary-evaluator rule counts. Every raw bitstring emits the complete negative literal on variable two in clause three, retains the following clause-terminator coordinate, and proves exact equality with encodedFormula.take (2 * (FormulaWidth + 26)). The external compiled bound is BuilderThirdClauseFirstLiteralPrefix.rawTimeBound + 1752 + 96*n + 48*FormulaWidth + 48*cursorWord.length. The complete 145-declaration audit has 46 empty closures, 32 using only propext, and 67 using only propext and Quot.sound, never Classical.choice or a project axiom. All eight pre-launch endpoints, malformed tally/output in all four appenders, malformed scratch in all four cursors, and one-step-short fuel remain timeout. This is the second fixed negative literal and does not emit the following Finish, complete clause three, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderThirdClausePrefix composes the complete third-clause second-literal prefix with one selected 59-rule Finish appender, two total symbol-preserving bridges, and the existing 45-rule cursor advance. Its literal table has 3126 plus eight inherited unary-evaluator rule counts. Every raw bitstring emits the complete third clause, advances to its first in-range padding coordinate, and proves exact equality with encodedFormula.take (2 * (FormulaWidth + 27)). The external compiled bound is BuilderThirdClauseSecondLiteralPrefix.rawTimeBound + 498 + 24*n + 12*FormulaWidth + 12*BuilderThirdClauseSeparatorStep.cursorWord.length. The combined 57-declaration audit has 14 empty closures, 10 using only propext, and 33 using only propext and Quot.sound, never Classical.choice or a project axiom. Both pre-launch endpoints, malformed appender tally/output, malformed cursor scratch, and one-step-short fuel remain timeout. This completes clause three only and does not traverse its padding, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderThirdClausePaddingRun composes the complete third-clause prefix with two structurally generated unary evaluators, the reused 25-rule PaddingCountdown machine, and three total symbol-preserving WorkChain bridges. Its literal table has 3178 plus the eight inherited/generated predecessor unary-evaluator rule counts and the two new evaluator rule counts. For every raw bitstring it evaluates and traverses exactly C - 8 remaining padding coordinates, reaches the fourth-clause start coordinate V + 1 + 3*C, proves every traversed slot is padding and the retained target slot is Sep, and preserves the exact output encodedFormula.take (2 * (FormulaWidth + 27)). The external compiled bound is BuilderThirdClausePrefix.rawTimeBound + 18 + 6*countEvaluator.timePoly + 6*(D*(2*R + 8) + D^2) + 6*targetEvaluator.timePoly, with D = C - 8 and R = countRootPrefixLength. The combined 68-declaration audit covers 65 new declarations and three reused countdown interfaces: 26 have empty closure, 9 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice. Both malformed countdown phases, the unlaunched predecessor endpoint, and one-step-short total fuel remain timeout. This traverses clause-three padding and reaches clause four only as a retained schedule coordinate; it does not emit the fourth-clause Sep, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderFourthClauseSeparatorStep composes the complete third-clause padding run with the reused 113-rule selected Sep appender/cursor machine through one outer total nine-symbol WorkChain bridge. Its literal table has 3300 plus ten inherited/generated unary-evaluator rule counts. Every raw bitstring emits exactly the fourth-clause Sep, preserves encodedFormula.take (2 * (FormulaWidth + 28)), and advances the retained unary coordinate to V + 1 + 3*C + 1, whose direct next token is the first fourth-clause negative sign F. The exact work trace is bounded by BuilderThirdClausePaddingRun.rawTimeBound + 426 + 24*n + 12*FormulaWidth + 12*cursorWord.length. The combined 56-declaration audit covers all 48 new public declarations plus eight reused separator/cursor and dead-state interfaces using only the approved Lean-standard closure, with no project axiom or Classical.choice. Malformed appender tally/output states, malformed cursor scratch, both unlaunched endpoints, and one-step-short fuel remain timeout. This emits only the fourth-clause separator; it does not emit the following F, complete clause four, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderFourthClauseFirstLiteralPrefix composes the complete fourth-clause separator prefix with the reused 357-rule F/T/F appender/cursor suffix through one outer total nine-symbol WorkChain bridge. Its literal table has 3666 plus ten inherited/generated unary-evaluator rule counts. Every raw bitstring emits the complete first negative literal on variable one in clause four, preserves encodedFormula.take (2 * (FormulaWidth + 31)), and advances the retained unary coordinate to V + 1 + 3*C + 4, whose direct next token is the second literal negative sign F. The exact work trace is bounded by BuilderFourthClauseSeparatorStep.rawTimeBound + 1422 + 72*n + 36*FormulaWidth + 36*cursorWord.length. The combined 115-declaration audit covers all 97 new public declarations, 16 reused suffix interfaces, and two cursor dead-state facts: 33 closures are empty, 25 use only propext, and 57 use only propext and Quot.sound, with no project axiom or Classical.choice. Malformed appender tally/output states, malformed cursor scratch, all six unlaunched endpoints, and one-step-short fuel remain timeout. This emits only the first fourth-clause literal; it does not emit the second literal, complete clause four, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderFourthClauseSecondLiteralPrefix composes the complete fourth-clause first-literal prefix with the reused 479-rule F/T/T/F appender/cursor suffix through one outer total nine-symbol WorkChain bridge. Its literal table has 4154 plus ten inherited/generated unary-evaluator rule counts. Every raw bitstring emits the complete second negative literal on variable two in clause four, preserves encodedFormula.take (2 * (FormulaWidth + 35)), and advances the retained unary coordinate to V + 1 + 3*C + 8, whose direct next token is Finish. The exact work trace is bounded by BuilderFourthClauseFirstLiteralPrefix.rawTimeBound + 2232 + 96*n + 48*FormulaWidth + 48*BuilderFourthClauseSeparatorStep.cursorWord.length. The combined 147-declaration audit covers all 124 new public declarations, 21 reused suffix interfaces, and two cursor dead-state facts: 46 closures are empty, 32 use only propext, and 69 use only propext and Quot.sound, with no project axiom or Classical.choice. Malformed appender tally/output states, malformed cursor scratch, all eight unlaunched endpoints, and one-step-short fuel remain timeout. This emits only the second fourth-clause literal; it does not emit the following Finish, complete clause four, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderFourthClausePrefix composes the fourth-clause second-literal prefix with a selected 59-rule Finish appender, one existing 45-rule cursor advance, and two total nine-symbol WorkChain bridges. Its selected suffix has 113 rules and its literal table has 4276 plus ten inherited/generated unary-evaluator rule counts. Every raw bitstring emits the Finish that completes clause four, preserves encodedFormula.take (2 * (FormulaWidth + 36)), and advances the retained unary coordinate to V + 1 + 3*C + 9, whose direct next token is padding. The exact work trace is bounded by BuilderFourthClauseSecondLiteralPrefix.rawTimeBound + 618 + 24*n + 12*FormulaWidth + 12*BuilderFourthClauseSeparatorStep.cursorWord.length. The combined 57-declaration audit covers all 55 new public declarations and two cursor dead-state facts: 14 closures are empty, 10 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice. Malformed appender tally/output states, malformed cursor scratch, both unlaunched endpoints, and one-step-short fuel remain timeout. This completes only the fourth populated clause; it does not traverse clause-four padding, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderFourthClausePaddingRun composes the complete fourth-clause prefix with two structurally generated unary evaluators, the reused 25-rule PaddingCountdown machine, and three total nine-symbol WorkChain bridges. Its literal table has 4328 plus twelve inherited/generated unary-evaluator rule counts. For every raw bitstring it traverses exactly C - 9 remaining padding opportunities without emitting a token, preserves encodedFormula.take (2 * (FormulaWidth + 36)), and retains V + 1 + 4*C, the first opportunity in an intentionally empty fifth fixed-width clause slot. Direct lookup and the specification cursor both prove that endpoint is padding. The exact work trace is bounded by BuilderFourthClausePrefix.rawTimeBound + 18 plus six times the count-evaluator work, countdown bound, and target-evaluator work. The combined 68-declaration audit covers all 65 new public declarations and three reused countdown interfaces: 26 closures are empty, 9 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice. Both malformed countdown phases, the unlaunched predecessor endpoint, and one-step-short fuel remain timeout. This traverses only clause-four padding; it does not traverse the empty fifth rectangle, reach the next constraint, emit another token, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderFifthClausePaddingRun composes the complete fourth-clause padding run with two structurally generated unary evaluators, the reused 25-rule PaddingCountdown machine, and three total nine-symbol WorkChain bridges. Its literal table has 4380 plus fourteen inherited/generated unary-evaluator rule counts. For every raw bitstring it traverses exactly C opportunities across the entire intentionally empty fifth fixed-width clause rectangle without emitting a token, preserves encodedFormula.take (2 * (FormulaWidth + 36)), and retains V + 1 + 5*C, the first opportunity in the intentionally empty sixth fixed-width clause rectangle. Direct lookup and the specification cursor prove every traversed opportunity and that endpoint are padding. The exact work trace is bounded by BuilderFourthClausePaddingRun.rawTimeBound + 18 plus six times the count-evaluator work, countdown bound, and target-evaluator work. The combined 68-declaration audit covers all 65 new public declarations and three reused countdown interfaces: 28 closures are empty, 9 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice. Both malformed countdown phases, the unlaunched predecessor endpoint, and one-step-short fuel remain timeout. This traverses only the empty fifth rectangle; it does not traverse the empty sixth rectangle, reach the next constraint, emit another token, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderFirstConstraintPaddingRun composes the complete fifth-clause padding run with two structurally generated unary evaluators, the reused 25-rule PaddingCountdown machine, and three total nine-symbol WorkChain bridges. Its literal table has 4432 plus sixteen inherited/generated unary-evaluator rule counts. For every raw bitstring it traverses exactly (V - 2)*(V + 2)*C = (Q - 5)*C remaining empty token opportunities of the first scheduled constraint, where Q = 1 + V*V, without emitting a token; it preserves encodedFormula.take (2 * (FormulaWidth + 36)) and retains V + 1 + Q*C. Direct lookup and the specification cursor prove every traversed opportunity is padding and the endpoint is the Sep beginning the second scheduled constraint. The exact work trace is bounded by BuilderFifthClausePaddingRun.rawTimeBound + 18 plus six times the count-evaluator work, countdown bound, and target-evaluator work. The combined 68-declaration audit covers all 65 new public declarations and three reused countdown interfaces: 26 closures are empty, 9 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice. Both malformed countdown phases, the unlaunched predecessor endpoint, and one-step-short fuel remain timeout. This observes but does not emit the second-constraint Sep; it does not emit that constraint's first literal, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintSeparatorStep composes the complete first-constraint padding run with the reused 113-rule selected Sep appender/cursor machine through one outer total nine-symbol WorkChain bridge. Its literal table has 4554 plus sixteen inherited/generated unary-evaluator rule counts. Every raw bitstring emits exactly the Sep beginning the second scheduled constraint, preserves encodedFormula.take (2 * (FormulaWidth + 37)), and advances the retained unary coordinate to V + 1 + Q*C + 1, where Q = formulaClauseSlotsPerConstraint; direct lookup and the specification cursor prove the following token is the positive sign T beginning that constraint's at-least-one clause. The exact work trace is bounded by BuilderFirstConstraintPaddingRun.rawTimeBound + 534 + 24*n + 12*FormulaWidth + 12*cursorWord.length. The combined 56-declaration audit covers all 48 new public declarations plus eight reused separator/cursor and dead-state interfaces: 14 closures are empty, 11 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice. Malformed appender tally/output states, malformed cursor scratch, both unlaunched endpoints, and one-step-short fuel remain timeout. This emits only the second-constraint separator; it does not emit the following T, emit that constraint's first literal, traverse the second constraint, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintFirstLiteralSignStep composes the complete second-constraint separator step with the reused 113-rule selected T appender/cursor machine through one outer total nine-symbol WorkChain bridge. Its literal table has 4676 plus sixteen inherited/generated unary-evaluator rule counts. Every raw bitstring emits exactly the positive T sign beginning the second scheduled constraint first literal, preserves encodedFormula.take (2 * (FormulaWidth + 38)), and advances the retained unary coordinate to V + 1 + Q*C + 2, where Q = formulaClauseSlotsPerConstraint; direct lookup and the specification cursor prove the following token is the first unary T of a nonzero variable index. The exact work trace is bounded by BuilderSecondConstraintSeparatorStep.rawTimeBound + 546 + 24*n + 12*FormulaWidth + 12*cursorWord.length. The combined 56-declaration audit covers all 48 new public declarations plus eight reused true-token/cursor and dead-state interfaces: 14 closures are empty, 11 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice. Malformed appender tally/output states, malformed cursor scratch, both unlaunched endpoints, and one-step-short fuel remain timeout. This emits only the second-constraint first-literal sign; it does not emit the following unary T, complete that literal, traverse the second constraint, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep composes the complete second-constraint first-literal sign step with the reused 113-rule selected T appender/cursor machine through one outer total nine-symbol WorkChain bridge. Its literal table has 4798 plus sixteen inherited/generated unary-evaluator rule counts. Every raw bitstring emits exactly the first unary T of the second scheduled constraint first variable index, preserves encodedFormula.take (2 * (FormulaWidth + 39)), and advances the retained unary coordinate to V + 1 + Q*C + 3, where Q = formulaClauseSlotsPerConstraint; a constructive schedule proof shows the index is at least three, so direct lookup and the specification cursor prove the following token is the second unary T. The exact work trace is bounded by BuilderSecondConstraintFirstLiteralSignStep.rawTimeBound + 558 + 24*n + 12*FormulaWidth + 12*cursorWord.length. The measured 56-declaration audit covers all 48 new public declarations plus eight reused true-token/cursor and dead-state interfaces: 14 closures are empty, 11 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice. Malformed appender tally/output states, malformed cursor scratch, both unlaunched endpoints, and one-step-short fuel remain timeout. This emits only the second-constraint first-literal first unary unit; it does not emit the following unary T, complete that literal, traverse the second constraint, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep composes the complete second-constraint first-literal first-unary-unit step with the reused 113-rule selected T appender/cursor machine through one outer total nine-symbol WorkChain bridge. Its literal table has 4920 plus sixteen inherited/generated unary-evaluator rule counts. Every raw bitstring emits exactly the second unary T of the second scheduled constraint first variable index, preserves encodedFormula.take (2 * (FormulaWidth + 40)), and advances the retained unary coordinate to V + 1 + Q*C + 4, where Q = formulaClauseSlotsPerConstraint; a constructive schedule proof shows the index is at least three, so direct lookup and the specification cursor prove the following token is the third unary T. The exact work trace is bounded by BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.rawTimeBound + 570 + 24*n + 12*FormulaWidth + 12*cursorWord.length. The measured 56-declaration audit covers all 48 new public declarations plus eight reused true-token/cursor and dead-state interfaces: 14 closures are empty, 11 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice. Malformed appender tally/output states, malformed cursor scratch, both unlaunched endpoints, and one-step-short fuel remain timeout. This emits only the second-constraint first-literal second unary unit; it does not emit the following third unary T, complete that literal, traverse the second constraint, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep composes the complete second-constraint first-literal second-unary-unit step with the reused 113-rule selected T appender/cursor machine through one outer total nine-symbol WorkChain bridge. Its literal table has 5042 plus sixteen inherited/generated unary-evaluator rule counts. Every raw bitstring emits exactly the third and final unary T of the second scheduled constraint first variable index, preserves encodedFormula.take (2 * (FormulaWidth + 41)), and advances the retained unary coordinate to V + 1 + Q*C + 5, where Q = formulaClauseSlotsPerConstraint. A constructive schedule case split proves the selected variable index is exactly three: the width-one branch has time bound zero and head index three, while the wider branch selects the position-one blank-symbol index three. Direct lookup and the specification cursor therefore prove the following token is the terminating F. The exact work trace is bounded by BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.rawTimeBound + 582 + 24*n + 12*FormulaWidth + 12*cursorWord.length. The measured 56-declaration audit covers all 48 new public declarations plus eight reused true-token/cursor and dead-state interfaces: 14 closures are empty, 11 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice. Malformed appender tally/output states, malformed cursor scratch, both unlaunched endpoints, and one-step-short fuel remain timeout. This emits only the second-constraint first-literal third unary unit; it does not emit the following terminating F, complete that literal, traverse the second constraint, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep composes the complete second-constraint first-literal third-unary-unit step with the reused 113-rule selected F appender/cursor machine through one outer total nine-symbol WorkChain bridge. Its literal table has 5164 plus sixteen inherited/generated unary-evaluator rule counts. Every raw bitstring emits exactly the terminating F of the second scheduled constraint first literal, preserves encodedFormula.take (2 * (FormulaWidth + 42)), and advances the retained unary coordinate to V + 1 + Q*C + 6, where Q = formulaClauseSlotsPerConstraint. A constructive schedule case split proves the following direct token is Finish when tapeWidth is one and the positive T beginning the next literal at wider widths. The exact work trace is bounded by BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.rawTimeBound + 594 + 24*n + 12*FormulaWidth + 12*cursorWord.length. The measured 56-declaration audit covers all 48 new public declarations plus eight reused false-token/cursor and dead-state interfaces: 14 closures are empty, 11 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice. Malformed appender tally/output states, malformed cursor scratch, both unlaunched endpoints, and one-step-short fuel remain timeout. This emits only the terminating F; it does not emit the following width-dependent Finish or T, traverse the second constraint, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep composes the complete first-literal terminator with a represented-width evaluator, one 93-rule finite branch table, one reused 59-rule token appender, and a retained-coordinate evaluator. Its literal table has 5284 plus eighteen inherited/generated unary-evaluator rule counts. Every raw bitstring emits exactly one width-selected successor token: Finish when tapeWidth is one and T at every wider width. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 43)), and the retained coordinate is V + 1 + Q*C + 7, where Q = formulaClauseSlotsPerConstraint. Direct lookup and the specification cursor prove that the following opportunity is padding at width one and unary T at wider widths. The exact work trace is bounded by BuilderSecondConstraintFirstLiteralTerminatorStep.rawTimeBound + 600 + 24*n + 12*FormulaWidth + 12*width + 12*widthRootPrefixLength + 6*widthWorkSteps + 6*targetWorkSteps. The measured 82-declaration audit covers all 80 new public declarations plus two strengthened predecessor boundary lemmas: 37 closures are empty, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice. The predecessor endpoint before launch and one-step-short fuel remain timeout, while component malformed-workspace results are retained. This emits only the width-selected successor; it does not emit the following padding or unary token, traverse the second constraint, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep composes the complete width-selected successor-token predecessor with a represented-width evaluator, one 93-rule optional-appender table, and a retained-coordinate evaluator. Its literal table has 5404 plus twenty inherited/generated unary-evaluator rule counts. At tapeWidth one the controller takes a direct skip bridge and emits no token; at every wider width it appends exactly the first unary T of the second literal. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 1)), and the retained coordinate is V + 1 + Q*C + 8, where Q = formulaClauseSlotsPerConstraint. Direct lookup and the specification cursor prove that the following slot is again padding at width one and the second unary T at wider widths. The exact work trace is bounded by BuilderSecondConstraintFirstLiteralSuccessorTokenStep.rawTimeBound + 612 + 24*n + 12*FormulaWidth + 12*width + 12*widthRootPrefixLength + 6*widthWorkSteps + 6*targetWorkSteps. The measured 82-declaration audit covers all 80 new public declarations plus two strengthened schedule lemmas: 37 closures are empty, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice. The predecessor endpoint before launch and one-step-short fuel remain timeout, while component malformed-workspace results are retained. This consumes only one fixed schedule opportunity; it does not consume the following padding or unary token, complete the second literal, traverse the second constraint, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep composes the complete first padding-or-unary-opportunity predecessor with a represented-width evaluator, the same reviewed 93-rule optional-appender table, and a retained-coordinate evaluator. Its literal table has 5524 plus twenty-two inherited/generated unary-evaluator rule counts. At tapeWidth one the controller takes a direct skip bridge and emits no token; at every wider width it appends exactly the second unary T of the second literal. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 2)), and the retained coordinate is V + 1 + Q*C + 9, where Q = formulaClauseSlotsPerConstraint. Direct lookup and the specification cursor prove that the following slot is again padding at width one and the third unary T at wider widths. The exact work trace is bounded by BuilderSecondConstraintPaddingOrUnaryOpportunityStep.rawTimeBound + 624 + 24*n + 12*FormulaWidth + 12*width + 12*widthRootPrefixLength + 6*widthWorkSteps + 6*targetWorkSteps. The measured 82-declaration audit covers all 66 new public declarations, fourteen reused optional-appender interfaces, and two strengthened schedule lemmas: 37 closures are empty, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice. The predecessor endpoint before launch and one-step-short fuel remain timeout, while component malformed-workspace results are retained. This consumes only one additional fixed schedule opportunity; it does not consume the following padding or unary token, complete the second literal, traverse the remainder of the second constraint, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep composes the complete second padding-or-unary-opportunity predecessor with a represented-width evaluator, the same reviewed 93-rule optional-appender table, and a retained-coordinate evaluator. Its literal table has 5644 plus twenty-four inherited/generated unary-evaluator rule counts. At tapeWidth one the controller takes a direct skip bridge and emits no token; at every wider width it appends exactly the third unary T of the second literal. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 3)), and the retained coordinate is V + 1 + Q*C + 10, where Q = formulaClauseSlotsPerConstraint. Direct lookup and the specification cursor prove that the following slot is again padding at width one and the fourth unary T at wider widths. The exact work trace is bounded by BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.rawTimeBound + 636 + 24*n + 12*FormulaWidth + 12*width + 12*widthRootPrefixLength + 6*widthWorkSteps + 6*targetWorkSteps. The measured 82-declaration audit covers all 66 new public declarations, fourteen reused optional-appender interfaces, and two strengthened schedule lemmas: 37 closures are empty, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice. The predecessor endpoint before launch and one-step-short fuel remain timeout, while component malformed-workspace results are retained. This consumes only one additional fixed schedule opportunity; it does not consume the following padding or unary token, complete the second literal, traverse the remainder of the second constraint, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep composes the complete third padding-or-unary-opportunity predecessor with a represented-width evaluator, the same reviewed 93-rule optional-appender table, and a retained-coordinate evaluator. Its literal table has 5764 plus twenty-six inherited/generated unary-evaluator rule counts. At tapeWidth one the controller takes a direct skip bridge and emits no token; at every wider width it appends exactly the fourth unary T of the second literal. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 4)), and the retained coordinate is V + 1 + Q*C + 11, where Q = formulaClauseSlotsPerConstraint. Direct lookup and the specification cursor prove that the following slot is padding at width one and the terminating F at wider widths. The exact work trace is bounded by BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.rawTimeBound + 648 + 24*n + 12*FormulaWidth + 12*width + 12*widthRootPrefixLength + 6*widthWorkSteps + 6*targetWorkSteps. The measured 82-declaration audit covers all 66 new public declarations, fourteen reused optional-appender interfaces, and two strengthened schedule lemmas: 37 closures are empty, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice. The predecessor endpoint before launch and one-step-short fuel remain timeout, while component malformed-workspace results are retained. This consumes only one additional fixed schedule opportunity; it does not consume the following padding or terminating F, complete the second literal, traverse the remainder of the second constraint, implement a general dynamic formula cursor, emit the remaining body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep composes the complete fourth padding-or-unary-opportunity predecessor with a represented-width evaluator, a new reviewed 93-rule optional-terminator table built from the audited token-appender rules, and a retained-coordinate evaluator. Its literal table has 5884 plus twenty-eight inherited/generated unary-evaluator rule counts. At tapeWidth one the controller takes a direct skip bridge and emits no token; at every wider width it appends exactly the terminating F of the second literal. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 5)), and the retained coordinate is V + 1 + Q*C + 12, where Q = formulaClauseSlotsPerConstraint. Direct lookup and the specification cursor prove that the following slot is padding at width one and the opening unary T of the following literal at wider widths. The exact work trace is bounded by BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.rawTimeBound + 660 + 24*n + 12*FormulaWidth + 12*width + 12*widthRootPrefixLength + 6*widthWorkSteps + 6*targetWorkSteps. The measured 82-declaration audit covers all 66 new public outer declarations, fourteen new optional-terminator interfaces, and two strengthened schedule lemmas: 37 closures are empty, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice. The predecessor endpoint before launch and one-step-short fuel remain timeout, while component malformed-workspace results are retained. This consumes only one additional fixed schedule opportunity; it does not consume the following padding or opening T, complete the following literal, traverse the remainder of the second constraint, implement a general dynamic formula cursor, emit the remaining formula body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep composes the complete fifth padding-or-terminator-opportunity predecessor with a represented-width evaluator, the reviewed 93-rule optional-appender table, and a retained-coordinate evaluator. Its literal table has 6004 plus thirty inherited/generated unary-evaluator rule counts. At tapeWidth one the controller takes a direct skip bridge and emits no token; at every wider width it appends exactly the opening positive T of the following literal. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 6)), and the retained coordinate is V + 1 + Q*C + 13, where Q = formulaClauseSlotsPerConstraint. Direct lookup and the specification cursor prove that the following slot is padding at width one and the first unary-index T of the following literal at wider widths. The exact work trace is bounded by BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.rawTimeBound + 672 + 24*n + 12*FormulaWidth + 12*width + 12*widthRootPrefixLength + 6*widthWorkSteps + 6*targetWorkSteps. The measured 82-declaration audit covers all 66 new public declarations, fourteen reused optional-appender interfaces, and two strengthened schedule lemmas: 37 closures are empty, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice. The predecessor endpoint before launch and one-step-short fuel remain timeout, while component malformed-workspace results are retained. This consumes only one additional fixed schedule opportunity; it does not consume the following padding or first unary-index T, complete the following literal, traverse the remainder of the second constraint, implement a general dynamic formula cursor, emit the remaining formula body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep composes the complete sixth padding-or-opening-unary-opportunity predecessor with a represented-width evaluator, the reviewed 93-rule optional-appender table, and a retained-coordinate evaluator. Its literal table has 6124 plus thirty-two inherited/generated unary-evaluator rule counts. At tapeWidth one the controller takes a direct skip bridge and emits no token; at every wider width it appends exactly the first unary-index T of the following literal. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 7)), and the retained coordinate is V + 1 + Q*C + 14, where Q = formulaClauseSlotsPerConstraint. Direct lookup and the specification cursor prove that the following slot is padding at width one and the second unary-index T of the following literal at wider widths. The exact work trace is bounded by BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.rawTimeBound + 684 + 24*n + 12*FormulaWidth + 12*width + 12*widthRootPrefixLength + 6*widthWorkSteps + 6*targetWorkSteps. The measured 82-declaration audit covers all 66 new public declarations, fourteen reused optional-appender interfaces, and two strengthened schedule lemmas: 37 closures are empty, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice. The predecessor endpoint before launch and one-step-short fuel remain timeout, while component malformed-workspace results are retained. This consumes only one additional fixed schedule opportunity; it does not consume the following padding or second unary-index T, complete the following literal, traverse the remainder of the second constraint, implement a general dynamic formula cursor, emit the remaining formula body, construct a complete builder, provide RawRefinement or PolynomialReduction, prove CNFSAT NP-complete or in P, or prove P = NP.", "The 9,300-pair canonical sweep, 40,020-pair extended sweep, 261,121-pair bounded differential sweep, and ten compiler-edge cases are finite regression evidence; the universal result comes from the audited Lean theorems, not from testing.", "The formalized direct-wire NAND semantics layer does not by itself prove enumeration, minimum size, replacement/slack, the locked NAND builder, its threshold, SAT, or P = NP.", "The exact-width syntactic NAND enumeration remains intentionally noncanonical and may contain duplicates.", "The exhaustive direct-wire truth-table and reference-minimum computation has no polynomial-runtime claim and does not formalize the report's residual-band minimizer.", "Replacement and global slack are proved only for the concrete serial framed-context construction, not arbitrary support profiles or the report's locked-NAND family.", "The typed local locked-NAND candidates, source-derived accounting, conditional square-baseline theorem, and five discharged local square baselines do not by themselves prove the later global semantic threshold, an encoded polynomial builder, the report-level locked-NAND theorem, or polynomial runtime.", "The report threshold word is multi-output: its baseline coordinates remain present alongside one final coordinate; a legacy single-output seed is not that construction.", "The legacy synthetic m=2 fixture is quarantined as internally inconsistent: honest source-derived baseline/displayed counts are 86/90, metadata-consistent counts are 95/99, and stored hybrid counts are 91/95.", "The proof-bearing conditional locked-NAND semantic boundary is now instantiated in all six fields by one answer-independent typed circuit candidate, but it is not the report threshold theorem or an encoded polynomial-time builder.", "The global typed full candidate now has residual slack at most four for every finite topological NAND circuit; this semantic bound does not construct the report's encoded family or prove its polynomial construction time.", "The locked-NAND carrier and TraceEquivalence milestone is unbounded over arbitrary finite topological NAND circuits: it proves the exact X/T/O/R/L/z partition, generates three checks per gate, constructs coherent traces, and proves accepted traces equal genuine evaluation. It does not assemble the complete exposed candidates or prove the threshold.", "The global locked-NAND candidate milestone constructs the exact square B-gate/B-output baseline and the B+4-gate/B+1-output extension for every finite typed circuit, preserves every baseline output, proves the final coordinate is z AND TraceChecks AND T_out, excludes internal constants, and proves every baseline coordinate is independent of z.", "The global locked-NAND BaselineDistinct milestone proves every exposed baseline coordinate is nonconstant, is not a positive carrier-input projection, and computes a Boolean function distinct from every other exposed coordinate. Consequently its exhaustive reference minimum is exactly B. By itself it does not prove either final branch law, residual slack at most four for the global family, or a uniform polynomial bitstring builder.", "The global locked-NAND unsatisfiable-final-zero milestone proves that unsatisfiability makes the full final coordinate false on every carrier valuation and makes the full exhaustive reference minimum exactly B. By itself it does not prove the satisfiable final-output conditions, residual slack at most four for the global family, or a uniform polynomial bitstring builder.", "The global locked-NAND semantic-threshold milestone discharges the satisfiable final-output conditions, all six typed premises, the exact satisfiable/unsatisfiable minimum split, and residual slack at most four for one answer-independent candidate over every finite topological NAND circuit. It uses exhaustive finite satisfiability decision only to eliminate a proposition-level case split and makes no polynomial-runtime claim.", "The encoded locked-NAND semantic boundary fixes a strict version-zero grammar, proves direct legacy output-normalization semantics, serializes the complete candidate and source-derived baseline, rejects malformed input, and proves the pure bitstring transformation correct. Its 48 audited declarations use only propext and Quot.sound. It is not a parser/validator machine, emitter machine, RawRefinement, PolynomialReduction, runtime bound, abstract PNP.LockedNANDThreshold discharge, CNFSAT-in-P theorem, or P = NP theorem.", "The strict-v0 locked-NAND source parser is one literal 228-state, 2,052-rule machine with unconditional exact behavior on every bitstring: valid sources accept and are preserved byte-for-byte, invalid sources reject with empty output, and the compiled six-for-one machine cannot time out within its explicit cubic bound. Its polynomial-time function and leaf RawRefinement validate source bytes only. The downstream target emitter now composes with it, but this parser alone supplies no PolynomialReduction, abstract threshold-language discharge, CNFSAT-in-P theorem, NP-hardness transport, or P = NP theorem.", "The strict-v0 locked-NAND target emitter is one fixed 1,387,921-rule grammar-only controller. It emits the exact direct raw target on every grammar-decoded circuit, including intrinsically invalid references, rejects malformed grammar with empty output, and has all-input polynomial runtime and output-size bounds. Composition with the strict source parser clears intrinsically invalid inputs and computes buildLockedNANDInstance with recursively compiled RawRefinement.", "The strict parser/emitter composition is packaged as a concrete PolynomialReduction from EncodedNANDSAT to EncodedLockedNANDThreshold, with exact output, all-bitstring language equivalence, a ReducesTo witness, and recursive RawRefinement. PNP.Main.locked_nand_threshold now consumes the composed CNFSAT reduction directly in the concrete charged-pipeline model.", "Against the hostile-review inventory, DirectWireOutputLowerBound, global MacroDistinct, semantic-carrier TraceEquivalence, whole-carrier ZeroOutputConvention, FinalLockSeparation, the encoded uniform polynomial builder, and its report-facing linkage are discharged.", "PNP.Main.locked_nand_threshold has exact type ReducesTo CNFSAT EncodedLockedNANDThreshold and depends only on Lean standard axioms propext and Quot.sound. It does not put the target in P, discharge residual-band minimization, ZeroSlack or PCCMin, prove concrete CNFSAT NP-hardness, activate the legacy string-handle bridge, or prove P = NP.", "The generic conditional module still quantifies an arbitrary satisfiable proposition and baseline natural number. The authoritative report-facing threshold theorem bypasses that legacy string-handle layer and links the concrete encoded CNFSAT builder directly to the concrete threshold language.", "The executable residual-route scan is complete only for the explicit finite implementation list supplied by its caller; unresolved excludes no unlisted gain and does not imply global minimality or ZeroSlack.", "An empty-list scan is formally shown to remain unresolved on a positive-slack implementation, so search failure cannot be promoted to zero residual slack.", "Exact and ZeroSlack route results require Lean proofs of semantic minimality and are never manufactured by the executable gain scanner; no BCEL, HN/BUD, selector, PCCMin-loop, or residual-band completeness follows.", "The residual-route equivalence check exhausts finite Boolean valuations and has no polynomial-runtime claim.", "The residual-gain chain checker validates every adjacent disclosed strict equivalent gain and proves that the number of accepted gains is at most the starting residual slack; it does not generate a route or prove that an undisclosed gain does not exist.", "The complete locked candidate has residual slack at most four, so every proof-bearing or executably accepted strict-gain chain from it has at most four steps. This is an iteration-count bound only, not route completeness, ZeroSlack, exact minimization, a polynomial checker, SAT in P, or P = NP.", "Global semantic stopping is now characterized for every finite direct-wire implementation: positive residual slack is equivalent to the existence of a strict equivalent gain, while zero slack and semantic minimality are equivalent to global absence of one.", "The global stopping theorem uses the exhaustive semantic reference minimum as a mathematical witness and requires a proof of global no-gain at a chain endpoint. It does not generate a route, derive global absence from a finite scan, construct the report ZeroSlack certificate, or establish polynomial runtime.", "The terminal full-carrier bridge preserves every input/output coordinate and exact gate count, proves an independently stated universal terminal minimum specification equal to the exhaustive reference minimum, and sends every cheaper whole-span realization to strict residual descent.", "This terminal bridge is direct-wire full-mode semantics only. It does not formalize the quotient carrier or mode firewall, proper supports, SaturatePositive, BCEL/BN2-BN6, selector completeness, ZeroSlack, PCCMin, route generation, or polynomial runtime.", "The terminal finite-profile mode firewall now computes role-indexed profile coordinates from each implementation, preserves the exact NAND implementation under projection, reconstructs full equality only from an explicit checked lift, and proves that a mismatch at any forgotten coordinate makes such a lift impossible.", "This mode-firewall milestone is terminal and forgetful-projection only. It does not formalize proper or governed support carriers, arbitrary quotient coarsening, full or quotient support minima, projection defect, SaturatePositive, BCEL/BN2-BN6, selector completeness, ZeroSlack, PCCMin, route generation, or polynomial runtime.", "The terminal projection-minimum layer exhaustively scans every direct-wire candidate size through the current implementation gate count. It computes attained full-profile and quotient-profile minima, proves both universal lower bounds and projection monotonicity, and characterizes zero projection defect exactly by a checked full lift at an attained quotient minimum.", "These terminal minima are finite reference computations, not polynomial algorithms. They do not construct proper or governed supports, SaturatePositive, Package E, BCEL/BN2-BN6, a complete residual route, ZeroSlack, PCCMin, SAT in P, or P = NP.", "The terminal projection-transfer layer proves the manuscript Section 5.2 signed four-corner balance law for every finite family sharing one computed terminal-profile observer and one explicit projection. Under constant-cut defect hypotheses, the projection excess equals the join defect and is positive when that defect is positive.", "This transfer theorem is arithmetic over supplied corners. It does not construct or certify a proper governed support square, prove SaturatePositive, discharge Package E or BCEL/BN2-BN6, generate a complete residual route, prove ZeroSlack or PCCMin, establish polynomial runtime, put SAT in P, or prove P = NP.", "The terminal saturation milestone reconstructs the manuscript Section 3 closure operator over every finite gate, boundary, interface, and computed-profile record universe with an explicit Boolean dependency relation carrying all ten rule tags. Its reflexive transitive closure is extensive, closed, least, monotone, and idempotent, and its fixed points are exactly the dependency-closed supports.", "Terminal executable saturation is a deterministic finite work-list implementation whose output is proved exactly equivalent to the inductive closure for every finite primitive-record universe and finite seed. The dependency relation is still explicit input data rather than the manuscript profile frontier extracted from a circuit.", "Terminal physical support completion computes the canonical incoming boundary and outgoing interface of every selected gate set in the actual finite direct-wire program, excluding constants and internal wires, and proves exact no-omission/no-addition compatibility. It does not construct proper positive support, a legitimate projection square, SaturatePositive, ZeroSlack, a polynomial algorithm, or P = NP.", "This saturation theorem does not extract dependencies from an arbitrary circuit, construct proper support, prove support completion or square legitimacy, instantiate a projection-compatible square, prove SaturatePositive or BCELReady, generate a complete residual route, prove ZeroSlack or PCCMin, establish polynomial runtime, put SAT in P, or prove P = NP.", "The terminal proper-positive support search is exhaustive only over the canonical finite primitive-record seed universe for a caller-supplied terminal dependency system. It computes exact reference-minimum local gain and returns a proof-bearing governed support when one exists in that universe; it does not derive the manuscript profile frontier, prove global gain completeness, construct a legitimate projection square, prove SaturatePositive or BCELReady, establish polynomial runtime, prove ZeroSlack or PCCMin, put SAT in P, or prove P = NP.", "The terminal saturated support-square milestone computes the exact closed left, right, meet, and join record sets for every pair of finite seeds under one caller-supplied dependency system. It proves the meet and join laws, computed physical compatibility, and exact open-support extraction for every corner. It does not derive the manuscript profile frontier, route completion obstructions, prove the frontier pushout or projection compatibility, establish BN2 square legitimacy or SaturatePositive, prove polynomial runtime, put SAT in P, or prove P = NP.", "Governed terminal support completion computes the physical boundary, ordered interface, and ten-role partition of every selected finite profile coordinate for saturated seeds and all four support-square corners. The dependency system remains explicit. It does not derive that system from an arbitrary circuit, prove the frontier pushout or projection compatibility, route every completion obstruction, establish BN2 square legitimacy or SaturatePositive, prove polynomial runtime, put SAT in P, or prove P = NP.", "Governed terminal frontier pushout proves the manuscript Section 3 gluing equation for every finite direct-wire candidate and computed saturated support square. Side boundary wires and interface producers are retained exactly when they remain external after union, otherwise their internalization is witnessed; meet profiles are the exact shared role coordinates and join profiles are their exact union. The dependency system remains explicit. This does not prove projection compatibility, BN2 square legitimacy, SaturatePositive, complete obstruction routing, polynomial runtime, SAT in P, or P = NP.", "Governed terminal projection squares prove the manuscript Section 3 structural projection-commutation law for every finite direct-wire candidate, computed saturated support square, and explicit forgetful terminal projection. Physical frontiers are retained, each of the ten role profiles is filtered exactly, projected meet is side overlap, and projected join is the side-only projected pushout. The dependency system remains explicit. This does not prove side-tight four-corner minima, BN2 square legitimacy, SaturatePositive, complete obstruction routing, polynomial runtime, SAT in P, or P = NP.", "Side-tight four-corner minimum arithmetic proves the manuscript Section 11.1 signed slack identity and a fail-closed exactness gate for every finite terminal projection four-corner family and independently attained full or quotient basis. Canonical minima pass and return the existing deltas. This does not construct a coherent four-corner basis, prove BN2 square legitimacy, maximize a tight family, establish SaturatePositive or BCELReady, complete obstruction routing, prove polynomial runtime, put SAT in P, or prove P = NP.", "Checked four-corner carrier transport derives all four governed and extracted endpoints from every finite computed saturated support square in common ambient coordinates. Profile overlap and union are exact; retained physical coordinates keep their identity, internalized coordinates carry concrete witnesses, and absent side queries fail closed. This is a prerequisite for the Section 11.1 four-corner optimum compatibility obligation, not transport of four optimum realizers, coherent completion, BN2 square legitimacy, SaturatePositive, BCELReady, polynomial runtime, SAT in P, or P = NP.", "Four-corner optimum carrier compatibility places all four canonical full and quotient minima on one reversible ambient carrier under one observer and projection, with exact localized minimum counts. It does not assert coherent transport along the square.", "The Section 11.1 optimum-coherence classifier traverses every finite computed support square in a deterministic order. It either returns a side-tight canonical tuple with exact commutation and incidence facts or the first exact semantic, profile, charge-profile, obligation, or mode-firewall mismatch. That classifier alone does not prove that the coherent branch always occurs or discharge a later no-outcome route.", "The sideTightCompletionExists milestone turns exact mode-appropriate local route silence into the checked side-tight coherent optimum tuple and its exact incidence value for every finite computed terminal support square. Full and quotient modes are discharged independently, and the quotient-promotion firewall remains separate.", "The complete BN2 tight-basis milestone enumerates every profile-constrained minimum implementation at all four corners, crosses and coherence-filters the full finite family, and proves that its signed maximum equals the selected delta under exact local route silence. This is a local all-finite maximum, not universal route silence, BN2 square legitimacy, the complete global no-outcome route system, terminal dependency extraction, SaturatePositive, BCELReady, ZeroSlack, PCCMin, polynomial runtime, SAT in P, or P = NP.", "Computed terminal BN2 square legitimacy now packages exact governed frontier transport, projection commutation, shared full and quotient quantities, and both exact tight-family maxima on every finite square built under one explicit terminal dependency system. The unconditional boundary returns the complete local conclusion under exact local route silence or the deterministic full-then-quotient proof-bearing first coherence failure. It does not derive the dependency system, prove universal route silence, connect local failures to the complete global no-outcome route system, identify a BCEL anchor square, establish SaturatePositive or BCELReady, prove ZeroSlack, PCCMin, polynomial runtime, SAT in P, or P = NP.", "The computed terminal BCEL anchor-nucleus milestone starts from one proof-bearing governed proper-positive support and an assumed positive whole-support projection defect. It exhaustively selects the canonical minimum positive anchor family, checks exact saturation meet/union algebra, recomputes all four defects on every proper cut, and returns either the exact first failure or constant-cut and local BN2 conclusions for every cut. It does not derive the positive premise or dependency system, identify absent manuscript activation or charge classes, connect local failures to the complete global route system, establish SaturatePositive or BCELReady, prove ZeroSlack, PCCMin, polynomial runtime, SAT in P, or P = NP.", "The computed terminal saturation-positivity firewall removes the caller-supplied whole-support positivity proof: zero projection defect returns an attained quotient minimum with a checked full lift, while positive defect delegates exactly to the existing BCEL anchor-nucleus classifier. This closes only projectionPositivityNotLostSilently. It does not discharge transparentSaturationCostBalanced, interfaceExposureRoutesToE, originKernelObligationClosureRouted, or firstNontransparentStepRecorded; derive the dependency system; establish full SaturatePositive or BCELReady; prove ZeroSlack, PCCMin, polynomial runtime, SAT in P, or P = NP.", "Candidate-derived terminal saturation computes physical incidence from the actual circuit, profile incidence from exhaustive context-sensitive observer influence, and a deterministic rule-labelled trace without accepting a dependency relation or extraction certificate. Exact per-event cost classification and linked-history theorems close the finite terminal forms of transparentSaturationCostBalanced and firstNontransparentStepRecorded. Nontransparent events are recorded but not routed: interfaceExposureRoutesToE and originKernelObligationClosureRouted remain open, as do full SaturatePositive, Package E, BCELReady, ZeroSlack, PCCMin, polynomial runtime, SAT in P, and P = NP.", "Finite terminal interface-exposure routing recognizes only candidate-derived interface-consumer events and returns either exact transparency evidence or a proof-bearing local E-route. The production trace classifier preserves the complete transparent prefix and routes only its exact first nontransparent interface event; non-interface failures remain fail-closed. This closes only the finite local form of interfaceExposureRoutesToE. The local route is not Package E VerifyDW, and originKernelObligationClosureRouted, full SaturatePositive, BCELReady, ZeroSlack, PCCMin, polynomial runtime, SAT in P, and P = NP remain open.", "Finite terminal origin, kernel, and obligation routing recognizes only exact candidate-derived role, rule, orientation, and edge combinations. Safety requires transparent cost, discharged obligations, and stability of forgotten profiles; failures retain deterministic proof-bearing reasons and the exact safe prefix. A proof-bearing candidate BCEL anchor problem with positive initial full slack composes these routes with preservation and the existing checked-lift or BCEL firewall. This closes a finite local routing edge and finite composition only. Local routes are not the complete global outcome system or Package E, the positive premise remains explicit, and full SaturatePositive, BCELReady, RankWF, ZeroSlack, PCCMin, polynomial runtime, SAT in P, and P = NP remain open.", "Residual RankWF is now formalized for the manuscript's fixed ten-coordinate natural lexicographic rank, with an executable comparison, exact Boolean equivalence, coordinate-priority witnesses, accessibility, induction, and kernel-checked well-foundedness. This does not map current finite routes into the complete global outcome system, prove that any route decreases the rank, establish route completeness or Package E, remove the explicit positive premise, establish full SaturatePositive or BCELReady, prove ZeroSlack, PCCMin, polynomial runtime, SAT in P, or P = NP.", "External review is optional audit evidence and is not a mathematical premise or release blocker.", "Historical releases and coordinates are preserved for auditability but are not current theorem-status authority.", "The designated legacy-v0 command replays pinned assertion-checker behavior only; it is neither current theorem authority nor a mathematical proof.", "The compiled Lean theorem inventory is declaration and axiom-dependency evidence; it does not widen any theorem beyond its exact type and stated scope.", "PNP.PEqualsNP uses abstract string-handle witnesses rather than a concrete standard complexity model and is categorically ineligible for public theorem activation.", "PNP.Main.ConcretePEqualsNP now names the inactive finite charged-pipeline target, while PNP.Main.p_eq_np remains absent.", "All five reviewed activation fingerprints remain intentionally unset, so target presence alone cannot open the concrete publication gate.", "The current canonical TeX and PDF are generated non-claiming reconstruction reports; the historical 56-page direct-claim report remains historical audit material only." ], "mathematicalTheoremEstablished": false, "publicTheoremEmissionAllowed": false, "publicTheoremStatement": null, "publicTheoremConclusion": null, "finalTheoremReady": false, "internalFinalTheoremReady": false, "unrestrictedFinalSoundnessDischarged": false, "uniformFinalSoundnessProved": false, "satInPConclusionAccepted": false, "pEqualsNPConclusionAccepted": false, "standardComplexityModelFormalized": true, "abstractPEqualsNPPublicationEligible": false, "publicationStatusDerivedOnlyFromConcreteGate": true, "concretePublicationGate": { "kind": "PNPConcretePublicationGate0", "version": 0, "compatibilityRootName": "PNP.Main.p_eq_np", "concreteTargetName": "PNP.Main.ConcretePEqualsNP", "expectedConcreteTargetKernelTypeSha256": null, "expectedConcreteTargetKernelValueSha256": null, "expectedRootKernelTypeSha256": null, "expectedAxiomClosureSha256": null, "expectedSourceClosureSha256": null, "actualConcreteTargetKernelTypeSha256": "b379ea53b6d32473c46c11da0f46fa34ead8d37dfb80c5f7b5d9a3f5f166369c", "actualConcreteTargetKernelValueSha256": "d10736cc9fc1d665512367ffe8fc9ff9345faa7ce4aa319cd5f976529f03a898", "actualRootKernelTypeSha256": null, "actualAxiomClosureSha256": "7de98496bd663ae170fcb7773362a3e89959de5d256b58837af49d746b96da3d", "actualSourceClosureSha256": "ce07f54409913eabe319f1e9c0adaa9738418fc9af406b6c0329dffa2dd0fa89", "axiomClosure": [], "allowedLeanStandardAxioms": [ "Classical.choice", "Quot.sound", "propext" ], "abstractPEqualsNPIsPublicationIneligible": true, "unsetFingerprintIsIntentionalFailClosedMigrationGate": true, "subchecks": { "standardComplexityModelEligible": true, "concreteTargetPresent": true, "concreteTargetIsDefinition": true, "concreteTargetKernelTypeFingerprintConfigured": false, "concreteTargetKernelTypeFingerprintMatches": false, "concreteTargetKernelValueFingerprintConfigured": false, "concreteTargetKernelValueFingerprintMatches": false, "compatibilityRootPresent": false, "compatibilityRootIsTheorem": false, "compatibilityRootHasExactConcreteType": false, "compatibilityRootKernelTypeFingerprintConfigured": false, "compatibilityRootKernelTypeFingerprintMatches": false, "axiomClosureFingerprintConfigured": false, "axiomClosureFingerprintMatches": false, "sourceClosureFingerprintConfigured": false, "sourceClosureFingerprintMatches": false, "axiomClosureUsesOnlyLeanStandardAllowlist": false }, "passed": false }, "leanTheoremInventoryCoordinate": "PNP-LEAN-THEOREM-INVENTORY-2026-08-12-134", "leanTheoremInventoryPath": "status/LEAN_THEOREM_INVENTORY.json", "leanTheoremInventoryPublicPath": "public/pnp-theorem-inventory.json", "leanTheoremInventorySha256": "5a1044bbf0438e67cf19749dae710c7a9a2d0fde5757ca57be997c175780cf86", "leanTheoremInventoryGeneratedFromCompiledEnvironment": true, "leanTheoremInventoryUsesEnvironmentConstants": true, "leanTheoremInventoryUsesCollectAxioms": true, "leanTheoremInventoryDeclarationCount": 27837, "leanTheoremInventoryExcludedPrivateDeclarationCount": 15011, "leanTheoremInventoryTheoremCount": 14470, "leanTheoremInventoryAssumptionFreeTheoremCount": 7347, "leanTheoremInventorySourceClosureModuleCount": 251, "leanSourceClosureSha256": "ce07f54409913eabe319f1e9c0adaa9738418fc9af406b6c0329dffa2dd0fa89", "formalPublicationMapCoordinate": "PNP-FORMAL-PUBLICATION-MAP-2026-08-12-134", "formalPublicationMapPath": "publication/FORMAL_PUBLICATION_MAP.json", "formalPublicationMapSha256": "d8e2bb51eb170e70bf30d9768922fc7b8d86195fc9b8f69abe458bfe3ab97da8", "formalPublicationMilestones": [ { "classification": "formalized-foundation-only", "id": "concrete-machine-cost-kernel", "nonClaim": "This closes the concrete complexity machine-link blocker only. It does not construct a deterministic polynomial-time CNF-SAT decider or an NP-completeness reduction. CNF-SAT in P, NP-completeness, and P = NP are not established; PNP.Main.p_eq_np remains absent and the publication gate remains false.", "requiredTheorems": [ "PNP.Concrete.BitString.decodePair_pair", "PNP.Concrete.DecisionProgram.RawRefinement.compile_haltsWithin", "PNP.Concrete.DecisionProgram.RawRefinement.compile_verdict_eq", "PNP.Concrete.FunctionProgram.RawRefinement.compile_haltsWithin", "PNP.Concrete.FunctionProgram.RawRefinement.compile_output_eq", "PNP.Concrete.NatPolynomial.eval_mono", "PNP.Concrete.PipelineCompiler.acceptingSuppliedTrace_workRunExact_of_rawRunExact", "PNP.Concrete.PipelineCompiler.outputBits_length_le_pipelineOutputSizeBound_of_rawRunExact", "PNP.Concrete.PipelineCompiler.pipelineOutputSizeBound_eval", "PNP.Concrete.PipelineCompiler.pipeline_accepts_iff", "PNP.Concrete.PipelineCompiler.pipeline_boundedDecide_eq", "PNP.Concrete.PipelineCompiler.pipeline_correct", "PNP.Concrete.PipelineCompiler.pipeline_machineOutput_eq", 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"PNP.Concrete.PipelineSequentialCompiler.run_sequential_accept_at_bound_of_rawRunExact", "PNP.Concrete.PipelineSequentialCompiler.run_sequential_reject_at_bound_of_rawRunExact", "PNP.Concrete.PipelineSequentialCompiler.sequentialOutputSizeBound_eval", "PNP.Concrete.PipelineSequentialCompiler.sequentialRawSteps_le_of_rawRunExact", "PNP.Concrete.PipelineSequentialCompiler.sequentialRawSteps_le_sequentialRawTimeBound", "PNP.Concrete.PipelineSequentialCompiler.sequential_accepts_iff", "PNP.Concrete.PipelineSequentialCompiler.sequential_boundedDecide_eq", "PNP.Concrete.PipelineSequentialCompiler.sequential_correct", "PNP.Concrete.PipelineSequentialCompiler.sequential_machineOutput_eq", "PNP.Concrete.PipelineSequentialCompiler.sequential_ne_timeout", "PNP.Concrete.PipelineSequentialCompiler.sequential_timeout_of_stuck_first_rawRunExact", "PNP.Concrete.PipelineSequentialStateNamespace.findWorkRule_sequential_first_of_some", "PNP.Concrete.PipelineSequentialStateNamespace.findWorkRule_sequential_second_of_some", "PNP.Concrete.PipelineSequentialStateNamespace.firstAcceptLaunch_workStep", "PNP.Concrete.PipelineSequentialStateNamespace.firstPipelineState_ne_secondPipelineState", "PNP.Concrete.PipelineSequentialStateNamespace.firstRejectLaunch_workStep", "PNP.Concrete.PipelineSequentialStateNamespace.sequentialWorkMachine_acceptState_ne_rejectState", "PNP.Concrete.PolynomialTimeDecider.compileToMachine_accepts_iff", "PNP.Concrete.PipelineInputFramer.boundedDecide_compilePairedInputFramer_accept", "PNP.Concrete.PipelineInputFramer.boundedDecide_compileTotalInputFramer_accept", "PNP.Concrete.PipelineInputFramer.boundedDecide_compileTotalInputFramer_ne_timeout", "PNP.Concrete.PipelineInputFramer.pairedInputFramerFinal_isHalted", "PNP.Concrete.PipelineInputFramer.pairedInputFramerFinal_represents", "PNP.Concrete.PipelineInputFramer.pairedInputFramerRawTimeBound_exact", "PNP.Concrete.PipelineInputFramer.pairedInputFramer_workRunExact", "PNP.Concrete.PipelineInputFramer.run_compilePairedInputFramer_rawTimeBound", "PNP.Concrete.PipelineInputFramer.run_compileTotalInputFramer_encoded_rawTimeBound", "PNP.Concrete.PipelineInputFramer.run_compileTotalInputFramer_rawTimeBound_blankEquivalent", "PNP.Concrete.PipelineInputFramer.totalInputFramerFinal_isHalted", "PNP.Concrete.PipelineInputFramer.totalInputFramerFinal_represents", "PNP.Concrete.PipelineInputFramer.totalInputFramerRawTimeBound_le", "PNP.Concrete.PipelineInputFramer.totalInputFramer_workRunExact", "PNP.Concrete.PipelineMachineSimulation.findIndexedRawRuleFrom_map_snd", "PNP.Concrete.PipelineMachineSimulation.find_liftMachine_entry", "PNP.Concrete.PipelineMachineSimulation.liftMachine_isHalted_eq_of_representsConfiguration", "PNP.Concrete.PipelineMachineSimulation.rawRunExact?_exists_le_run", "PNP.Concrete.PipelineMachineSimulation.run_compileWorkMachine_eighteen_mul_of_rawRunExact", "PNP.Concrete.PipelineMachineSimulation.run_compileWorkMachine_eighteen_mul_of_run_halted", "PNP.Concrete.PipelineMachineSimulation.run_compileWorkMachine_eighteen_of_step", "PNP.Concrete.PipelineMachineSimulation.workRunExact_three_mul_of_rawRunExact", "PNP.Concrete.PipelineMachineSimulation.workRunExact_three_of_step", "PNP.Concrete.PipelineMachineSimulation.workRun_three_mul_of_run_halted", "PNP.Concrete.PipelineOutputHandoff.framedOutputHandoffFinal_isHalted", "PNP.Concrete.PipelineOutputHandoff.framedOutputHandoffRawTimeBound_exact", "PNP.Concrete.PipelineOutputHandoff.framedOutputHandoff_workRunExact_of_represents", "PNP.Concrete.PipelineOutputHandoff.run_compileFramedOutputHandoff_of_represents", "PNP.Concrete.PipelinePairedCompiler.machineOutput_length_le_input_add_fuel", "PNP.Concrete.PipelinePairedCompiler.outputBits_length_le_pairedPipelineOutputSizeBound_of_rawRunExact", "PNP.Concrete.PipelinePairedCompiler.pairedPipelineOutputSizeBound_eval", 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"PNP.Concrete.PipelineStageBridges.bridgedReject_workRunExact_of_rawRunExact", "PNP.Concrete.PipelineStageBridges.bridgedWorkSteps_eq", "PNP.Concrete.PipelineStageBridges.findWorkRule_bridged_acceptingHandoff_of_some", "PNP.Concrete.PipelineStageBridges.findWorkRule_bridged_input_of_some", "PNP.Concrete.PipelineStageBridges.findWorkRule_bridged_rejectingHandoff_of_some", "PNP.Concrete.PipelineStageBridges.findWorkRule_bridged_simulation_of_some", "PNP.Concrete.PipelineStageBridges.inputLaunch_workStep", "PNP.Concrete.PipelineStageBridges.rejectingLaunch_workStep", "PNP.Concrete.PipelineStageBridges.run_compileBridgedMachine_accept_of_rawRunExact", "PNP.Concrete.PipelineStageBridges.run_compileBridgedMachine_reject_of_rawRunExact", "PNP.Concrete.PipelineStageBridges.workBoundedDecide_bridged_accept_of_rawRunExact", "PNP.Concrete.PipelineStageBridges.workBoundedDecide_bridged_reject_of_rawRunExact", "PNP.Concrete.PipelineStageBridges.workBoundedDecide_bridged_timeout_of_stuck_rawRunExact", "PNP.Concrete.PipelineStateNamespace.findWorkRule_composedRules_handoff", "PNP.Concrete.PipelineStateNamespace.findWorkRule_composedRules_input", "PNP.Concrete.PipelineStateNamespace.findWorkRule_composedRules_simulation", "PNP.Concrete.PipelineStateNamespace.findWorkRule_rename", "PNP.Concrete.PipelineStateNamespace.renamedInputFramer_workRunExact", "PNP.Concrete.PipelineStateNamespace.renamedLiftMachine_workRunExact_of_rawRunExact", "PNP.Concrete.PipelineStateNamespace.renamedOutputHandoff_workRunExact_of_represents", "PNP.Concrete.PipelineStateNamespace.stageState_injective", "PNP.Concrete.PipelineStateNamespace.workBoundedDecide_rename", "PNP.Concrete.PipelineStateNamespace.workRunExact?_rename", "PNP.Concrete.PipelineTape.frameWithGarbage_represents", "PNP.Concrete.PipelineTape.handoffTarget_withGarbage_represents", "PNP.Concrete.PipelineTape.represents_expandLeft_of_nil", "PNP.Concrete.PipelineTape.represents_expandRight_of_nil", "PNP.Concrete.PipelineTape.represents_moveLeft_of_cons", "PNP.Concrete.PipelineTape.represents_moveRight_of_cons", "PNP.Concrete.PipelineTape.represents_write", "PNP.Concrete.PipelineTerminalBridge.acceptingPackerLaunch_workStep", "PNP.Concrete.PipelineTerminalBridge.acceptingSuppliedTrace_workRunExact_of_rawRunExact", "PNP.Concrete.PipelineTerminalBridge.acceptingTerminalFinal_isHalted", "PNP.Concrete.PipelineTerminalBridge.acceptingTerminal_workRunExact_of_represents", "PNP.Concrete.PipelineTerminalBridge.bridged_workRunExact_of_exact", "PNP.Concrete.PipelineTerminalBridge.bridged_workStep?_of_some", "PNP.Concrete.PipelineTerminalBridge.findWorkRule_terminalBridge_acceptingPacker_of_some", "PNP.Concrete.PipelineTerminalBridge.findWorkRule_terminalBridge_rejectingPacker_of_some", "PNP.Concrete.PipelineTerminalBridge.machineOutput_compileTerminalBridge_accept_of_rawRunExact", "PNP.Concrete.PipelineTerminalBridge.machineOutput_compileTerminalBridge_reject_of_rawRunExact", "PNP.Concrete.PipelineTerminalBridge.outputBits_compileTerminalBridge_accepting_of_represents", "PNP.Concrete.PipelineTerminalBridge.outputBits_compileTerminalBridge_rejecting_of_represents", "PNP.Concrete.PipelineTerminalBridge.rejectingPackerLaunch_workStep", "PNP.Concrete.PipelineTerminalBridge.rejectingSuppliedTrace_workRunExact_of_rawRunExact", "PNP.Concrete.PipelineTerminalBridge.rejectingTerminalFinal_isHalted", "PNP.Concrete.PipelineTerminalBridge.rejectingTerminal_workRunExact_of_represents", "PNP.Concrete.PipelineTerminalBridge.run_compileTerminalBridge_accept_of_rawRunExact", "PNP.Concrete.PipelineTerminalBridge.run_compileTerminalBridge_accepting_of_represents", "PNP.Concrete.PipelineTerminalBridge.run_compileTerminalBridge_accepting_of_represents_at_bound", "PNP.Concrete.PipelineTerminalBridge.run_compileTerminalBridge_reject_of_rawRunExact", "PNP.Concrete.PipelineTerminalBridge.run_compileTerminalBridge_rejecting_of_represents", "PNP.Concrete.PipelineTerminalBridge.run_compileTerminalBridge_rejecting_of_represents_at_bound", "PNP.Concrete.PipelineTerminalBridge.simulationPrefix_terminalBridge_workBoundedDecide_timeout", "PNP.Concrete.PipelineTerminalBridge.suppliedTraceTerminalWorkSteps_eq", "PNP.Concrete.PipelineTerminalBridge.terminalBridgeMachine_acceptState_ne_rejectState", "PNP.Concrete.PipelineTerminalBridge.terminalBridge_runtime_le", "PNP.Concrete.PipelineTerminalBridge.workBoundedDecide_terminalBridge_accept_of_rawRunExact", "PNP.Concrete.PipelineTerminalBridge.workBoundedDecide_terminalBridge_reject_of_rawRunExact", "PNP.Concrete.PipelineTerminalBridge.workBoundedDecide_terminalBridge_timeout_of_stuck_rawRunExact", "PNP.Concrete.PolynomialTimeMachine.verdict_accepts_iff", "PNP.Concrete.PolynomialTimeMachine.verdict_ne_timeout", "PNP.Concrete.Tape.outputBits_handoffTarget", "PNP.Concrete.Tape.outputBits_moveRight_moveLeft", "PNP.Concrete.Tape.outputBits_ofInput", "PNP.Concrete.Tape.outputBits_right_append_blank", "PNP.Concrete.TerminalOutputPacker.machineOutput_compileTerminalOutputPacker_eq", "PNP.Concrete.TerminalOutputPacker.run_compileTerminalOutputPacker", "PNP.Concrete.TerminalOutputPacker.run_compileTerminalOutputPacker_exact", "PNP.Concrete.TerminalOutputPacker.terminalOutputPackerFinal_isHalted", "PNP.Concrete.TerminalOutputPacker.terminalOutputPacker_one_step_short_timeout", "PNP.Concrete.TerminalOutputPacker.terminalOutputPacker_output_eq", "PNP.Concrete.TerminalOutputPacker.terminalOutputPacker_runtime_le", "PNP.Concrete.TerminalOutputPacker.terminalOutputPacker_workRunExact", "PNP.Concrete.run_succ", "PNP.Concrete.run_zero" ], "scope": "One literal finite four-stage pipeline handles every raw bitstring. The total framer uses exactly 4 work steps on empty input, 4 * k * k + 9 * k + 7 for complete two-bit cells, and 4 * k * k + 9 * k + 5 when the final cell is partial; its compiled raw cost is bounded by 6 * m * m + 39 * m + 75. Every supplied exact n-step target run lifts to exactly 3 * n work steps. PipelineCompiler preserves one raw target's verdict and ordinary output at an explicit external polynomial. PipelineSequentialCompiler composes two raw targets in one literal finite table, passes either first verdict onward, preserves the second verdict and output, retains stuck-first timeout, and proves R(m) = PipelineRaw(p)(m) + 6 + PipelineRaw(q)(m + p(m) + 1). PipelineRefinement recursively applies that compiler to every FunctionProgram composition and DecisionProgram precomposition node. Its 16 audited declarations have empty axiom closure, and PolynomialTimeDecider.compileToMachine exposes the complete tree as one raw polynomial-time machine.", "title": "Concrete machine and cost kernel", "theoremRows": [ { "name": "PNP.Concrete.BitString.decodePair_pair", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "b54894853e84209d5bdf4d35b02e8e45bf9c77272a8f59868f08da3255cd0e29", "expectedKernelTypeSha256": "b54894853e84209d5bdf4d35b02e8e45bf9c77272a8f59868f08da3255cd0e29", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.DecisionProgram.RawRefinement.compile_haltsWithin", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "4057fc9d48be85dd7f961ce7acf5bef68ddb4ed0c8b6798617b31deb9da8c7c5", "expectedKernelTypeSha256": "4057fc9d48be85dd7f961ce7acf5bef68ddb4ed0c8b6798617b31deb9da8c7c5", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.DecisionProgram.RawRefinement.compile_verdict_eq", "present": 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Elementary at-least-one and pair-exclusion CNF clauses have exact propositional semantics. 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It does not yet quantify a variable certificate, encode tape windows or tableau rows as CNF, compile a formula emitter, define a polynomial reduction to CNFSAT, or establish CNFSAT NP-completeness, CNFSAT in P, or P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.FixedTableauInstance.exists_accepting_iff_boundedDecide_accept", "PNP.Concrete.CookLevin.FixedTableauInstance.tableauVerdict_of_valid", "PNP.Concrete.CookLevin.FixedTableauInstance.valid_iff_eq_canonical", "PNP.Concrete.CookLevin.canonicalTableau_valid", "PNP.Concrete.CookLevin.run_succ_eq_run_advance", "PNP.Concrete.CookLevin.tableauEndpoint_of_valid", "PNP.Concrete.CookLevin.trace_length", "PNP.Concrete.CookLevin.validTableau_iff_eq_trace" ], "scope": "For one fixed source input, certificate, finite-rule machine, and fuel budget, the canonical raw execution trace has exactly fuel + 1 configurations. Intrinsic first-match transition validity is equivalent to equality with that trace, every valid endpoint equals the ordinary run endpoint, and accept/reject/timeout verdicts agree exactly with boundedDecide. 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Every certificate inside the explicit certificate polynomial shares one answer-independent raw fuel obtained by evaluating the compiled raw-time polynomial at the maximum encoded input size. Per-certificate fuel is below that bound, padding follows a proved halt, exact verdicts are preserved, and language membership is equivalent to an existential bounded accepting semantic tableau. 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It does not enumerate the verifier tableau, encode initialization, transition windows, certificate length, or acceptance as a complete formula; prove an all-input emitter size/runtime polynomial; define a reduction to CNFSAT; or establish CNFSAT NP-completeness, CNFSAT in P, or P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BoundedLiteral.emit_variable_lt", "PNP.Concrete.CookLevin.LocalConstraint.emit_holds_iff", "PNP.Concrete.CookLevin.LocalProgram.emitted_clause_count", "PNP.Concrete.CookLevin.LocalProgram.toFormula_satisfiable_iff", "PNP.Concrete.CookLevin.LocalProgram.toFormula_satisfied_iff", "PNP.Concrete.CookLevin.assignmentAt_assignmentOf", "PNP.Concrete.CookLevin.exactlyOneBoundedClauses_holds_iff", "PNP.Concrete.CookLevin.implicationClauses_holds_iff", "PNP.Concrete.CookLevin.localProgram_formula_wellScoped" ], "scope": "Width-indexed literals materialize exact Boolean assignments and compile to the concrete CNF semantics without out-of-range variables. Unit constraints, finite implications, and pairwise exactly-one groups have exact satisfaction equivalences; local programs have exact recursive clause counts, satisfiability reflection, and a proved well-scoped formula. All 75 declarations and nine reviewed theorem types have empty axiom closure.", "title": "Finite local Cook-Levin CNF compiler", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BoundedLiteral.emit_variable_lt", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "c089b0fedc7320ec14f46b5b74f7ec26158cdad85ecbe647b4c728660dcb6a68", "expectedKernelTypeSha256": "c089b0fedc7320ec14f46b5b74f7ec26158cdad85ecbe647b4c728660dcb6a68", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.LocalConstraint.emit_holds_iff", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "a772353bb9cc6f114f6d795f18afa0e3f48b9a2de99d3034a7c2cb73f8a0618e", "expectedKernelTypeSha256": "a772353bb9cc6f114f6d795f18afa0e3f48b9a2de99d3034a7c2cb73f8a0618e", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.LocalProgram.emitted_clause_count", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "da3a74f1e3782b6d0ce56c3128f108ef348ac7f60d158f85759f392bdfd21dc3", "expectedKernelTypeSha256": "da3a74f1e3782b6d0ce56c3128f108ef348ac7f60d158f85759f392bdfd21dc3", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.LocalProgram.toFormula_satisfiable_iff", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "45b4527c3005c490c7433bc397eed6c40dd9d8555cc23a847cbec86b1da811cb", "expectedKernelTypeSha256": "45b4527c3005c490c7433bc397eed6c40dd9d8555cc23a847cbec86b1da811cb", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.LocalProgram.toFormula_satisfied_iff", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "56ae700d17d52a3666b6c77276fa21098919af5993577b6e53caa226f8cb29f8", "expectedKernelTypeSha256": "56ae700d17d52a3666b6c77276fa21098919af5993577b6e53caa226f8cb29f8", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.assignmentAt_assignmentOf", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "138ace06e46c2ba101542489b6bb13a2fbff763efa13eb8575ab60939cc709cc", "expectedKernelTypeSha256": "138ace06e46c2ba101542489b6bb13a2fbff763efa13eb8575ab60939cc709cc", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.exactlyOneBoundedClauses_holds_iff", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "aa5597d9c84ba7561bef3f0a0f9dea54f47c30bf8fedd2bb8c7df4c9643e9578", "expectedKernelTypeSha256": "aa5597d9c84ba7561bef3f0a0f9dea54f47c30bf8fedd2bb8c7df4c9643e9578", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.implicationClauses_holds_iff", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "59d5498dd4beb8f5749ffd1e76d9d8a5f060fb69d798c43ce51ef3eb40911f53", "expectedKernelTypeSha256": "59d5498dd4beb8f5749ffd1e76d9d8a5f060fb69d798c43ce51ef3eb40911f53", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.localProgram_formula_wellScoped", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "f8c84399020ccdd34509711adba113c4db47f775a76b65f14387ea3ddc1d7fde", "expectedKernelTypeSha256": "f8c84399020ccdd34509711adba113c4db47f775a76b65f14387ea3ddc1d7fde", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": true, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-tableau-cnf", "nonClaim": "This is finite formula syntax and local-clause reflection only. It does not yet prove that formula satisfiability is equivalent to an accepting raw verifier execution, prove external encoded-output-size or construction-runtime polynomials, define a concrete polynomial reduction to CNFSAT, or establish CNFSAT NP-completeness, CNFSAT in P, or P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.VerifierTableauProblem.certificateBitWidth_eq_of_paired", "PNP.Concrete.CookLevin.VerifierTableauProblem.certificateLengthWidth_eq_of_paired", "PNP.Concrete.CookLevin.VerifierTableauProblem.decode_encodedFormula", "PNP.Concrete.CookLevin.VerifierTableauProblem.dimensions_encodedInputLength", "PNP.Concrete.CookLevin.VerifierTableauProblem.encodedFormula_mem_CNFSAT_iff", "PNP.Concrete.CookLevin.VerifierTableauProblem.findRule_some_mem", "PNP.Concrete.CookLevin.VerifierTableauProblem.formula_clauseCount", "PNP.Concrete.CookLevin.VerifierTableauProblem.formula_satisfiable_iff", "PNP.Concrete.CookLevin.VerifierTableauProblem.formula_satisfied_iff", "PNP.Concrete.CookLevin.VerifierTableauProblem.formula_wellScoped" ], "scope": "A uniform bounded verifier-tableau problem is compiled answer-independently into one finite, well-scoped concrete CNF formula. The program emits one-hot symbol/head/state rows, first-match control updates, untouched-cell preservation, exact input-only or symbolic paired-certificate initialization, and a designated final accept constraint. Canonical encoding/decoding, exact local satisfaction reflection, and exact emitted clause counting are proved. All 81 declarations and ten reviewed theorem types have empty axiom closure.", "title": "Finite whole-tableau Cook-Levin CNF syntax", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.VerifierTableauProblem.certificateBitWidth_eq_of_paired", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "8f909f88887d1f94a0da2fb83c8296231a21e7d1dd33bb9d0d2f1a64e4819027", "expectedKernelTypeSha256": "8f909f88887d1f94a0da2fb83c8296231a21e7d1dd33bb9d0d2f1a64e4819027", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.VerifierTableauProblem.certificateLengthWidth_eq_of_paired", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "9d525c3ac7896848309a5572432e16e66a161da988512e7b0d078c88a34fec9e", "expectedKernelTypeSha256": "9d525c3ac7896848309a5572432e16e66a161da988512e7b0d078c88a34fec9e", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.VerifierTableauProblem.decode_encodedFormula", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "958b9c618da5bd37240505d2cbc6d15c5018eff3d23e5b4278da07d3826b8fb9", "expectedKernelTypeSha256": "958b9c618da5bd37240505d2cbc6d15c5018eff3d23e5b4278da07d3826b8fb9", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.VerifierTableauProblem.dimensions_encodedInputLength", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "c3ea1b40ab0bd79ff824d504a807107ebe08de6e3af69fd09ed1ea7be02140ea", "expectedKernelTypeSha256": "c3ea1b40ab0bd79ff824d504a807107ebe08de6e3af69fd09ed1ea7be02140ea", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.VerifierTableauProblem.encodedFormula_mem_CNFSAT_iff", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "94a0172b80bf24b4f755fb3993e88b9c6bd5f39a3e6e60d9325daa4aaf415f58", "expectedKernelTypeSha256": "94a0172b80bf24b4f755fb3993e88b9c6bd5f39a3e6e60d9325daa4aaf415f58", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.VerifierTableauProblem.findRule_some_mem", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "501b96f885967a0a2598f9eac5c4b57cebe5485bba6a1f3bda8fa2bec074af42", "expectedKernelTypeSha256": "501b96f885967a0a2598f9eac5c4b57cebe5485bba6a1f3bda8fa2bec074af42", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.VerifierTableauProblem.formula_clauseCount", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "69b8c747c903a1c0c8706b340cb8d7aa6b25af05742de951a63e106ffd2869f8", "expectedKernelTypeSha256": "69b8c747c903a1c0c8706b340cb8d7aa6b25af05742de951a63e106ffd2869f8", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.VerifierTableauProblem.formula_satisfiable_iff", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "514c1e52e50664c459c11c09a61510b0550c3b26bc85294bc603b9780dedeb3f", "expectedKernelTypeSha256": "514c1e52e50664c459c11c09a61510b0550c3b26bc85294bc603b9780dedeb3f", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.VerifierTableauProblem.formula_satisfied_iff", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "f285ab7fab210401cf8f8c67066633480d0171e2511330151199e981b2abd356", "expectedKernelTypeSha256": "f285ab7fab210401cf8f8c67066633480d0171e2511330151199e981b2abd356", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.VerifierTableauProblem.formula_wellScoped", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "d4582f49cd67afef6907d9db323fe1151362f8e72dc246fc25c8d31b029ce935", "expectedKernelTypeSha256": "d4582f49cd67afef6907d9db323fe1151362f8e72dc246fc25c8d31b029ce935", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": true, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-tableau-cnf-semantics", "nonClaim": "This milestone itself proves equivalence only with an intrinsic finite-row model; the following raw-tape milestone supplies the execution bridge. External encoded-output-size and construction-runtime polynomials, a concrete polynomial reduction to CNFSAT, CNFSAT NP-completeness, CNFSAT in P, and P = NP remain absent.", "requiredTheorems": [ "PNP.Concrete.CookLevin.VerifierTableauProblem.decodedInitialRow_eq_paired", "PNP.Concrete.CookLevin.VerifierTableauProblem.decodedRow_next", "PNP.Concrete.CookLevin.VerifierTableauProblem.decodedTableau_transitions", "PNP.Concrete.CookLevin.VerifierTableauProblem.encodedFormula_mem_CNFSAT_iff_finiteAccepting", "PNP.Concrete.CookLevin.VerifierTableauProblem.formula_satisfiable_iff_finiteAccepting", "PNP.Concrete.CookLevin.VerifierTableauProblem.tableauAssignment_transitionProgram_holds" ], "scope": "Every satisfying assignment decodes to a unique-symbol, unique-head, unique-state finite row at each bounded time. The decoded initial row is exact in both verifier modes, each adjacent row is the literal deterministic first-match successor, and the final state is accepting. Conversely, every such intrinsic finite accepting tableau constructs a satisfying assignment. Formula satisfiability and concrete CNFSAT membership are therefore equivalent to this finite semantics. 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It does not yet prove external encoded-formula-size or formula-construction-runtime polynomials, package a concrete PolynomialReduction, establish CNFSAT NP-completeness, CNFSAT in P, or P = NP.", "requiredTheorems": [ "PNP.Concrete.Tape.symbolAt_moveLeft", "PNP.Concrete.Tape.symbolAt_moveRight", "PNP.Concrete.CookLevin.VerifierTableauProblem.FiniteRow.next_represents_advance", "PNP.Concrete.CookLevin.VerifierTableauProblem.certificateOf_certificate", "PNP.Concrete.CookLevin.VerifierTableauProblem.encodedFormula_mem_CNFSAT_iff_language", "PNP.Concrete.CookLevin.VerifierTableauProblem.finiteRun_head_bounds", "PNP.Concrete.CookLevin.VerifierTableauProblem.finiteRun_represents_run", "PNP.Concrete.CookLevin.VerifierTableauProblem.hasFiniteAccepting_iff_language", "PNP.Concrete.CookLevin.VerifierTableauProblem.pairedInitialRowFor_represents" ], "scope": "The finite Cook-Levin rows are proved exactly equivalent to ordinary focused raw Tape execution. Absolute two-sided tape observations cover explicit and implicit blanks; the local successor matches designated halts, missing-rule stutter, and the literal first matching raw rule; a centered head invariant rules out finite-window clamping; and both verifier input modes have exact initial rows, including reversible bounded-certificate reconstruction. Formula satisfiability and concrete CNFSAT membership are therefore equivalent to the concrete verifier language. 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"sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-formula-size", "nonClaim": "This does not implement or time a raw finite formula builder, package a concrete PolynomialReduction, establish CNFSAT NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.encodeCNF_size_exact", "PNP.Concrete.CookLevin.encodeCNF_size_le", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaVariableCountPolynomial_eval", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaWidthPolynomial_eval", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaConstraintCountPolynomial_eval", "PNP.Concrete.CookLevin.VerifierTableauProblem.program_length_le_formulaConstraintCountPolynomial", "PNP.Concrete.CookLevin.VerifierTableauProblem.formula_clauseCount_le_formulaClauseCountPolynomial", 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Exact codec lengths, both verifier input modes, every concrete tableau constraint family, program and emitted-clause counts, clause width, and the mode-sensitive exact formula-variable width polynomial are covered. 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It does not interpret a slot as a constant-time raw-machine action, implement or time a raw finite formula builder, construct a FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.VerifierTableauProblem.initialCellAtCoordinate_eq_initialCellAt", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaConstraintSchedule_emit_eq_program", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaConstraintSchedule_length", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaClauseSchedule_length", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaClauseSchedule_emit_eq_formulaClauses", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaTokenSchedule_length", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaBitSchedule_length", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaBitSchedule_emit_eq_encodedFormula" ], "scope": "For every concrete verifier-tableau problem, a finite answer-independent rectangular schedule allocates exact constraint, clause, token, and raw-bit slots. Removing empty slots in order reproduces the existing canonical local program, bounded clauses, CNF tokens, and encoded formula. The raw-bit schedule length is exactly the previously proved encodedFormulaSizePolynomial evaluated at external source-input length. 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It does not prove constant-time raw slot interpretation, implement or time a raw finite formula builder, construct a FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaConstraintSlotDirect_eq", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaClauseSlotDirect_eq", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaTokenSlotDirect_eq", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaBitSlotDirect_eq", "PNP.Concrete.CookLevin.VerifierTableauProblem.formulaBitSlotCountDirect_eq_polynomial", "PNP.Concrete.CookLevin.VerifierTableauProblem.FormulaBitCursor.run_prefix", "PNP.Concrete.CookLevin.VerifierTableauProblem.FormulaBitCursor.run_to_end", "PNP.Concrete.CookLevin.VerifierTableauProblem.FormulaBitCursor.run_full", "PNP.Concrete.CookLevin.VerifierTableauProblem.FormulaBitCursor.step_at_end", "PNP.Concrete.CookLevin.VerifierTableauProblem.FormulaBitCursor.run_one_step_short", "PNP.Concrete.CookLevin.VerifierTableauProblem.FormulaBitCursor.step_after_one_step_short", "PNP.Concrete.CookLevin.VerifierTableauProblem.FormulaBitCursor.run_excess", "PNP.Concrete.CookLevin.VerifierTableauProblem.FormulaBitCursor.run_full_emit_eq_encodedFormula", "PNP.Concrete.CookLevin.VerifierTableauProblem.FormulaTokenCursor.step_at_end", "PNP.Concrete.CookLevin.VerifierTableauProblem.FormulaTokenCursor.step_of_done", "PNP.Concrete.CookLevin.VerifierTableauProblem.FormulaTokenCursor.step_of_lt" ], "scope": "For every concrete verifier-tableau problem and natural coordinate, direct constraint, clause, token, and raw-bit decoders agree pointwise with the canonical rectangular schedules while preserving out-of-range, valid-padding, and populated states. Fuelled bit traversal proves exact bounded prefixes, complete traversal, one-step-short behavior, terminal behavior, excess-fuel stability, the external encoded-size polynomial, and canonical emitted output. A token-level specification cursor additionally exposes exact in-range, done, and terminal single-step laws. 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input-length preparation stage. It does not emit formula bits, interpret direct cursor slots as raw transitions, compose a complete raw formula builder, construct a FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderInputLength.finalTape_represents", "PNP.Concrete.CookLevin.BuilderInputLength.inputTape_eq_totalInputFramerFinalTape", "PNP.Concrete.CookLevin.BuilderInputLength.malformedScanSymbol_timeout", "PNP.Concrete.CookLevin.BuilderInputLength.rawTimeBound_exact", "PNP.Concrete.CookLevin.BuilderInputLength.run_compile", "PNP.Concrete.CookLevin.BuilderInputLength.tallySizeBound_exact", "PNP.Concrete.CookLevin.BuilderInputLength.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderInputLength.workRunExact", "PNP.Concrete.CookLevin.BuilderInputLength.workRunExact_after_totalInputFramer", "PNP.Concrete.CookLevin.BuilderInputLength.work_one_step_short_timeout" ], "scope": "A fixed 19-rule work machine starts at the proved all-input framer endpoint, preserves every external source bit, appends exactly one unary tally symbol per source bit in fresh right-side workspace, and returns to the source head. It accepts after exactly 2*n*n + 4*n + 2 work steps; the compiled raw machine reaches the encoded endpoint after exactly 12*n*n + 24*n + 12 steps. Malformed internal scan symbols and one-step-short fuel both time out. 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It emits no formula bit, performs no raw formula-cursor interpretation, supplies no complete formula builder or construction-time RawRefinement, packages no concrete PolynomialReduction, and establishes neither CNFSAT NP-completeness, CNFSAT in P, nor P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderInputPrefix.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderInputPrefix.finalTape_represents", "PNP.Concrete.CookLevin.BuilderInputPrefix.finalTape_tally_length", "PNP.Concrete.CookLevin.BuilderInputPrefix.findWorkRule_framer_of_some", "PNP.Concrete.CookLevin.BuilderInputPrefix.findWorkRule_tally_of_some", "PNP.Concrete.CookLevin.BuilderInputPrefix.framerState_ne_tallyState", "PNP.Concrete.CookLevin.BuilderInputPrefix.launch_workStep", "PNP.Concrete.CookLevin.BuilderInputPrefix.malformedTallyScanSymbol_timeout", "PNP.Concrete.CookLevin.BuilderInputPrefix.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderInputPrefix.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderInputPrefix.run_compile_exact", "PNP.Concrete.CookLevin.BuilderInputPrefix.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderInputPrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderInputPrefix.work_one_step_short_timeout" ], "scope": "One literal finite work machine contains collision-free renamed copies of the total all-input framer and fixed 19-rule unary tally, with a total nine-symbol tape/head-preserving launch between them. Every raw bitstring reaches the exact preserved-input tally endpoint in totalInputFramerWorkSteps(input) + 1 + 2*n*n + 4*n + 2 work transitions and within the external compiled polynomial 18*n*n + 63*n + 93. 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It does not compute the remaining width header, interpret a dynamic formula cursor coordinate, emit a complete formula, construct a builder RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderTokenAppender.appendToken_workRunExact", "PNP.Concrete.CookLevin.BuilderTokenAppender.firstHeaderToken_after_builderInputPrefix", "PNP.Concrete.CookLevin.BuilderTokenAppender.firstHeaderToken_bits_eq_encodedFormula_take_two", "PNP.Concrete.CookLevin.BuilderTokenAppender.firstHeaderToken_one_step_short_timeout", "PNP.Concrete.CookLevin.BuilderTokenAppender.firstHeaderToken_workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderTokenAppender.firstHeaderToken_workRunExact", "PNP.Concrete.CookLevin.BuilderTokenAppender.firstTokenRawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderTokenAppender.firstTokenRawTimeBound_le", "PNP.Concrete.CookLevin.BuilderTokenAppender.formulaBitSlotDirect_one", "PNP.Concrete.CookLevin.BuilderTokenAppender.formulaBitSlotDirect_zero", "PNP.Concrete.CookLevin.BuilderTokenAppender.malformedOutputSymbol_timeout", "PNP.Concrete.CookLevin.BuilderTokenAppender.malformedTallySymbol_timeout", "PNP.Concrete.CookLevin.BuilderTokenAppender.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderTokenAppender.run_compile_firstHeaderToken_rawTimeBound", "PNP.Concrete.CookLevin.BuilderTokenAppender.tokenSymbol_bits", "PNP.Concrete.CookLevin.BuilderTokenAppender.workspaceTape_empty_eq_builderInputLength_finalTape", "PNP.Concrete.CookLevin.BuilderTokenAppender.workspaceTape_represents" ], "scope": "A fixed 59-rule work machine appends any of the four canonical two-bit CNF tokens selected only by its finite control state after a represented source word and exact unary input-length tally. For source length n and k existing tokens, it accepts at the exact endpoint after 2 * (max 1 n + n + k + 3) work steps. Its distinguished start state appends T, the first canonical formula header token, within the external compiled bound 24*n + 48; the two emitted bits are proved equal to the first two bits of every concrete verifier-tableau encoding. Malformed tally/output phases and one-step-short fuel remain timeouts. All 68 public declarations are axiom-audited: 42 have empty closure, 13 use only propext, and 13 use only propext and Quot.sound.", "title": "Standalone Cook-Levin builder token appender", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderTokenAppender.appendToken_workRunExact", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "948f8fd82b0b7afb85ae562995bebfcf59e50896cb46765fd4fbb807dd6652ad", "expectedKernelTypeSha256": "948f8fd82b0b7afb85ae562995bebfcf59e50896cb46765fd4fbb807dd6652ad", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderTokenAppender.firstHeaderToken_after_builderInputPrefix", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "b071c352d94a07f866d7dd896f2560811ab683ebebc12e8cb5dd6028ef14b67c", "expectedKernelTypeSha256": "b071c352d94a07f866d7dd896f2560811ab683ebebc12e8cb5dd6028ef14b67c", "kernelTypeFingerprintMatches": true }, { 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"axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-first-token-prefix", "nonClaim": "This emits exactly the fixed answer-independent first T token, hence only the first two canonical formula bits. It does not compute the remaining width header, implement a dynamic cursor, emit a complete formula, construct a builder RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.appenderState_injective", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.appender_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.finalTape_represents", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.finalTokenBits_eq_encodedFormula_take_two", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.findWorkRule_appender_of_some", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.findWorkRule_prefix_of_some", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.launch_workStep", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.malformedPrefixTally_timeout", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.prefixState_injective", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.prefixState_ne_appenderState", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.rules_length", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.run_compile_exact", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.work_one_step_short_timeout" ], "scope": "One literal 184-rule finite work machine contains all 116 executable input-prefix rules, nine total symbol-preserving bridge rules, and all 59 token-appender rules under injective disjoint state renamings. Every raw bitstring runs through total framing, exact unary input tallying, the bridge, and the appender to preserve the workspace and emit exactly the first T header token. The exact all-input work trace compiles within 18*n*n + 87*n + 147 raw steps; the final two bits equal the first two bits of every canonical encoded verifier-tableau formula. Prefix-endpoint, malformed tally/output, and one-step-short negative cases time out. All 37 public declarations are axiom-audited: 21 have empty closure, three use only propext, and 13 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Composed Cook-Levin builder first-token prefix", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.appenderState_injective", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "4d586fd6d8c0376788f22ecca4287c3afd7a288414818605ef7123e618dd75ef", "expectedKernelTypeSha256": "4d586fd6d8c0376788f22ecca4287c3afd7a288414818605ef7123e618dd75ef", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderFirstTokenPrefix.appender_workRunExact", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "0ed4be02d600690bdfc2e63df388293f7f95b9595dad123d95d9982c3002b1c4", "expectedKernelTypeSha256": "0ed4be02d600690bdfc2e63df388293f7f95b9595dad123d95d9982c3002b1c4", "kernelTypeFingerprintMatches": 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It does not implement the dynamic formula cursor or body emission, construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderCompleteHeader.HeaderController.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderCompleteHeader.HeaderController.rules_length", "PNP.Concrete.CookLevin.BuilderCompleteHeader.HeaderController.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderCompleteHeader.HeaderController.workRunExact", "PNP.Concrete.CookLevin.BuilderCompleteHeader.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderCompleteHeader.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderCompleteHeader.controllerF_launch_workStep", "PNP.Concrete.CookLevin.BuilderCompleteHeader.controllerState_injective", "PNP.Concrete.CookLevin.BuilderCompleteHeader.controllerT_launch_workStep", "PNP.Concrete.CookLevin.BuilderCompleteHeader.evaluatorController_launch_workStep", "PNP.Concrete.CookLevin.BuilderCompleteHeader.evaluatorState_injective", "PNP.Concrete.CookLevin.BuilderCompleteHeader.fAppenderState_injective", "PNP.Concrete.CookLevin.BuilderCompleteHeader.finalTape_represents", "PNP.Concrete.CookLevin.BuilderCompleteHeader.finalTokenBits_eq_encodedFormula_header", "PNP.Concrete.CookLevin.BuilderCompleteHeader.findWorkRule_controller_of_some", "PNP.Concrete.CookLevin.BuilderCompleteHeader.findWorkRule_evaluator_of_some", "PNP.Concrete.CookLevin.BuilderCompleteHeader.findWorkRule_fAppender_of_some", "PNP.Concrete.CookLevin.BuilderCompleteHeader.findWorkRule_prefix_of_some", "PNP.Concrete.CookLevin.BuilderCompleteHeader.findWorkRule_tAppender_of_some", "PNP.Concrete.CookLevin.BuilderCompleteHeader.headerTokens_eq_encodeUnaryTokens", "PNP.Concrete.CookLevin.BuilderCompleteHeader.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderCompleteHeader.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderCompleteHeader.prefixEvaluator_launch_workStep", "PNP.Concrete.CookLevin.BuilderCompleteHeader.prefixState_injective", "PNP.Concrete.CookLevin.BuilderCompleteHeader.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderCompleteHeader.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderCompleteHeader.rules_length", "PNP.Concrete.CookLevin.BuilderCompleteHeader.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderCompleteHeader.run_compile_exact", "PNP.Concrete.CookLevin.BuilderCompleteHeader.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderCompleteHeader.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderCompleteHeader.tAppenderState_injective", "PNP.Concrete.CookLevin.BuilderCompleteHeader.tController_launch_workStep", "PNP.Concrete.CookLevin.BuilderCompleteHeader.width_eq_FormulaWidth", "PNP.Concrete.CookLevin.BuilderCompleteHeader.width_positive", "PNP.Concrete.CookLevin.BuilderCompleteHeader.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderCompleteHeader.workRunExact", "PNP.Concrete.CookLevin.BuilderCompleteHeader.work_one_step_short_timeout", "PNP.Concrete.CookLevin.BuilderUnaryPolynomial.compilerStepsPolynomial_eval", "PNP.Concrete.CookLevin.BuilderUnaryPolynomial.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderUnaryPolynomial.root_prefix_length", "PNP.Concrete.CookLevin.BuilderUnaryPolynomial.root_register_length", "PNP.Concrete.CookLevin.BuilderUnaryPolynomial.rule_source_lt_acceptState", "PNP.Concrete.CookLevin.BuilderUnaryPolynomial.rules_length", "PNP.Concrete.CookLevin.BuilderUnaryPolynomial.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderUnaryPolynomial.scratchWord_symbol", "PNP.Concrete.CookLevin.BuilderUnaryPolynomial.workRunExact", "PNP.Concrete.CookLevin.BuilderUnaryPolynomial.workTimePolynomial_eval" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine contains the 184-rule raw-input/first-token prefix, a structurally generated unary NatPolynomial evaluator, a 16-rule unary-root controller, two complete 59-rule token appenders, and five total nine-symbol bridges under injective pairwise-disjoint state renamings. 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It does not implement a dynamic formula cursor or subsequent body emission, construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.appenderState_injective", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.appender_workRunExact", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.bodyStartTokens_eq_canonical_prefix", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.cursorAppender_launch_workStep", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.cursorDeadState_timeout", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.cursorEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.cursorState_injective", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.cursorState_ne_appenderState", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.cursor_workRunExact", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.finalOutside_contains_nextTokenSlot", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.finalTape_represents", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.finalTokenBits_eq_encodedFormula_bodyStart", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.findWorkRule_appender_of_some", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.findWorkRule_cursor_of_some", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.findWorkRule_header_of_some", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.firstBodyTokenSlotDirect_eq_separator", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.headerCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.headerEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.headerRejectEndpoint_timeout", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.headerState_injective", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.headerState_ne_appenderState", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.headerState_ne_cursorState", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.header_workRunExact", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.nextBitCursor_nextSlot", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.nextTokenSlot_eq_formulaVariableSlotBound_add_two", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.rule_source_ne_rejectState", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.rules_length", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.run_compile_exact", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderBodyStartPrefix.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine contains the complete width-header machine, a structurally generated unary evaluator for the next canonical token slot, one complete 59-rule token appender, and two total nine-symbol bridges under injective pairwise-disjoint state renamings. Its table has exactly 440 plus the width-evaluator and next-token-slot-evaluator rule counts. Every raw input follows an exact all-input trace, preserves the unary input tally and header workspace, retains next token coordinate FormulaVariableSlotBound + 2 and bit cursor twice that coordinate, and emits exactly FormulaWidth copies of T followed by F and Sep. The resulting bits are the canonical encoded-formula prefix through the first body separator; the compiled run is bounded by the complete-header bound plus 72, six times the unary next-slot evaluator work count, 24*n, and 12*width. 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after the body-opening separator and retains the following coordinate as data. It does not implement a dynamic formula cursor or remaining body emission, construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.evaluatorDeadState_timeout", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.evaluatorEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.evaluatorState_injective", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.evaluatorState_ne_fAppenderState", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.evaluatorState_ne_tAppenderState", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.evaluatorT_launch_workStep", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.evaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.fAppenderState_injective", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.fAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.finalOutside_contains_nextTokenSlot", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.finalTape_represents", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.finalTokenBits_eq_encodedFormula_firstLiteral", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.findWorkRule_evaluator_of_some", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.findWorkRule_fAppender_of_some", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.findWorkRule_prefix_of_some", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.findWorkRule_tAppender_of_some", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.firstLiteralSignSlotDirect_eq_t", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.firstLiteralTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.firstLiteralTokens_eq_canonical_prefix", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.firstLiteralZeroTerminatorSlotDirect_eq_f", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.malformedFAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.malformedFAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.nextBitCursor_nextSlot", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.nextTokenSlot_eq_formulaVariableSlotBound_add_four", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.prefixEvaluator_launch_workStep", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.prefixRejectEndpoint_timeout", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.prefixState_injective", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.prefixState_ne_evaluatorState", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.prefixState_ne_fAppenderState", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.prefixState_ne_tAppenderState", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.rules_length", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.run_compile_exact", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.tAppenderEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.tAppenderState_injective", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.tAppenderState_ne_fAppenderState", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.tAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.tF_launch_workStep", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine contains the complete body-start-prefix machine, a structurally generated unary evaluator for token coordinate FormulaVariableSlotBound + 4, two complete 59-rule token appenders, and three total nine-symbol bridges under four injective pairwise-disjoint state renamings. Its table has exactly 585 plus the width, body-start next-slot, and first-literal next-slot evaluator rule counts. Every raw input follows an exact all-input trace, preserves the unary tally and earlier token workspace, retains the corresponding doubled bit cursor, and emits exactly FormulaWidth copies of T followed by F, Sep, T, and F. The final T/F pair is constructively pinned to the positive sign and unary-zero terminator of the first scheduled literal, positive variable zero, and all emitted bits equal encodedFormula.take (2 * (FormulaWidth + 4)). The compiled run is bounded by the body-start bound plus 174, six times the new unary evaluator work count, 48*n, and 24*width. All 74 current public declarations are axiom-audited: 21 have empty closure, three use only propext, and 50 use only propext and Quot.sound, with no project axiom or Classical.choice; the added halt-separation interface supports the reviewed first-clause composition.", "title": "Composed Cook-Levin first-literal prefix", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.boundedDecide_compile_accept", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "abde80b5fd506e56a8e12c9f065228395b67c5f43463c306e74405a004657ef1", "expectedKernelTypeSha256": "abde80b5fd506e56a8e12c9f065228395b67c5f43463c306e74405a004657ef1", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.boundedDecide_compile_ne_timeout", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "c0e26a45f7d69b953efe8d788ccbee094f52e1ace4c7dbac159cdcf1a0c8e8f3", 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It does not implement a dynamic formula cursor or remaining formula-body emission, construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.FirstClauseTailAppender.finalTape_represents", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.FirstClauseTailAppender.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.FirstClauseTailAppender.rules_length", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.FirstClauseTailAppender.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.FirstClauseTailAppender.tailTokens_length", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.FirstClauseTailAppender.workRunExact", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.WorkChain.firstState_injective", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.WorkChain.firstState_ne_secondState", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.WorkChain.launch_workStep", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.WorkChain.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.WorkChain.secondState_injective", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.WorkChain.workRunExact", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.evaluatorEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.evaluatorTail_launch_workStep", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.evaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.finalTape_represents", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.finalTokenBits_eq_encodedFormula_firstClause", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.firstClauseTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.firstClauseTokens_eq_canonical_prefix", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.launch_workStep", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.nextBitCursor_nextSlot", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.nextTokenSlot_eq_formulaVariableSlotBound_add_twelve", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.rules_length", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.run_compile_exact", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.tail_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.work_one_step_short_timeout", "PNP.Concrete.CookLevin.BuilderFirstLiteralPrefix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.nextTokenSlot_direct_eq_padding", "PNP.Concrete.CookLevin.BuilderFirstClausePrefix.rule_source_ne_acceptState" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine contains the complete first-literal-prefix machine, a structurally generated unary evaluator for token coordinate FormulaVariableSlotBound + 12, and a fixed 535-rule tail made from eight complete 59-rule token appenders and seven total nine-symbol bridges under disjoint injective state renamings. Its table has exactly 1138 plus the width, body-start next-slot, first-literal next-slot, and first-clause next-slot evaluator rule counts. Every raw input follows an exact all-input trace, preserves the unary tally and earlier token workspace, retains the corresponding doubled bit cursor, and emits exactly FormulaWidth copies of T followed by F, Sep, T, F, T, T, F, T, T, T, F, and Finish. The final eight tokens are constructively pinned to the remainder of the complete positive at-least-one shape clause on variables zero, one, and two; the following direct token slot is proved to be valid padding. All emitted bits equal encodedFormula.take (2 * (FormulaWidth + 12)). The compiled run is bounded by the first-literal bound plus 1158, six times the new unary evaluator work count, 192*n, and 96*width. 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after the complete first clause. It is not a general dynamic cursor loop or raw decoder for arbitrary schedule coordinates, emits no additional formula token, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.advance_workRunExact", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.rules_length", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.cursor_workRunExact", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.directOutcome_is_padding", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.finalTokenBits_eq_encodedFormula_firstClause", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.finalTokenSlot_eq_formulaVariableSlotBound_add_thirteen", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.launch_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.malformedCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.rules_length", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.specification_step", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.workRunExact", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete first-clause prefix with a total nine-symbol bridge and a fixed 45-rule cursor-advance table under disjoint nested state images. Its table has exactly 1192 plus the four inherited unary-evaluator rule counts. Every raw input follows an exact all-input trace, preserves the complete first-clause output, and advances the retained unary token coordinate from FormulaVariableSlotBound + 12 to FormulaVariableSlotBound + 13. The direct decoder proves that the consumed coordinate is the first in-range padding slot, so the token-level specification cursor advances without emitting a token. The cursor suffix takes exactly 2*cursorWord.length + 8 work steps including launch, and the compiled run is bounded by FirstClausePrefix.rawTimeBound + 48 + 12*cursorWord.length. All 47 public declarations, including two reviewed dead-state dispatch facts used downstream, are axiom-audited: 14 have empty closure, seven use only propext, and 26 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Literal Cook-Levin token-cursor padding step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.advance_workRunExact", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "73d62bec018a0b0559ac1ce94fd6c50ffa9ec274f2b4f50acd499504f77a6105", "expectedKernelTypeSha256": "73d62bec018a0b0559ac1ce94fd6c50ffa9ec274f2b4f50acd499504f77a6105", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.machine_acceptState_ne_rejectState", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "c57dfe6eb354a46c2c164fec7b67af45c82353a861830c7952067fdabb1c0759", 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}, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-first-clause-padding-run", "nonClaim": "This milestone executes exactly the remaining first-clause padding block and identifies the second-clause separator boundary. It is not a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the second clause or remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loopSteps_le", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loop_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loopback_workStep", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.rules_length", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.countEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.countdownBoundPolynomial_eval", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.countdown_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.finalTape_represents", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.finalTokenBits_eq_encodedFormula_firstClause", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.finalTokenSlot_eq_secondClauseStart", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.formulaVariablePredecessorPolynomial_eval", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.formulaVariablePredecessorPolynomial_eval_add_one", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.formulaVariableSlotBound_at_least_three", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.launch_workStep", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.malformedCountdownRoot_timeout", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.malformedCountdownScratch_timeout", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.paddingSlot_direct_eq_padding", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.predecessorSlot_add_remainingPaddingCount", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.remainingPaddingCount_eq", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.remainingPaddingCount_eq_formulaTokensPerClause_sub_twelve", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.remainingPaddingCount_positive", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.rules_length", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.run_compile_exact", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.secondClauseStart_direct_eq_sep", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.secondClauseStart_eq", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.specification_padding_run", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.specification_target_step", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.workRunExact", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.work_one_step_short_timeout", "PNP.Concrete.CookLevin.BuilderCompleteHeader.HeaderController.workRunExact_of_unit_or_separator" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the preceding first-clause cursor step with structurally generated unary evaluators for D = (FormulaVariableSlotBound - 1) * (FormulaVariableSlotBound + 6) and the second-clause coordinate, plus one fixed 25-rule countdown controller. Its table has exactly 1244 plus the six inherited and generated evaluator rule counts. Every raw input follows an exact all-input trace, preserves the complete first-clause output, traverses all D remaining first-clause padding coordinates without emitting a token, and reaches FormulaVariableSlotBound + 1 + FormulaTokensPerClause, where direct schedule lookup is proved to return Sep. The recursive specification run agrees at every padding coordinate and the following specification step observes Sep. The compiled exact run has an explicit external input-size polynomial bound, accepts within that bound, and remains fail-closed for malformed countdown scratch/root states, the unlaunched prefix endpoint, and one-step-short fuel. 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only the fixed populated Sep transition beginning clause two and advances to the following F coordinate. It is not a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit that F or the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.appenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.appender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.cursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.finalTokenBits_eq_encodedFormula_secondClauseStart", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.finalTokenSlot_eq_secondClauseStart_add_one", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.malformedCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.nextTokenSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.prefixSeparator_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.secondClauseStartTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.separatorCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.specification_next_step", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.specification_separator_step", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.work_one_step_short_timeout", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete first-clause padding run with a selected 59-rule Sep appender, two total nine-symbol bridges, and the existing fixed 45-rule cursor advance under collision-free nested state images. Its table has exactly 1366 plus the six inherited and generated unary-evaluator rule counts. Every raw input follows an exact all-input trace, emits the canonical token prefix through the Sep beginning clause two, and advances the retained unary coordinate by one. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 13)), and direct schedule lookup proves that the following token is F. The compiled run is bounded by BuilderFirstClausePaddingRun.rawTimeBound + 246 + 24*n + 12*FormulaWidth + 12*cursorWord.length. Malformed appender tally/output, malformed cursor scratch, both unlaunched endpoints, and one-step-short total fuel remain timeout. All 54 new public declarations plus two reviewed predecessor dispatch facts are axiom-audited: 15 have empty closure, 11 use only propext, and 30 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-clause separator step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.machine_acceptState_ne_rejectState", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "eee6920f72c0fa3566bb3cc5fec3951b18a9b2aa1b50b5cf34436d0183ab750b", "expectedKernelTypeSha256": "eee6920f72c0fa3566bb3cc5fec3951b18a9b2aa1b50b5cf34436d0183ab750b", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.rule_source_ne_acceptState", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "7374475130eed22d8cc19092f72edf903e2c68e024a2f87b3ab4220e8cce98ef", 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negative-sign coordinate. It does not complete clause two, implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, emit the remaining formula body, construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.FalseTokenCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.FalseTokenCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.FalseTokenCursor.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.FalseTokenCursor.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.FirstLiteralSuffix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.FirstLiteralSuffix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.FirstLiteralSuffix.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.FirstLiteralSuffix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.finalTokenBits_eq_encodedFormula_secondClauseFirstLiteral", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.finalTokenSlot_eq_secondClauseStart_add_three", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.firstAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.firstAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.firstCursorEndpoint_before_secondAppender_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.firstCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.firstFalseTokenCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.firstFalseTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.firstLiteralSignSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.firstLiteralSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.firstLiteralZeroTerminatorSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.malformedFirstAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.malformedFirstAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.malformedFirstCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.malformedSecondAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.malformedSecondAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.malformedSecondCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.nextTokenSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.prefixFirstLiteral_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.secondAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.secondAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.secondClauseFirstLiteralTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.secondCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.secondFalseTokenCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.secondFalseTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.specification_firstLiteral_sign_step", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.specification_firstLiteral_terminator_step", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.specification_next_step", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.work_one_step_short_timeout", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete second-clause-separator prefix with two selected 59-rule F appenders, four total symbol-preserving bridges, and two copies of the existing 45-rule cursor advance under collision-free nested state images. One F/cursor component has 113 rules, the two-component suffix has 235 rules, and the global table has exactly 1610 plus the six inherited and generated unary-evaluator rule counts. Every raw input follows an exact all-input trace, emits the canonical prefix through the complete negative literal on variable zero in clause two, and retains the negative-sign coordinate on variable one. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 15)); direct schedule lookup proves the sign, unary-zero terminator, and following sign are all F. The compiled run is bounded by BuilderSecondClauseSeparatorStep.rawTimeBound + 564 + 48*n + 24*FormulaWidth + 24*cursorWord.length. All four pre-launch endpoints, malformed tally/output in both appenders, malformed scratch in both cursors, and one-step-short fuel remain timeout. All 85 new public declarations plus two reviewed cursor dead-loop facts are axiom-audited: 25 have empty closure, 18 use only propext, and 44 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-clause first-literal prefix", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.FalseTokenCursor.machine_acceptState_ne_rejectState", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "100d797c35b0c6669b7073cdb854d4081c9d53d8688205e876159170ec7db7ec", "expectedKernelTypeSha256": "100d797c35b0c6669b7073cdb854d4081c9d53d8688205e876159170ec7db7ec", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderSecondClauseFirstLiteralPrefix.FalseTokenCursor.rule_source_ne_acceptState", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "bf196201e146f446a56416af0eb05e5a515f1ac5832d224c313b343ca814ba15", 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"concrete-cook-levin-builder-second-clause-second-literal-prefix", "nonClaim": "This milestone emits only the fixed negative literal on variable one in clause two and advances to the following Finish coordinate. It does not emit the clause terminator, complete clause two, implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, emit the remaining formula body, construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.SecondLiteralSuffix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.SecondLiteralSuffix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.SecondLiteralSuffix.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.SecondLiteralSuffix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueFalseSuffix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueFalseSuffix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueFalseSuffix.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueFalseSuffix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.finalTokenBits_eq_encodedFormula_secondClauseSecondLiteral", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.finalTokenSlot_eq_secondClauseStart_add_six", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.firstAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.firstCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.malformedSignAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.malformedSignAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.malformedSignCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.malformedTerminatorAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.malformedTerminatorAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.malformedTerminatorCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.malformedUnaryAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.malformedUnaryAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.malformedUnaryCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.nextTokenSlot_direct_eq_finish", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.prefixSecondLiteral_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.secondAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.secondClauseSecondLiteralTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.secondCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.secondLiteralSignSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.secondLiteralSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.secondLiteralTerminatorSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.secondLiteralUnaryUnitSlot_direct_eq_t", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.signAppenderCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.signAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.signCursorEndpoint_before_unary_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.signTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.specification_next_step", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.specification_secondLiteral_sign_step", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.specification_secondLiteral_terminator_step", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.specification_secondLiteral_unaryUnit_step", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.terminatorAppenderCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.terminatorAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.terminatorTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.thirdAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.thirdCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.trueFalseSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.trueFalseSuffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.unaryAppenderCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.unaryAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.unaryCursorEndpoint_before_terminator_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.unaryTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete clause-two first-literal prefix with selected 59-rule F, T, and F appenders, six total symbol-preserving bridges, and three copies of the existing 45-rule cursor advance under collision-free nested state images. The selected T/cursor component has 113 rules, the T/F tail has 235 rules, the complete three-token suffix has 357 rules, and the global table has exactly 1976 plus the six inherited and generated unary-evaluator rule counts. Every raw input follows an exact all-input trace, emits the canonical prefix through the complete negative literal on variable one in clause two, and retains the following Finish coordinate. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 18)); direct schedule lookup proves the sign F, unary unit T, terminator F, and following Finish. The compiled run is bounded by BuilderSecondClauseFirstLiteralPrefix.rawTimeBound + 1026 + 72*n + 36*FormulaWidth + 36*cursorWord.length. All six pre-launch endpoints, malformed tally/output in all three appenders, malformed scratch in all three cursors, and one-step-short fuel remain timeout. All 113 new public declarations plus two reviewed cursor dead-loop facts are axiom-audited: 34 have empty closure, 25 use only propext, and 56 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-clause second-literal prefix", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "expectedKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "bfde87bc9e3bc501eb3ca0f3edb197ab2b120fef845b632574bdf14209c72292", 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It does not traverse clause-two padding, reach clause three, implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, emit the remaining formula body, construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.FinishTokenCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.FinishTokenCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.FinishTokenCursor.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.FinishTokenCursor.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.appenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.appender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.clauseTerminatorSlot_direct_eq_finish", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.cursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.finalTokenBits_eq_encodedFormula_secondClause", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.finalTokenSlot_eq_secondClauseStart_add_seven", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.finishTokenCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.malformedCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.nextTokenSlot_direct_eq_padding", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.prefixFinish_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.secondClauseTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.specification_next_step", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.specification_terminator_step", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClausePrefix.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete clause-two second-literal prefix with a selected 59-rule Finish appender, two total symbol-preserving bridges, and one copy of the existing 45-rule cursor advance under collision-free state images. The fixed suffix has exactly 113 rules and the global table has exactly 2098 plus the six inherited and generated unary-evaluator rule counts. Every raw input follows an exact all-input trace, emits the canonical prefix through the complete second clause, and retains its first in-range padding coordinate. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 19)); direct schedule lookup proves the executed opportunity is Finish and the retained next opportunity is padding. The compiled run is bounded by BuilderSecondClauseSecondLiteralPrefix.rawTimeBound + 390 + 24*n + 12*FormulaWidth + 12*cursorWord.length. Both pre-launch endpoints, malformed appender tally/output, malformed cursor scratch, and one-step-short fuel remain timeout. All 55 new public declarations plus two reviewed cursor dead-loop facts are axiom-audited: 15 have empty closure, 10 use only propext, and 32 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin complete second-clause prefix", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "expectedKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "bfde87bc9e3bc501eb3ca0f3edb197ab2b120fef845b632574bdf14209c72292", "expectedKernelTypeSha256": 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"classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-second-clause-padding-run", "nonClaim": "This milestone executes exactly the remaining second-clause padding block and identifies the third-clause separator boundary without emitting a token. It reaches clause three only as a retained coordinate. It is not a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the third-clause separator or remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loopSteps_le", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loop_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.countEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.countdownBoundPolynomial_eval", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.countdown_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.finalTokenBits_eq_encodedFormula_secondClause", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.finalTokenSlot_eq_thirdClauseStart", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.malformedCountdownRoot_timeout", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.malformedCountdownScratch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.paddingSlot_direct_eq_padding", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.predecessorSlot_add_remainingPaddingCount", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.remainingPaddingCount_eq", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.remainingPaddingCount_eq_formulaTokensPerClause_sub_seven", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.remainingPaddingCount_positive", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.specification_padding_run", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.specification_target_step", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.thirdClauseStart_direct_eq_sep", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.thirdClauseStart_eq", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondClausePaddingRun.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete second-clause prefix with structurally generated unary evaluators for D = (FormulaVariableSlotBound - 1) * (FormulaVariableSlotBound + 6) + 5 and the third-clause coordinate, plus the reused fixed 25-rule PaddingCountdown controller. Three total symbol-preserving WorkChain bridges yield a table with exactly 2150 plus the six inherited/generated predecessor evaluator rule counts and the two new evaluator rule counts. Every raw input follows an exact all-input trace, preserves the complete second-clause output, traverses all D = FormulaTokensPerClause - 7 remaining second-clause padding coordinates without emitting a token, and reaches FormulaVariableSlotBound + 1 + 2 * FormulaTokensPerClause, where direct schedule lookup is proved to return Sep. The recursive specification run agrees at every padding coordinate and the following specification step observes Sep. The emitted bits remain encodedFormula.take (2 * (FormulaWidth + 19)). The compiled exact run is bounded by BuilderSecondClausePrefix.rawTimeBound + 18 + 6 * countEvaluator.workSteps + 6 * (D * (2 * countRootPrefixLength + 8) + D * D) + 6 * targetEvaluator.workSteps, accepts within that external input-size polynomial, and remains fail-closed for malformed countdown scratch/root states, the unlaunched prefix endpoint, and one-step-short fuel. All 65 new public declarations plus the three reviewed reused-countdown declarations are axiom-audited: 26 have empty closure, 9 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-clause remaining-padding run", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loopSteps_le", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "972da09f44b6d35d56ce775bd4a46687be45609dacd0670bc983d603715bf278", "expectedKernelTypeSha256": "972da09f44b6d35d56ce775bd4a46687be45609dacd0670bc983d603715bf278", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loop_workRunExact", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "ff60669772a7f137755431c1300c4368e8004a05b1d36972a768b50d1846cf5a", 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It is not a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit that F or the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.appenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.appender_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.cursor_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.finalTokenBits_eq_encodedFormula_thirdClauseStart", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.finalTokenSlot_eq_thirdClauseStart_add_one", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.malformedCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.nextTokenSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.prefixSeparator_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.rules_length", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.separatorCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.specification_next_step", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.specification_separator_step", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.thirdClauseStartTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSeparatorStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete second-clause padding run with the reused selected 59-rule Sep appender, two total nine-symbol bridges, and the existing fixed 45-rule cursor advance under collision-free nested state images. Its table has exactly 2272 plus eight inherited unary-evaluator rule counts. Every raw input follows an exact all-input trace, emits the canonical token prefix through the Sep beginning clause three, and advances the retained unary coordinate by one. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 20)), and direct schedule lookup proves that the following token is F. The compiled run is bounded by BuilderSecondClausePaddingRun.rawTimeBound + 330 + 24*n + 12*FormulaWidth + 12*cursorWord.length. 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"PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.finalTape_represents", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.finalTokenBits_eq_encodedFormula_thirdClauseFirstLiteral", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.finalTokenSlot_eq_thirdClauseStart_add_three", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.firstAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.firstAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.firstCursorEndpoint_before_secondAppender_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.firstCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.firstFalseTokenCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.firstFalseTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.firstLiteralSignSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.firstLiteralSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.firstLiteralZeroTerminatorSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.malformedFirstAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.malformedFirstAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.malformedFirstCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.malformedSecondAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.malformedSecondAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.malformedSecondCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.nextTokenSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.prefixFirstLiteral_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.rules_length", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.run_compile_exact", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.secondAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.secondAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.secondCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.secondFalseTokenCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.secondFalseTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.specification_firstLiteral_sign_step", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.specification_firstLiteral_terminator_step", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.specification_next_step", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.thirdClauseFirstLiteralTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseFirstLiteralPrefix.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete third-clause-separator prefix with the reused 235-rule two-F appender/cursor suffix behind one total nine-symbol bridge under collision-free nested state images. One F/cursor component has 113 rules, the two-component suffix has 235 rules, and the global table has exactly 2516 plus the eight inherited and generated unary-evaluator rule counts. Every raw input follows an exact all-input trace, emits the canonical prefix through the complete negative literal on variable zero in clause three, and retains the negative-sign coordinate on variable two. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 22)); direct schedule lookup proves the sign, unary-zero terminator, and following sign are all F, with the third excluded pair constructively identified as variables zero and two. The compiled run is bounded by BuilderThirdClauseSeparatorStep.rawTimeBound + 732 + 48*n + 24*FormulaWidth + 24*cursorWord.length. All four pre-launch endpoints, malformed tally/output in both appenders, malformed scratch in both cursors, and one-step-short fuel remain timeout. All 74 new public declarations plus eleven reviewed suffix declarations and two reviewed cursor dead-loop facts are axiom-audited: 24 have empty closure, 18 use only propext, and 45 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin third-clause first-literal prefix", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "expectedKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": 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true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-third-clause-second-literal-prefix", "nonClaim": "This milestone emits only the fixed negative literal on variable two in clause three and advances to the following Finish coordinate. It does not emit the following Finish, complete clause three, implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, emit the remaining formula body, construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.SecondLiteralSuffix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.SecondLiteralSuffix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.SecondLiteralSuffix.rules_length", 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"PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.finalTokenBits_eq_encodedFormula_thirdClauseSecondLiteral", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.finalTokenSlot_eq_thirdClauseStart_add_seven", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.firstAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.firstCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.firstUnaryAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.firstUnaryCursorEndpoint_before_secondUnary_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.fourthAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.fourthCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.malformedFirstUnaryAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.malformedFirstUnaryAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.malformedFirstUnaryCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.malformedSecondUnaryAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.malformedSecondUnaryAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.malformedSecondUnaryCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.malformedSignAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.malformedSignAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.malformedSignCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.malformedTerminatorAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.malformedTerminatorAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.malformedTerminatorCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.nextTokenSlot_direct_eq_finish", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.prefixSecondLiteral_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.rule_source_ne_acceptState", 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"PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.secondLiteralSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.secondLiteralTerminatorSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.secondUnaryAppenderCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.secondUnaryAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.secondUnaryCursorEndpoint_before_terminator_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.secondUnaryTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.signAppenderCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.signAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.signCursorEndpoint_before_firstUnary_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.signTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.specification_next_step", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.specification_secondLiteral_secondUnaryUnit_step", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.specification_secondLiteral_sign_step", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.specification_secondLiteral_terminator_step", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.specification_secondLiteral_unaryUnit_step", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.terminatorAppenderCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.terminatorAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.terminatorTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.thirdAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.thirdClauseSecondLiteralTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.thirdCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.trueFalseSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.trueFalseSuffix_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.trueTrueFalseSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.trueTrueFalseSuffix_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.unaryAppenderCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.unaryTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete third-clause-first-literal prefix with a fixed 479-rule F/T/T/F appender/cursor suffix behind one total nine-symbol bridge under collision-free nested state images. The nested component tables have 113, 235, 357, and 479 rules, and the global table has exactly 3004 plus the eight inherited and generated unary-evaluator rule counts. Every raw input follows an exact all-input trace, emits the canonical prefix through the complete negative literal on variable two in clause three, and retains the following Finish coordinate. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 26)); direct schedule lookup proves the sign F, both unary units T, the terminator F, and the following Finish. The compiled run is bounded by BuilderThirdClauseFirstLiteralPrefix.rawTimeBound + 1752 + 96*n + 48*FormulaWidth + 48*cursorWord.length. All eight pre-launch endpoints, malformed tally/output in all four appenders, malformed scratch in all four cursors, and one-step-short fuel remain timeout. All 145 public declarations are axiom-audited: 46 have empty closure, 32 use only propext, and 67 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin third-clause second-literal prefix", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "expectedKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "bfde87bc9e3bc501eb3ca0f3edb197ab2b120fef845b632574bdf14209c72292", "expectedKernelTypeSha256": 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It does not traverse clause-three padding, implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, emit the remaining formula body, construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.FinishTokenCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.FinishTokenCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.FinishTokenCursor.rules_length", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.FinishTokenCursor.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.appenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.appender_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.clauseTerminatorSlot_direct_eq_finish", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.cursor_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.finalTape_represents", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.finalTokenBits_eq_encodedFormula_thirdClause", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.finalTokenSlot_eq_thirdClauseStart_add_eight", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.finishTokenCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.malformedCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.nextTokenSlot_direct_eq_padding", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.prefixFinish_launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.rules_length", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.run_compile_exact", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.specification_next_step", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.specification_terminator_step", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.thirdClauseTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClausePrefix.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete clause-three second-literal prefix with a selected 59-rule Finish appender, two total symbol-preserving bridges, and one copy of the existing 45-rule cursor advance under collision-free state images. The fixed suffix has exactly 113 rules and the global table has exactly 3126 plus the eight inherited unary-evaluator rule counts. Every raw input follows an exact all-input trace, emits the canonical prefix through the complete third clause, and retains its first in-range padding coordinate. The emitted bits equal encodedFormula.take (2 * (FormulaWidth + 27)); direct schedule lookup proves the executed opportunity is Finish and the retained next opportunity is padding. The compiled run is bounded by BuilderThirdClauseSecondLiteralPrefix.rawTimeBound + 498 + 24*n + 12*FormulaWidth + 12*BuilderThirdClauseSeparatorStep.cursorWord.length. Both pre-launch endpoints, malformed appender tally/output, malformed cursor scratch, and one-step-short fuel remain timeout. All 55 new public declarations plus two reviewed cursor dead-loop facts are axiom-audited: 14 have empty closure, 10 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin complete third-clause prefix", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "expectedKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "bfde87bc9e3bc501eb3ca0f3edb197ab2b120fef845b632574bdf14209c72292", "expectedKernelTypeSha256": 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"formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-third-clause-padding-run", "nonClaim": "This milestone executes exactly the remaining third-clause padding block and identifies the fourth-clause separator boundary without emitting a token. It reaches clause four only as a retained coordinate. It is not a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the fourth-clause separator or remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loopSteps_le", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loop_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.countEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.countdownBoundPolynomial_eval", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.countdown_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.finalTape_represents", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.finalTokenBits_eq_encodedFormula_thirdClause", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.finalTokenSlot_eq_fourthClauseStart", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.fourthClauseStart_direct_eq_sep", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.fourthClauseStart_eq", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.launch_workStep", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.malformedCountdownRoot_timeout", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.malformedCountdownScratch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.paddingSlot_direct_eq_padding", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.predecessorSlot_add_remainingPaddingCount", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.remainingPaddingCount_eq", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.remainingPaddingCount_eq_formulaTokensPerClause_sub_eight", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.remainingPaddingCount_positive", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.rules_length", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.run_compile_exact", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.specification_padding_run", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.specification_target_step", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.workRunExact", "PNP.Concrete.CookLevin.BuilderThirdClausePaddingRun.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete third-clause prefix with structurally generated unary evaluators for D = (FormulaVariableSlotBound - 1) * (FormulaVariableSlotBound + 6) + 4 and the fourth-clause coordinate, plus the reused fixed 25-rule PaddingCountdown controller. Three total symbol-preserving WorkChain bridges yield a table with exactly 3178 plus the eight inherited/generated predecessor evaluator rule counts and the two new evaluator rule counts. Every raw input follows an exact all-input trace, preserves the complete third-clause output, traverses all D = FormulaTokensPerClause - 8 remaining third-clause padding coordinates without emitting a token, and reaches FormulaVariableSlotBound + 1 + 3 * FormulaTokensPerClause, where direct schedule lookup is proved to return Sep. The recursive specification run agrees at every padding coordinate and the following specification step observes Sep. The emitted bits remain encodedFormula.take (2 * (FormulaWidth + 27)). The compiled exact run is bounded by BuilderThirdClausePrefix.rawTimeBound + 18 + 6 * countEvaluator.workSteps + 6 * (D * (2 * countRootPrefixLength + 8) + D * D) + 6 * targetEvaluator.workSteps, accepts within that external input-size polynomial, and remains fail-closed for malformed countdown scratch/root states, the unlaunched prefix endpoint, and one-step-short fuel. All 65 new public declarations plus the three reviewed reused-countdown declarations are axiom-audited: 26 have empty closure, 9 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin third-clause remaining-padding run", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loopSteps_le", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "972da09f44b6d35d56ce775bd4a46687be45609dacd0670bc983d603715bf278", "expectedKernelTypeSha256": "972da09f44b6d35d56ce775bd4a46687be45609dacd0670bc983d603715bf278", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loop_workRunExact", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "ff60669772a7f137755431c1300c4368e8004a05b1d36972a768b50d1846cf5a", 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It observes but does not emit the following F, does not complete clause four, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.appenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.appender_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.cursor_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.finalTokenBits_eq_encodedFormula_fourthClauseStart", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.finalTokenSlot_eq_fourthClauseStart_add_one", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.fourthClauseStartTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.malformedCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.nextTokenSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.prefixSeparator_launch_workStep", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.rules_length", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.separatorCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.specification_next_step", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.specification_separator_step", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseSeparatorStep.work_one_step_short_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.rules_pairwise_query_distinct" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete third-clause padding run with the reused selected 59-rule Sep appender and 45-rule cursor advance through one outer total nine-symbol WorkChain bridge. The global table has exactly 3300 plus the ten inherited/generated unary-evaluator rule counts. Every raw input follows exact prefix, appender, cursor, bridge, suffix, and combined traces; emits exactly the fourth-clause Sep; preserves encodedFormula.take (2 * (FormulaWidth + 28)); and retains coordinate FormulaVariableSlotBound + 1 + 3 * FormulaTokensPerClause + 1, whose direct next schedule token is F. The external compiled bound evaluates to BuilderThirdClausePaddingRun.rawTimeBound + 426 + 24 * n + 12 * FormulaWidth + 12 * cursorWord.length. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, malformed-workspace, both unlaunched-endpoint, and one-step-short results are proved. All 48 new public declarations plus eight reviewed reused separator/cursor interfaces are axiom-audited: 14 have empty closure, 11 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin fourth-clause separator step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "expectedKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "bfde87bc9e3bc501eb3ca0f3edb197ab2b120fef845b632574bdf14209c72292", 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"nonClaim": "This milestone emits exactly the fixed first negative literal on variable one in clause four. It observes but does not emit the following second-literal F, does not complete clause four, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.finalTape_represents", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.finalTokenBits_eq_encodedFormula_fourthClauseFirstLiteral", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.finalTokenSlot_eq_fourthClauseStart_add_four", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.firstAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.firstCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.firstLiteralSignSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.firstLiteralSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.firstLiteralTerminatorSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.firstLiteralUnaryUnitSlot_direct_eq_t", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.fourthClauseFirstLiteralTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.malformedSignAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.malformedSignAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.malformedSignCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.malformedTerminatorAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.malformedTerminatorAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.malformedTerminatorCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.malformedUnaryAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.malformedUnaryAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.malformedUnaryCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.nextTokenSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.prefixFirstLiteral_launch_workStep", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.rules_length", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.run_compile_exact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.secondAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.secondCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.signAppenderCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.signAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.signCursorEndpoint_before_unary_launch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.signTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.specification_firstLiteral_sign_step", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.specification_firstLiteral_terminator_step", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.specification_firstLiteral_unaryUnit_step", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.specification_next_step", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.terminatorAppenderCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.terminatorAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.terminatorTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.thirdAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.thirdCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.trueFalseSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.trueFalseSuffix_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.unaryAppenderCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.unaryAppenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.unaryCursorEndpoint_before_terminator_launch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.unaryTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseFirstLiteralPrefix.work_one_step_short_timeout", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.SecondLiteralSuffix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.SecondLiteralSuffix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.SecondLiteralSuffix.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.SecondLiteralSuffix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueFalseSuffix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueFalseSuffix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueFalseSuffix.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueFalseSuffix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rules_pairwise_query_distinct" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete fourth-clause separator prefix with the reused 357-rule F/T/F appender/cursor suffix through one outer total nine-symbol WorkChain bridge. The global table has exactly 3666 plus the ten inherited/generated unary-evaluator rule counts. Every raw input follows exact prefix, appender, cursor, bridge, suffix, and combined traces; emits the complete first negative literal on variable one in clause four; preserves encodedFormula.take (2 * (FormulaWidth + 31)); and retains coordinate FormulaVariableSlotBound + 1 + 3 * FormulaTokensPerClause + 4, whose direct next schedule token is F. The external compiled bound evaluates to BuilderFourthClauseSeparatorStep.rawTimeBound + 1422 + 72 * n + 36 * FormulaWidth + 36 * cursorWord.length. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, malformed-workspace, all six unlaunched-endpoint, and one-step-short results are proved. All 97 new public declarations, 16 reviewed reused suffix interfaces, and two cursor dead-state facts are axiom-audited: 33 have empty closure, 25 use only propext, and 57 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin fourth-clause first-literal prefix", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "expectedKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": 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"PNP.Concrete.CookLevin.BuilderFourthClauseSecondLiteralPrefix.unaryAppenderCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderFourthClauseSecondLiteralPrefix.unaryTokenCursor_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseSecondLiteralPrefix.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderFourthClauseSecondLiteralPrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClauseSecondLiteralPrefix.work_one_step_short_timeout", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.SecondLiteralSuffix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.SecondLiteralSuffix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.SecondLiteralSuffix.rules_length", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.SecondLiteralSuffix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.TrueFalseSuffix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.TrueFalseSuffix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.TrueFalseSuffix.rules_length", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.TrueFalseSuffix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.TrueTokenCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.TrueTokenCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.TrueTokenCursor.rules_length", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.TrueTokenCursor.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.TrueTrueFalseSuffix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.TrueTrueFalseSuffix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.TrueTrueFalseSuffix.rules_length", "PNP.Concrete.CookLevin.BuilderThirdClauseSecondLiteralPrefix.TrueTrueFalseSuffix.rules_pairwise_query_distinct" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete fourth-clause first-literal prefix with the reused 479-rule F/T/T/F appender/cursor suffix through one outer total nine-symbol WorkChain bridge. The global table has exactly 4154 plus the ten inherited/generated unary-evaluator rule counts. Every raw input follows exact prefix, appender, cursor, bridge, suffix, and combined traces; emits the complete second negative literal on variable two in clause four; preserves encodedFormula.take (2 * (FormulaWidth + 35)); and retains coordinate FormulaVariableSlotBound + 1 + 3 * FormulaTokensPerClause + 8, whose direct next schedule token is Finish. The external compiled bound evaluates to BuilderFourthClauseFirstLiteralPrefix.rawTimeBound + 2232 + 96 * n + 48 * FormulaWidth + 48 * BuilderFourthClauseSeparatorStep.cursorWord.length. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, malformed-workspace, all eight unlaunched-endpoint, and one-step-short results are proved. All 124 new public declarations, 21 reviewed reused suffix interfaces, and two cursor dead-state facts are axiom-audited: 46 have empty closure, 32 use only propext, and 69 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin fourth-clause second-literal prefix", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "expectedKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": 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It does not traverse clause-four padding, implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, emit the remaining formula body, construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.FinishTokenCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.FinishTokenCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.FinishTokenCursor.rules_length", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.FinishTokenCursor.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.appenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.appender_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.clauseTerminatorSlot_direct_eq_finish", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.cursor_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.finalTape_represents", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.finalTokenBits_eq_encodedFormula_fourthClause", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.finalTokenSlot_eq_fourthClauseStart_add_nine", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.finishTokenCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.fourthClauseTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.malformedCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.nextTokenSlot_direct_eq_padding", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.prefixFinish_launch_workStep", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.rules_length", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.run_compile_exact", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.specification_next_step", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.specification_terminator_step", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClausePrefix.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the fourth-clause second-literal prefix with a selected 59-rule Finish appender, one existing 45-rule cursor advance, and two total nine-symbol WorkChain bridges. The selected suffix has exactly 113 rules and the global table has exactly 4276 plus the ten inherited/generated unary-evaluator rule counts. Every raw input follows exact prefix, appender, cursor, bridge, suffix, and combined traces; emits the Finish that completes clause four; preserves encodedFormula.take (2 * (FormulaWidth + 36)); and retains coordinate FormulaVariableSlotBound + 1 + 3 * FormulaTokensPerClause + 9, whose direct next schedule token is padding. The external compiled bound evaluates to BuilderFourthClauseSecondLiteralPrefix.rawTimeBound + 618 + 24 * n + 12 * FormulaWidth + 12 * BuilderFourthClauseSeparatorStep.cursorWord.length. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, malformed-workspace, both unlaunched-endpoint, and one-step-short results are proved. 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"allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-fourth-clause-padding-run", "nonClaim": "This milestone traverses only the remaining padding in clause four and retains the first opportunity in the intentionally empty fifth clause rectangle. It does not traverse that empty rectangle, reach the next constraint, emit another token, implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, emit the remaining formula body, construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loopSteps_le", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loop_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.countEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.countdownBoundPolynomial_eval", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.countdown_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.fifthClauseSlotStart_direct_eq_padding", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.fifthClauseSlotStart_eq", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.finalTape_represents", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.finalTokenBits_eq_encodedFormula_fourthClause", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.finalTokenSlot_eq_fifthClauseSlotStart", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.launch_workStep", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.malformedCountdownRoot_timeout", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.malformedCountdownScratch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.paddingSlot_direct_eq_padding", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.predecessorSlot_add_remainingPaddingCount", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.remainingPaddingCount_eq", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.remainingPaddingCount_eq_formulaTokensPerClause_sub_nine", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.remainingPaddingCount_positive", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.rules_length", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.run_compile_exact", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.specification_padding_run", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.specification_target_step", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.workRunExact", "PNP.Concrete.CookLevin.BuilderFourthClausePaddingRun.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete fourth-clause prefix with two structurally generated unary evaluators, the reused 25-rule PaddingCountdown machine, and three total nine-symbol WorkChain bridges. Its table has exactly 4328 plus twelve inherited/generated unary-evaluator rule counts. For every raw bitstring it traverses exactly FormulaTokensPerClause - 9 padding opportunities without emitting a token, preserves encodedFormula.take (2 * (FormulaWidth + 36)), and retains coordinate FormulaVariableSlotBound + 1 + 4 * FormulaTokensPerClause. Direct schedule lookup and the specification cursor both prove that this first opportunity in the intentionally empty fifth fixed-width clause slot is padding. The external compiled bound is BuilderFourthClausePrefix.rawTimeBound + 18 + six times the count-evaluator work, countdown bound, and target-evaluator work. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, malformed-countdown, unlaunched-predecessor, and one-step-short results are proved. All 65 new public declarations and three reused countdown interfaces are axiom-audited: 26 have empty closure, 9 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin fourth-clause remaining-padding run", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loopSteps_le", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "972da09f44b6d35d56ce775bd4a46687be45609dacd0670bc983d603715bf278", "expectedKernelTypeSha256": "972da09f44b6d35d56ce775bd4a46687be45609dacd0670bc983d603715bf278", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loop_workRunExact", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "ff60669772a7f137755431c1300c4368e8004a05b1d36972a768b50d1846cf5a", "expectedKernelTypeSha256": 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"kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-fifth-clause-padding-run", "nonClaim": "This milestone traverses only the intentionally empty fifth clause rectangle and retains the first opportunity in the intentionally empty sixth clause rectangle. It does not traverse that sixth rectangle, reach the next constraint, emit another token, implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, emit the remaining formula body, construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loopSteps_le", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loop_workRunExact", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.countEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.countdownBoundPolynomial_eval", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.countdown_workRunExact", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.sixthClauseSlotStart_direct_eq_padding", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.sixthClauseSlotStart_eq", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.finalTape_represents", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.finalTokenBits_eq_encodedFormula_fourthClause", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.finalTokenSlot_eq_sixthClauseSlotStart", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.launch_workStep", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.malformedCountdownRoot_timeout", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.malformedCountdownScratch_timeout", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.paddingSlot_direct_eq_padding", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.predecessorSlot_add_paddingCount", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.paddingCount_eq", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.paddingCount_eq_formulaTokensPerClause", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.paddingCount_positive", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.rules_length", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.run_compile_exact", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.specification_padding_run", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.specification_target_step", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.workRunExact", "PNP.Concrete.CookLevin.BuilderFifthClausePaddingRun.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the fourth-clause padding predecessor with two structurally generated unary evaluators, the reused 25-rule PaddingCountdown machine, and three total nine-symbol WorkChain bridges. Its table has exactly 4380 plus fourteen inherited/generated unary-evaluator rule counts. For every raw bitstring it traverses exactly FormulaTokensPerClause padding opportunities in the intentionally empty fifth fixed-width clause rectangle without emitting a token, preserves encodedFormula.take (2 * (FormulaWidth + 36)), and retains coordinate FormulaVariableSlotBound + 1 + 5 * FormulaTokensPerClause. Direct schedule lookup and the specification cursor prove that every traversed fifth-slot opportunity and the first opportunity in the intentionally empty sixth slot are padding. The external compiled bound is BuilderFourthClausePaddingRun.rawTimeBound + 18 + six times the count-evaluator work, countdown bound, and target-evaluator work. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, malformed-countdown, unlaunched-predecessor, and one-step-short results are proved. All 65 new public declarations and three reused countdown interfaces are axiom-audited: 28 have empty closure, 9 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin empty fifth-clause padding run", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loopSteps_le", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "972da09f44b6d35d56ce775bd4a46687be45609dacd0670bc983d603715bf278", "expectedKernelTypeSha256": "972da09f44b6d35d56ce775bd4a46687be45609dacd0670bc983d603715bf278", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loop_workRunExact", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "ff60669772a7f137755431c1300c4368e8004a05b1d36972a768b50d1846cf5a", "expectedKernelTypeSha256": 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"kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-first-constraint-padding-run", "nonClaim": "This milestone traverses only the remaining empty clause rectangles of the first scheduled constraint and retains the Sep beginning the second scheduled constraint. It observes but does not emit that separator, does not emit the next constraint's first literal, implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, emit the remaining formula body, construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loopSteps_le", "PNP.Concrete.CookLevin.BuilderFirstClausePaddingRun.PaddingCountdown.loop_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.countEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.countdownBoundPolynomial_eval", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.countdown_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.secondConstraintStart_direct_eq_sep", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.secondConstraintStart_eq", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.finalTape_represents", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.finalTokenBits_eq_encodedFormula_fourthClause", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.finalTokenSlot_eq_secondConstraintStart", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.launch_workStep", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.malformedCountdownRoot_timeout", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.malformedCountdownScratch_timeout", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.paddingSlot_direct_eq_padding", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.predecessorSlot_add_paddingCount", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.paddingCount_eq", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.paddingCount_eq_remaining_first_constraint", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.paddingCount_positive", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.rules_length", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.run_compile_exact", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.specification_padding_run", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.specification_target_step", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.workRunExact", "PNP.Concrete.CookLevin.BuilderFirstConstraintPaddingRun.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the fifth-clause padding predecessor with two structurally generated unary evaluators, the reused 25-rule PaddingCountdown machine, and three total nine-symbol WorkChain bridges. Its table has exactly 4432 plus sixteen inherited/generated unary-evaluator rule counts. For every raw bitstring it traverses exactly (FormulaVariableSlotBound - 2) * (FormulaVariableSlotBound + 2) * FormulaTokensPerClause = (FormulaClauseSlotsPerConstraint - 5) * FormulaTokensPerClause remaining empty token opportunities of the first scheduled constraint without emitting a token, preserves encodedFormula.take (2 * (FormulaWidth + 36)), and retains coordinate FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause. Direct schedule lookup and the specification cursor prove that every traversed opportunity is padding and the endpoint is the Sep beginning the second scheduled constraint. The external compiled bound is BuilderFifthClausePaddingRun.rawTimeBound + 18 + six times the count-evaluator work, countdown bound, and target-evaluator work. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, malformed-countdown, unlaunched-predecessor, and one-step-short results are proved. 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It observes but does not emit the following T, does not emit the first literal or traverse the second constraint, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSeparatorStep.SeparatorCursor.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.appenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.appender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.cursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.finalTokenBits_eq_encodedFormula_secondConstraintStart", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.finalTokenSlot_eq_secondConstraintStart_add_one", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.malformedCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.nextTokenSlot_direct_eq_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.prefixSeparator_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.secondConstraintStartTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.separatorCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.specification_next_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.specification_separator_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeparatorStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete first-constraint padding run with the reused selected 59-rule Sep appender and 45-rule cursor advance through one outer total nine-symbol WorkChain bridge. The global table has exactly 4554 plus the sixteen inherited/generated unary-evaluator rule counts. Every raw input follows exact prefix, appender, cursor, bridge, suffix, and combined traces; emits exactly the Sep beginning the second scheduled constraint; preserves encodedFormula.take (2 * (FormulaWidth + 37)); and retains coordinate FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 1, whose direct next schedule token is T. The external compiled bound evaluates to BuilderFirstConstraintPaddingRun.rawTimeBound + 534 + 24 * n + 12 * FormulaWidth + 12 * cursorWord.length. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, malformed-workspace, both unlaunched-endpoint, and one-step-short results are proved. All 48 new public declarations plus eight reviewed reused separator/cursor interfaces are axiom-audited: 14 have empty closure, 11 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-constraint separator step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "expectedKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "bfde87bc9e3bc501eb3ca0f3edb197ab2b120fef845b632574bdf14209c72292", 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], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-second-constraint-first-literal-sign-step", "nonClaim": "This milestone emits exactly one token: the fixed positive T sign that begins the second scheduled constraint's first literal. It observes but does not emit the following unary T, does not complete that literal or traverse the second constraint, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.appenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.appender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.cursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.finalTokenBits_eq_encodedFormula_secondConstraintFirstLiteralSign", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.finalTokenSlot_eq_secondConstraintStart_add_two", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.malformedCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.nextTokenSlot_direct_eq_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.prefixSign_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.secondConstraintFirstLiteralSignTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.specification_next_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.specification_sign_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.trueTokenCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSignStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete second-constraint separator step with the reused selected 59-rule T appender and 45-rule cursor advance through one outer total nine-symbol WorkChain bridge. The global table has exactly 4676 plus the sixteen inherited/generated unary-evaluator rule counts. Every raw input follows exact prefix, appender, cursor, bridge, suffix, and combined traces; emits exactly the positive sign beginning the second scheduled constraint's first literal; preserves encodedFormula.take (2 * (FormulaWidth + 38)); and retains coordinate FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 2, whose direct next schedule token is the first unary T of a nonzero variable index. The external compiled bound evaluates to BuilderSecondConstraintSeparatorStep.rawTimeBound + 546 + 24 * n + 12 * FormulaWidth + 12 * cursorWord.length. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, malformed-workspace, both unlaunched-endpoint, and one-step-short results are proved. All 48 new public declarations plus eight reviewed reused true-token/cursor interfaces are axiom-audited: 14 have empty closure, 11 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-constraint first-literal sign step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "expectedKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "bfde87bc9e3bc501eb3ca0f3edb197ab2b120fef845b632574bdf14209c72292", 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It observes but does not emit the following second unary T, does not complete that literal or traverse the second constraint, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.appenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.appender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.cursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.finalTokenBits_eq_encodedFormula_secondConstraintFirstLiteralFirstUnary", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.finalTokenSlot_eq_secondConstraintStart_add_three", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.malformedCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.nextTokenSlot_direct_eq_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.prefixFirstUnaryUnit_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.secondConstraintFirstLiteralFirstUnaryTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.specification_firstUnaryUnit_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.specification_next_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.trueTokenCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete second-constraint first-literal sign step with the reused selected 59-rule T appender and 45-rule cursor advance through one outer total nine-symbol WorkChain bridge. The global table has exactly 4798 plus the sixteen inherited/generated unary-evaluator rule counts. Every raw input follows exact prefix, appender, cursor, bridge, suffix, and combined traces; emits exactly the first unary T of the second scheduled constraint's first variable index; preserves encodedFormula.take (2 * (FormulaWidth + 39)); and retains coordinate FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 3. A constructive schedule proof establishes that the index is at least three, so the direct next schedule token is the second unary T. The external compiled bound evaluates to BuilderSecondConstraintFirstLiteralSignStep.rawTimeBound + 558 + 24 * n + 12 * FormulaWidth + 12 * cursorWord.length. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, malformed-workspace, both unlaunched-endpoint, and one-step-short results are proved. All 48 new public declarations plus eight reviewed reused true-token/cursor interfaces are axiom-audited: 14 have empty closure, 11 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-constraint first-literal first unary-unit step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "expectedKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": 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constraint's first variable index. It observes but does not emit the following third unary T, does not complete that literal or traverse the second constraint, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.appenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.appender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.cursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.finalTokenBits_eq_encodedFormula_secondConstraintFirstLiteralSecondUnary", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.finalTokenSlot_eq_secondConstraintStart_add_four", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.malformedCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.nextTokenSlot_direct_eq_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.prefixSecondUnaryUnit_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.secondConstraintFirstLiteralSecondUnaryTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.specification_next_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.specification_secondUnaryUnit_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.trueTokenCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete second-constraint first-literal first-unary-unit step with the reused selected 59-rule T appender and 45-rule cursor advance through one outer total nine-symbol WorkChain bridge. The global table has exactly 4920 plus the sixteen inherited/generated unary-evaluator rule counts. Every raw input follows exact prefix, appender, cursor, bridge, suffix, and combined traces; emits exactly the second unary T of the second scheduled constraint's first variable index; preserves encodedFormula.take (2 * (FormulaWidth + 40)); and retains coordinate FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 4. A constructive schedule proof establishes that the index is at least three, so the direct next schedule token is the third unary T. The external compiled bound evaluates to BuilderSecondConstraintFirstLiteralFirstUnaryUnitStep.rawTimeBound + 570 + 24 * n + 12 * FormulaWidth + 12 * cursorWord.length. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, malformed-workspace, both unlaunched-endpoint, and one-step-short results are proved. All 48 new public declarations plus eight reviewed reused true-token/cursor interfaces are axiom-audited: 14 have empty closure, 11 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-constraint first-literal second unary-unit step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "expectedKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": 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"earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-second-constraint-first-literal-third-unary-unit-step", "nonClaim": "This milestone emits exactly one token: the third and final unary T of the second scheduled constraint's first variable index. It observes but does not emit the following terminating F, does not complete that literal or traverse the second constraint, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rules_length", "PNP.Concrete.CookLevin.BuilderSecondClauseSecondLiteralPrefix.TrueTokenCursor.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.appenderEndpoint_before_cursor_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.appender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.cursor_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.finalTokenBits_eq_encodedFormula_secondConstraintFirstLiteralThirdUnary", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.finalTokenSlot_eq_secondConstraintStart_add_five", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.malformedAppenderOutput_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.malformedAppenderTally_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.malformedCursorScratch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.nextTokenSlot_direct_eq_f", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.prefixThirdUnaryUnit_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.secondConstraintFirstLiteralThirdUnaryTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.specification_next_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.specification_thirdUnaryUnit_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.trueTokenCursor_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete second-constraint first-literal second-unary-unit step with the reused selected 59-rule T appender and 45-rule cursor advance through one outer total nine-symbol WorkChain bridge. The global table has exactly 5042 plus the sixteen inherited/generated unary-evaluator rule counts. Every raw input follows exact prefix, appender, cursor, bridge, suffix, and combined traces; emits exactly the third and final unary T of the second scheduled constraint's first variable index; preserves encodedFormula.take (2 * (FormulaWidth + 41)); and retains coordinate FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 5. A constructive schedule case split proves the selected variable index is exactly three in both the width-one head-variable and wider position-one blank-symbol branches, so the direct next schedule token is the terminating F. The external compiled bound evaluates to BuilderSecondConstraintFirstLiteralSecondUnaryUnitStep.rawTimeBound + 582 + 24 * n + 12 * FormulaWidth + 12 * cursorWord.length. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, malformed-workspace, both unlaunched-endpoint, and one-step-short results are proved. 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"PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.nextTokenSlot_direct_eq_finish_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.prefixTerminator_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.secondConstraintFirstLiteralTerminatorTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.specification_next_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.specification_terminator_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete second-constraint first-literal third-unary-unit step with the reused selected 59-rule F appender and 45-rule cursor advance through one outer total nine-symbol WorkChain bridge. The global table has exactly 5164 plus the sixteen inherited/generated unary-evaluator rule counts. Every raw input follows exact prefix, appender, cursor, bridge, suffix, and combined traces; emits exactly the terminating F of the second scheduled constraint's first literal; preserves encodedFormula.take (2 * (FormulaWidth + 42)); and retains coordinate FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 6. A constructive schedule case split proves the direct next schedule token is Finish when tapeWidth is one and the positive T beginning the next literal at wider widths. The external compiled bound evaluates to BuilderSecondConstraintFirstLiteralThirdUnaryUnitStep.rawTimeBound + 594 + 24 * n + 12 * FormulaWidth + 12 * cursorWord.length. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, malformed-workspace, both unlaunched-endpoint, and one-step-short results are proved. All 48 new public declarations plus eight reviewed reused false-token/cursor and dead-state interfaces are axiom-audited: 14 have empty closure, 11 use only propext, and 31 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-constraint first-literal terminator step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.deadState_workStep", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "expectedKernelTypeSha256": "681cac71bc2a0b9c3c42250a3b3e4ef28bb743f8b409b0b440a64230655eabaf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderDynamicTokenCursorStep.CursorAdvance.malformedScratch_enters_dead", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": 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scheduled constraint's first literal: Finish at width one or positive T at wider widths. It observes but does not emit the following padding opportunity at width one or unary T at wider widths, does not emit the remainder of the second constraint or traverse that constraint, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.WidthBranchAppender.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.WidthBranchAppender.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.WidthBranchAppender.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.WidthBranchAppender.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.WidthBranchAppender.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.branchAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.finalTokenBits_eq_encodedFormula_secondConstraintFirstLiteralSuccessor", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.finalTokenSlot_eq_secondConstraintStart_add_seven", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.followingTokenSlot_direct_eq_padding_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.prefixSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.secondConstraintFirstLiteralSuccessorTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.specification_following_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.specification_successor_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.successorTokenSlot_eq_secondConstraintStart_add_seven", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.widthEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.width_eq_tapeWidth", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.width_positive", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.work_one_step_short_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.followingTokenSlot_direct_eq_padding_or_t" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete second-constraint first-literal terminator with the represented-width unary evaluator, one 93-rule finite width branch that enters a single reused 59-rule token appender at either Finish or T, and the retained-coordinate unary evaluator through three total nine-symbol WorkChain bridges. The global table has exactly 5284 plus the eighteen inherited/generated unary-evaluator rule counts. Every raw input follows exact predecessor, width-evaluator, branch/appender, retained-coordinate, bridge, suffix, and combined traces; emits Finish exactly when tapeWidth is one and T at every wider width; preserves encodedFormula.take (2 * (FormulaWidth + 43)); and retains coordinate FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 7. Direct lookup and the specification cursor prove the following opportunity is padding at width one and unary T at wider widths. The external compiled bound evaluates to BuilderSecondConstraintFirstLiteralTerminatorStep.rawTimeBound + 600 + 24 * n + 12 * FormulaWidth + 12 * width + 12 * widthRootPrefixLength + 6 * widthWorkSteps + 6 * targetWorkSteps. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, predecessor-unlaunched-endpoint, and one-step-short results are proved while component malformed-workspace behavior remains fail-closed. All 80 new public declarations plus two strengthened predecessor boundary lemmas are axiom-audited: 37 have empty closure, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-constraint first-literal successor token step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.WidthBranchAppender.machine_acceptState_ne_rejectState", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "26f035f528660d27a4ab29f6fa83ff0be3bf211d51b877c5d8f55a6bca4bf0bd", "expectedKernelTypeSha256": "26f035f528660d27a4ab29f6fa83ff0be3bf211d51b877c5d8f55a6bca4bf0bd", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralSuccessorTokenStep.WidthBranchAppender.rule_source_ne_acceptState", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": 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"sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-second-constraint-padding-or-unary-opportunity-step", "nonClaim": "This milestone consumes exactly one width-selected schedule opportunity after the second scheduled constraint first literal: padding with no emitted token at width one or the first unary T of the second literal at wider widths. It observes but does not consume the following padding opportunity at width one or second unary T at wider widths, does not complete the second literal or traverse the remainder of the second constraint, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor_and_optional_unary", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.secondFollowingTokenSlot_direct_eq_padding_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.finalTokenBits_eq_encodedFormula_secondConstraintPaddingOrUnary", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.finalTokenSlot_eq_secondConstraintStart_add_eight", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.followingTokenSlot_direct_eq_padding_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.opportunitySlot_eq_secondConstraintStart_add_eight", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.optionalAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.prefixSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.secondConstraintPaddingOrUnaryTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.specification_following_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.specification_opportunity_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.widthEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.width_eq_tapeWidth", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.width_positive", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete second-constraint first-literal successor-token predecessor with the represented-width unary evaluator, one 93-rule finite optional-appender table, and the retained-coordinate unary evaluator through three total nine-symbol WorkChain bridges. The global table has exactly 5404 plus the twenty inherited/generated unary-evaluator rule counts. Every raw input follows exact predecessor, width-evaluator, width-one skip or wider-width T-appender, retained-coordinate, bridge, suffix, and combined traces. At tapeWidth one it consumes padding and emits no token; at every wider width it appends exactly the first unary T of the second literal. The output equals encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 1)) and the retained coordinate is FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 8. Direct lookup and the specification cursor prove the following slot is again padding at width one and the second unary T at wider widths. The external compiled bound evaluates to BuilderSecondConstraintFirstLiteralSuccessorTokenStep.rawTimeBound + 612 + 24 * n + 12 * FormulaWidth + 12 * width + 12 * widthRootPrefixLength + 6 * widthWorkSteps + 6 * targetWorkSteps. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, predecessor-unlaunched-endpoint, and one-step-short results are proved while component malformed-workspace behavior remains fail-closed. All 80 new public declarations plus two strengthened schedule lemmas are axiom-audited: 37 have empty closure, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-constraint padding-or-unary opportunity step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor_and_optional_unary", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "e2430440ee1e6c9a103e78f021f0c91e7b6903385ee8cc9ca053c3303a743b2e", "expectedKernelTypeSha256": "e2430440ee1e6c9a103e78f021f0c91e7b6903385ee8cc9ca053c3303a743b2e", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.secondFollowingTokenSlot_direct_eq_padding_or_t", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": 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"concrete-cook-levin-builder-second-constraint-second-padding-or-unary-opportunity-step", "nonClaim": "This milestone consumes exactly one additional width-selected schedule opportunity after the second scheduled constraint first literal: padding with no emitted token at width one or the second unary T of the second literal at wider widths. It observes but does not consume the following padding opportunity at width one or third unary T at wider widths, does not complete the second literal or traverse the remainder of the second constraint, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor_and_two_optional_unary", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.thirdFollowingTokenSlot_direct_eq_padding_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.finalTokenBits_eq_encodedFormula_secondConstraintSecondPaddingOrUnary", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.finalTokenSlot_eq_secondConstraintStart_add_nine", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.followingTokenSlot_direct_eq_padding_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.opportunitySlot_eq_secondConstraintStart_add_nine", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.optionalAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.prefixSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.secondConstraintSecondPaddingOrUnaryTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.specification_following_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.specification_opportunity_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.widthEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.width_eq_tapeWidth", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.width_positive", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete first padding-or-unary-opportunity predecessor with the represented-width unary evaluator, the same reviewed 93-rule finite optional-appender table, and the retained-coordinate unary evaluator through three total nine-symbol WorkChain bridges. The global table has exactly 5524 plus the twenty-two inherited/generated unary-evaluator rule counts. Every raw input follows exact predecessor, width-evaluator, width-one skip or wider-width T-appender, retained-coordinate, bridge, suffix, and combined traces. At tapeWidth one it consumes padding and emits no token; at every wider width it appends exactly the second unary T of the second literal. The output equals encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 2)) and the retained coordinate is FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 9. Direct lookup and the specification cursor prove the following slot is again padding at width one and the third unary T at wider widths. The external compiled bound evaluates to BuilderSecondConstraintPaddingOrUnaryOpportunityStep.rawTimeBound + 624 + 24 * n + 12 * FormulaWidth + 12 * width + 12 * widthRootPrefixLength + 6 * widthWorkSteps + 6 * targetWorkSteps. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, predecessor-unlaunched-endpoint, and one-step-short results are proved while component malformed-workspace behavior remains fail-closed. All 66 new public declarations, fourteen reused optional-appender interfaces, and two strengthened schedule lemmas are axiom-audited: 37 have empty closure, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-constraint second padding-or-unary opportunity step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor_and_two_optional_unary", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "4309cbc40036e6dda6d29606d9a319c356abdbb382c43a5a909987a8dbc1c0f3", "expectedKernelTypeSha256": "4309cbc40036e6dda6d29606d9a319c356abdbb382c43a5a909987a8dbc1c0f3", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.thirdFollowingTokenSlot_direct_eq_padding_or_t", "present": true, "kind": "theorem", "axioms": 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"axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-second-constraint-third-padding-or-unary-opportunity-step", "nonClaim": "This milestone consumes exactly one additional width-selected schedule opportunity after the second scheduled constraint first literal: padding with no emitted token at width one or the third unary T of the second literal at wider widths. It observes but does not consume the following padding opportunity at width one or fourth unary T at wider widths, does not complete the second literal or traverse the remainder of the second constraint, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor_and_three_optional_unary", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.fourthFollowingTokenSlot_direct_eq_padding_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.finalTokenBits_eq_encodedFormula_secondConstraintThirdPaddingOrUnary", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.finalTokenSlot_eq_secondConstraintStart_add_ten", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.followingTokenSlot_direct_eq_padding_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.opportunitySlot_eq_secondConstraintStart_add_ten", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.optionalAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.prefixSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.secondConstraintThirdPaddingOrUnaryTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.specification_following_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.specification_opportunity_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.widthEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.width_eq_tapeWidth", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.width_positive", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete second padding-or-unary-opportunity predecessor with the represented-width unary evaluator, the same reviewed 93-rule finite optional-appender table, and the retained-coordinate unary evaluator through three total nine-symbol WorkChain bridges. The global table has exactly 5644 plus the twenty-four inherited/generated unary-evaluator rule counts. Every raw input follows exact predecessor, width-evaluator, width-one skip or wider-width T-appender, retained-coordinate, bridge, suffix, and combined traces. At tapeWidth one it consumes padding and emits no token; at every wider width it appends exactly the third unary T of the second literal. The output equals encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 3)) and the retained coordinate is FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 10. Direct lookup and the specification cursor prove the following slot is again padding at width one and the fourth unary T at wider widths. The external compiled bound evaluates to BuilderSecondConstraintSecondPaddingOrUnaryOpportunityStep.rawTimeBound + 636 + 24 * n + 12 * FormulaWidth + 12 * width + 12 * widthRootPrefixLength + 6 * widthWorkSteps + 6 * targetWorkSteps. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, predecessor-unlaunched-endpoint, and one-step-short results are proved while component malformed-workspace behavior remains fail-closed. All 66 new public declarations, fourteen reused optional-appender interfaces, and two strengthened schedule lemmas are axiom-audited: 37 have empty closure, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-constraint third padding-or-unary opportunity step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor_and_three_optional_unary", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "92700cf1a70adfb254512d9ac5492a634ba0e083e939b218f99361e52762da75", "expectedKernelTypeSha256": "92700cf1a70adfb254512d9ac5492a634ba0e083e939b218f99361e52762da75", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.fourthFollowingTokenSlot_direct_eq_padding_or_t", "present": true, "kind": "theorem", 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"PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.prefixSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.secondConstraintFourthPaddingOrUnaryTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.specification_following_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.specification_opportunity_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.widthEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.width_eq_tapeWidth", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.width_positive", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete third padding-or-unary-opportunity predecessor with the represented-width unary evaluator, the same reviewed 93-rule finite optional-appender table, and the retained-coordinate unary evaluator through three total nine-symbol WorkChain bridges. The global table has exactly 5764 plus the twenty-six inherited/generated unary-evaluator rule counts. Every raw input follows exact predecessor, width-evaluator, width-one skip or wider-width T-appender, retained-coordinate, bridge, suffix, and combined traces. At tapeWidth one it consumes padding and emits no token; at every wider width it appends exactly the fourth unary T of the second literal. The output equals encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 4)) and the retained coordinate is FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 11. Direct lookup and the specification cursor prove the following slot is padding at width one and the terminating F at wider widths. The external compiled bound evaluates to BuilderSecondConstraintThirdPaddingOrUnaryOpportunityStep.rawTimeBound + 648 + 24 * n + 12 * FormulaWidth + 12 * width + 12 * widthRootPrefixLength + 6 * widthWorkSteps + 6 * targetWorkSteps. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, predecessor-unlaunched-endpoint, and one-step-short results are proved while component malformed-workspace behavior remains fail-closed. All 66 new public declarations, fourteen reused optional-appender interfaces, and two strengthened schedule lemmas are axiom-audited: 37 have empty closure, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-constraint fourth padding-or-unary opportunity step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor_and_four_optional_unary", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "90d6dbf3d5226e04b2e37831bcefa905586c90db15377024a3e218d7edb1de05", "expectedKernelTypeSha256": "90d6dbf3d5226e04b2e37831bcefa905586c90db15377024a3e218d7edb1de05", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.fifthFollowingTokenSlot_direct_eq_padding_or_f", "present": true, "kind": "theorem", "axioms": 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"axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-second-constraint-fifth-padding-or-terminator-opportunity-step", "nonClaim": "This milestone consumes exactly one additional width-selected schedule opportunity after the second scheduled constraint first literal: padding with no emitted token at width one or the terminating F of the second literal at wider widths. It observes but does not consume the following padding opportunity at width one or opening unary T at wider widths, does not complete the following literal or traverse the remainder of the second constraint, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor_and_four_optional_unary_and_optional_terminator", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.sixthFollowingTokenSlot_direct_eq_padding_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.WidthOptionalTerminatorAppender.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.WidthOptionalTerminatorAppender.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.WidthOptionalTerminatorAppender.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.WidthOptionalTerminatorAppender.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.WidthOptionalTerminatorAppender.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.finalTokenBits_eq_encodedFormula_secondConstraintFifthPaddingOrTerminator", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.finalTokenSlot_eq_secondConstraintStart_add_twelve", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.followingTokenSlot_direct_eq_padding_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.opportunitySlot_eq_secondConstraintStart_add_twelve", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.optionalAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.prefixSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.secondConstraintFifthPaddingOrTerminatorTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.specification_following_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.specification_opportunity_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.widthEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.width_eq_tapeWidth", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.width_positive", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete fourth padding-or-unary-opportunity predecessor with the represented-width unary evaluator, a new reviewed 93-rule finite optional-terminator table over the audited token-appender rules, and the retained-coordinate unary evaluator through three total nine-symbol WorkChain bridges. The global table has exactly 5884 plus the twenty-eight inherited/generated unary-evaluator rule counts. Every raw input follows exact predecessor, width-evaluator, width-one skip or wider-width F-appender, retained-coordinate, bridge, suffix, and combined traces. At tapeWidth one it consumes padding and emits no token; at every wider width it appends exactly the terminating F of the second literal. The output equals encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 5)) and the retained coordinate is FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 12. Direct lookup and the specification cursor prove the following slot is padding at width one and the opening unary T of the following literal at wider widths. The external compiled bound evaluates to BuilderSecondConstraintFourthPaddingOrUnaryOpportunityStep.rawTimeBound + 660 + 24 * n + 12 * FormulaWidth + 12 * width + 12 * widthRootPrefixLength + 6 * widthWorkSteps + 6 * targetWorkSteps. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, predecessor-unlaunched-endpoint, and one-step-short results are proved while component malformed-workspace behavior remains fail-closed. The measured audit covers all 66 new public outer declarations, fourteen new optional-terminator interfaces, and two strengthened schedule lemmas: 37 closures are empty, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-constraint fifth padding-or-terminator opportunity step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor_and_four_optional_unary_and_optional_terminator", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "8f4243419d3a6b2b4811b8c456180b78c134b58f4c29cfb79c616d317ed7eeff", "expectedKernelTypeSha256": "8f4243419d3a6b2b4811b8c456180b78c134b58f4c29cfb79c616d317ed7eeff", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.sixthFollowingTokenSlot_direct_eq_padding_or_t", 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"5860e69e2bcd994de08fb3af34997889d3819e9fdb73f3a48ea9b8a7c925fbed", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-second-constraint-sixth-padding-or-opening-unary-opportunity-step", "nonClaim": "This milestone consumes exactly one additional width-selected schedule opportunity after the second scheduled constraint second literal: padding with no emitted token at width one or the opening positive T of the following literal at wider widths. It observes but does not consume the following padding opportunity at width one or first unary-index T at wider widths, does not complete the following literal or traverse the remainder of the second constraint, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor_and_four_optional_unary_and_optional_terminator_and_optional_opening_unary", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.seventhFollowingTokenSlot_direct_eq_padding_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.finalTokenBits_eq_encodedFormula_secondConstraintSixthPaddingOrOpeningUnary", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.finalTokenSlot_eq_secondConstraintStart_add_thirteen", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.followingTokenSlot_direct_eq_padding_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.opportunitySlot_eq_secondConstraintStart_add_thirteen", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.optionalAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.prefixSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.secondConstraintSixthPaddingOrOpeningUnaryTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.specification_following_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.specification_opportunity_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.widthEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.width_eq_tapeWidth", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.width_positive", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete fifth padding-or-terminator-opportunity predecessor with the represented-width unary evaluator, the reviewed 93-rule finite optional-appender table, and the retained-coordinate unary evaluator through three total nine-symbol WorkChain bridges. The global table has exactly 6004 plus the thirty inherited/generated unary-evaluator rule counts. Every raw input follows exact predecessor, width-evaluator, width-one skip or wider-width opening-T appender, retained-coordinate, bridge, suffix, and combined traces. At tapeWidth one it consumes padding and emits no token; at every wider width it appends exactly the opening positive T of the following literal. The output equals encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 6)) and the retained coordinate is FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 13. Direct lookup and the specification cursor prove the following slot is padding at width one and the first unary-index T of the following literal at wider widths. The external compiled bound evaluates to BuilderSecondConstraintFifthPaddingOrTerminatorOpportunityStep.rawTimeBound + 672 + 24 * n + 12 * FormulaWidth + 12 * width + 12 * widthRootPrefixLength + 6 * widthWorkSteps + 6 * targetWorkSteps. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, predecessor-unlaunched-endpoint, and one-step-short results are proved while component malformed-workspace behavior remains fail-closed. The measured audit covers all 66 new public declarations, fourteen reused optional-appender interfaces, and two strengthened schedule lemmas: 37 closures are empty, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-constraint sixth padding-or-opening-unary opportunity step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor_and_four_optional_unary_and_optional_terminator_and_optional_opening_unary", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "9d7f366b3a42520747cd82097e2b7976d347f6c105e30f297726801b6865daed", "expectedKernelTypeSha256": "9d7f366b3a42520747cd82097e2b7976d347f6c105e30f297726801b6865daed", "kernelTypeFingerprintMatches": true }, { "name": 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"97ccd90e58fd9753562bb482bc812290ed7235b4d4bf09d8c3beda77acd19df8", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-foundation-only" }, { "classification": "formalized-foundation-only", "id": "concrete-cook-levin-builder-second-constraint-seventh-padding-or-unary-opportunity-step", "nonClaim": "This milestone consumes exactly one additional width-selected schedule opportunity after the second scheduled constraint following literal opening: padding with no emitted token at width one or the first unary-index T of the following literal at wider widths. It observes but does not consume the following padding opportunity at width one or second unary-index T at wider widths, does not complete the following literal or traverse the remainder of the second constraint, does not implement a general dynamic formula cursor or raw decoder for arbitrary schedule coordinates, does not emit the remaining formula body, and does not construct a complete formula builder or FunctionProgram.RawRefinement, package a concrete PolynomialReduction, establish CNFSAT NP-hardness or NP-completeness, establish CNFSAT in P, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor_and_four_optional_unary_and_optional_terminator_and_optional_opening_unary_and_optional_first_unary", "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.eighthFollowingTokenSlot_direct_eq_padding_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintPaddingOrUnaryOpportunityStep.WidthOptionalAppender.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.boundedDecide_compile_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.boundedDecide_compile_ne_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.finalConfiguration_state", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.finalOutside_contains_finalTokenSlot", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.finalTape_represents", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.finalTokenBits_eq_encodedFormula_secondConstraintSeventhPaddingOrUnary", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.finalTokenSlot_eq_secondConstraintStart_add_fourteen", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.followingTokenSlot_direct_eq_padding_or_t", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.machine_acceptState_ne_rejectState", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.opportunitySlot_eq_secondConstraintStart_add_fourteen", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.optionalAppender_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.prefixEndpoint_before_launch_timeout", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.prefixSuffix_launch_workStep", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.prefix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.rawTimeBound_eval", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.rawTimeBound_le", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.rule_source_ne_acceptState", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.rules_length", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.rules_pairwise_query_distinct", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.run_compile_exact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.run_compile_rawTimeBound", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.run_compile_rawTimeBound_blankEquivalent", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.secondConstraintSeventhPaddingOrUnaryTokens_eq_canonical_formula_prefix", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.specification_following_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.specification_opportunity_step", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.suffix_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.targetEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.widthEvaluator_workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.width_eq_tapeWidth", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.width_positive", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.workBoundedDecide_accept", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.workRunExact", "PNP.Concrete.CookLevin.BuilderSecondConstraintSeventhPaddingOrUnaryOpportunityStep.work_one_step_short_timeout" ], "scope": "For every fixed concrete polynomial-time verifier, one literal finite work machine composes the complete sixth padding-or-opening-unary-opportunity predecessor with the represented-width unary evaluator, the reviewed 93-rule finite optional-appender table, and the retained-coordinate unary evaluator through three total nine-symbol WorkChain bridges. The global table has exactly 6124 plus the thirty-two inherited/generated unary-evaluator rule counts. Every raw input follows exact predecessor, width-evaluator, width-one skip or wider-width first-unary-index-T appender, retained-coordinate, bridge, suffix, and combined traces. At tapeWidth one it consumes padding and emits no token; at every wider width it appends exactly the first unary-index T of the following literal. The output equals encodedFormula.take (2 * (FormulaWidth + 43 + if tapeWidth = 1 then 0 else 7)) and the retained coordinate is FormulaVariableSlotBound + 1 + FormulaClauseSlotsPerConstraint * FormulaTokensPerClause + 14. Direct lookup and the specification cursor prove the following slot is padding at width one and the second unary-index T of the following literal at wider widths. The external compiled bound evaluates to BuilderSecondConstraintSixthPaddingOrOpeningUnaryOpportunityStep.rawTimeBound + 684 + 24 * n + 12 * FormulaWidth + 12 * width + 12 * widthRootPrefixLength + 6 * widthWorkSteps + 6 * targetWorkSteps. Compiled exact, bounded, blank-equivalent, accepting, non-timeout, predecessor-unlaunched-endpoint, and one-step-short results are proved while component malformed-workspace behavior remains fail-closed. The measured audit covers all 66 new public declarations, fourteen reused optional-appender interfaces, and two strengthened schedule lemmas: 37 closures are empty, 12 use only propext, and 33 use only propext and Quot.sound, with no project axiom or Classical.choice.", "title": "Cook-Levin second-constraint seventh padding-or-unary opportunity step", "theoremRows": [ { "name": "PNP.Concrete.CookLevin.BuilderSecondConstraintFirstLiteralTerminatorStep.encodeCNFTokens_eq_terminator_then_successor_and_four_optional_unary_and_optional_terminator_and_optional_opening_unary_and_optional_first_unary", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "d84e664bc9f9a319f4be056013df50f32cae94aad1c237f79058397d97884cb8", "expectedKernelTypeSha256": "d84e664bc9f9a319f4be056013df50f32cae94aad1c237f79058397d97884cb8", "kernelTypeFingerprintMatches": true }, { "name": 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"classification": "formalized", "id": "locked-nand-global-unsatisfiable-final-zero", "nonClaim": "This does not prove satisfiable FinalLockSeparation, instantiate the complete threshold package, establish the global locked-NAND threshold or residual-slack bound, or construct the uniform polynomial builder.", "requiredTheorems": [ "PNP.DirectWire.LockedNANDGlobalCandidates.fullCandidate_final_eq_false_of_unsatisfiable", "PNP.DirectWire.LockedNANDGlobalCandidates.fullCandidate_referenceMinimum_eq_baseline_of_unsatisfiable" ], "scope": "For every finite topologically ordered NAND circuit, unsatisfiability makes the full final coordinate identically false on the whole carrier and fixes the exhaustive reference minimum at B.", "title": "Locked-NAND global unsatisfiable final-zero branch", "theoremRows": [ { "name": "PNP.DirectWire.LockedNANDGlobalCandidates.fullCandidate_final_eq_false_of_unsatisfiable", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], 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construct or compile the report's encoded polynomial-time SAT-to-locked-NAND builder, establish CNFSAT in P, prove NP-hardness transport, discharge the abstract locked-NAND threshold axiom, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.LockedNANDGlobalCandidates.fullCandidate_final_nonconstant_of_satisfiable", "PNP.DirectWire.LockedNANDGlobalCandidates.fullCandidate_final_notPositiveProjection_of_satisfiable", "PNP.DirectWire.LockedNANDGlobalCandidates.fullCandidate_final_distinctFromBaseline_of_satisfiable", "PNP.DirectWire.LockedNANDGlobalCandidates.fullCandidate_satisfiableFinalConditions", "PNP.DirectWire.LockedNANDGlobalCandidates.fullCandidate_referenceMinimum_bounds_of_satisfiable", "PNP.DirectWire.LockedNANDGlobalCandidates.fullCandidate_residualSlack_le_four", "PNP.DirectWire.LockedNANDGlobalCandidates.fullCandidate_satisfiable_iff_referenceMinimum_ge_succ", "PNP.DirectWire.LockedNANDGlobalCandidates.fullCandidate_referenceMinimum_eq_baseline_of_unsatisfiable" ], "scope": "For every finite topologically ordered NAND circuit, one answer-independent full candidate instantiates all six semantic premises, has residual slack at most four, and crosses the exact source-derived minimum threshold exactly when the source circuit is satisfiable.", "title": "Locked-NAND satisfiable separation and typed semantic threshold", "theoremRows": [ { "name": "PNP.DirectWire.LockedNANDGlobalCandidates.fullCandidate_final_nonconstant_of_satisfiable", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "d6ad3c12cf8b3d36a3491340149b78d60aa3b983765a409b56e990012c87cfc1", "expectedKernelTypeSha256": "d6ad3c12cf8b3d36a3491340149b78d60aa3b983765a409b56e990012c87cfc1", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.LockedNANDGlobalCandidates.fullCandidate_final_notPositiveProjection_of_satisfiable", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": 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"concrete-locked-nand-encoded-semantic-boundary", "nonClaim": "This is not a parser/validator machine, emitter machine, RawRefinement, PolynomialReduction, construction-runtime or output-size bound, abstract locked-NAND threshold discharge, CNFSAT-in-P result, or P = NP.", "requiredTheorems": [ "PNP.Concrete.LockedNAND.RawCircuit.normalize_idempotent", "PNP.Concrete.LockedNAND.RawCircuit.normalize_eval", "PNP.Concrete.LockedNAND.RawCircuit.elaborate_ofCircuit", "PNP.Concrete.LockedNAND.RawCandidate.elaborate_ofCandidate", "PNP.Concrete.LockedNAND.RawLockedInstance.elaborate_ofCandidate", "PNP.Concrete.LockedNAND.decodeTokens_encodeTokens", "PNP.Concrete.LockedNAND.decodeCircuit_encodeCircuit", "PNP.Concrete.LockedNAND.decodeLockedInstance_encodeLockedInstance", "PNP.Concrete.LockedNAND.decodeElaboratedCircuit_encodeCircuit_ofCircuit", "PNP.Concrete.LockedNAND.encoded_fullCandidate_threshold_iff_satisfiable", "PNP.Concrete.LockedNAND.buildLockedNANDInstance_correct" ], "scope": "A strict version-zero bit grammar round-trips normalized NAND circuits and complete locked-NAND candidates; the pure all-bitstring transformation is fail-closed and preserves source satisfiability at the exact target threshold.", "title": "Encoded locked-NAND semantic boundary", "theoremRows": [ { "name": "PNP.Concrete.LockedNAND.RawCircuit.normalize_idempotent", "present": true, "kind": "theorem", "axioms": [ "propext" ], "actualKernelTypeSha256": "7001ae372f765c79569bf135c4fb74a284d215ecac39babf32492c38334fa2b8", "expectedKernelTypeSha256": "7001ae372f765c79569bf135c4fb74a284d215ecac39babf32492c38334fa2b8", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.LockedNAND.RawCircuit.normalize_eval", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "f18064d9bb1f00d62077374aa7ac7cbf4340557f6b7d9c85137077aa329f93a4", "expectedKernelTypeSha256": "f18064d9bb1f00d62077374aa7ac7cbf4340557f6b7d9c85137077aa329f93a4", 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"name": "PNP.Concrete.LockedNAND.buildLockedNANDInstance_correct", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "0d97147c27f39b417f82efd1e45e0defb1bf26bdc2e5fbd27312dd36b5ce23a8", "expectedKernelTypeSha256": "0d97147c27f39b417f82efd1e45e0defb1bf26bdc2e5fbd27312dd36b5ce23a8", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-semantic-boundary" }, { "classification": "formalized-foundation-only", "id": "concrete-locked-nand-source-parser", "nonClaim": "This source parser alone does not emit the locked-NAND target or establish the source-to-target PolynomialReduction. The downstream emitter now supplies its own runtime/output bounds and strict composition, but the abstract locked-NAND threshold assumption, CNFSAT-in-P result, NP-hardness or NP-completeness transport, and P = NP remain absent.", "requiredTheorems": [ "PNP.Concrete.LockedNAND.SourceParser.acceptedTape_outputBits", "PNP.Concrete.LockedNAND.SourceParser.allInput_exact", "PNP.Concrete.LockedNAND.SourceParser.canonicalSteps_le_validWorkBound", "PNP.Concrete.LockedNAND.SourceParser.compiledBoundedDecide_accept_iff", "PNP.Concrete.LockedNAND.SourceParser.compiledBoundedDecide_ne_timeout", "PNP.Concrete.LockedNAND.SourceParser.compiledMachineOutput_eq_validatedSourceBytes", "PNP.Concrete.LockedNAND.SourceParser.compiledStart_blankEquivalent", "PNP.Concrete.LockedNAND.SourceParser.decodeCircuitTokens_eq_none_iff_failure", "PNP.Concrete.LockedNAND.SourceParser.illFormed_exact", "PNP.Concrete.LockedNAND.SourceParser.machine_acceptState_ne_rejectState", "PNP.Concrete.LockedNAND.SourceParser.malformed_exact", "PNP.Concrete.LockedNAND.SourceParser.rules_length", "PNP.Concrete.LockedNAND.SourceParser.rules_pairwise_query_distinct", "PNP.Concrete.LockedNAND.SourceParser.statePrograms_length", "PNP.Concrete.LockedNAND.SourceParser.validFinalConfiguration_isHalted", "PNP.Concrete.LockedNAND.SourceParser.validFinalConfiguration_state", "PNP.Concrete.LockedNAND.SourceParser.validRawBound_eq", "PNP.Concrete.LockedNAND.SourceParser.validRawTimePolynomial_eval", "PNP.Concrete.LockedNAND.SourceParser.validatedSourceBytesPolynomialTimeFunction_output", "PNP.Concrete.LockedNAND.SourceParser.wellFormed_exact" ], "scope": "One literal nine-symbol finite work machine validates every strict version-zero source bitstring: it accepts exactly ValidEncodedCircuit, preserves valid bytes, clears invalid bytes, cannot time out within the proved compiled cubic bound, and supplies polynomial-time machine/function witnesses plus the validator's exact leaf 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The standalone emitter does not itself package the language equivalence as PolynomialReduction; the downstream concrete reduction milestone now does. The abstract locked-NAND threshold assumption, CNFSAT-in-P result, NP-hardness transport, and P = NP remain absent.", "requiredTheorems": [ "PNP.Concrete.LockedNAND.RawBuilder.rawLockedInstance_of_elaborate", "PNP.Concrete.LockedNAND.RawBuilder.targetBytes_of_elaborated", "PNP.Concrete.LockedNAND.TargetEmitterSpec.targetBytes_validatedSourceBytes_eq_buildLockedNANDInstance", "PNP.Concrete.LockedNAND.TargetEmitterSpec.targetBytes_size_le", "PNP.Concrete.LockedNAND.TargetEmitterController.rules_length_literal", "PNP.Concrete.LockedNAND.TargetEmitterController.rules_pairwise", "PNP.Concrete.LockedNAND.TargetEmitterController.machine_accept_ne_reject", "PNP.Concrete.LockedNAND.TargetEmitterController.graph_wellFormed", "PNP.Concrete.LockedNAND.TargetEmitterControllerTotalTrace.malformed_bounded_exact", "PNP.Concrete.LockedNAND.TargetEmitterControllerTotalTrace.decoded_bounded_exact", "PNP.Concrete.LockedNAND.TargetEmitterControllerTotalTrace.allInput_bounded_exact", "PNP.Concrete.LockedNAND.TargetEmitterControllerPolynomialBound.controllerWorkTimePolynomial_eval", "PNP.Concrete.LockedNAND.TargetEmitterControllerPolynomialBound.allInputWorkTimePolynomial_eval", "PNP.Concrete.LockedNAND.TargetEmitterControllerPolynomialBound.compiledRawTimePolynomial_eval", "PNP.Concrete.LockedNAND.TargetEmitterControllerPolynomialBound.controller_complete_path_polynomial", "PNP.Concrete.LockedNAND.TargetEmitterControllerPolynomialBound.controllerUniformEnvelope_le_workBound", "PNP.Concrete.LockedNAND.TargetEmitterControllerCompiled.compiledStart_blankEquivalent", "PNP.Concrete.LockedNAND.TargetEmitterControllerCompiled.compiledMachineOutput_eq_targetBytes", "PNP.Concrete.LockedNAND.TargetEmitterControllerCompiled.compiledBoundedDecide_accept_iff", "PNP.Concrete.LockedNAND.TargetEmitterControllerCompiled.compiledBoundedDecide_ne_timeout", "PNP.Concrete.LockedNAND.TargetEmitterControllerCompiled.rawTargetBytesPolynomialTimeFunction_output", 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"The downstream all-input CNF compiler now identifies CNFSAT with this concrete source language through a fixed finite machine and a direct polynomial reduction, then composes with this reduction. This milestone does not discharge the abstract target-language assumption, prove the report-level locked-NAND threshold theorem, put CNFSAT in P, complete residual minimization or ZeroSlack, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.LockedNAND.strictLockedNANDPolynomialReduction_function", "PNP.Concrete.LockedNAND.strictLockedNANDPolynomialReduction_output", "PNP.Concrete.LockedNAND.strictLockedNANDPolynomialReduction_correct", "PNP.Concrete.LockedNAND.encodedNANDSAT_reducesTo_encodedLockedNANDThreshold", "PNP.Concrete.LockedNAND.strictLockedNANDPolynomialReduction_hasRawRefinement" ], "scope": "The existing strict parser/emitter composition is packaged as a concrete polynomial many-one reduction from EncodedNANDSAT to EncodedLockedNANDThreshold, with exact function identity, exact output, all-bitstring language equivalence, a ReducesTo witness, and recursive raw-machine refinement.", "title": "Concrete strict-v0 locked-NAND polynomial reduction", "theoremRows": [ { "name": "PNP.Concrete.LockedNAND.strictLockedNANDPolynomialReduction_function", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "3e8000fd18d8836c1ae1ded6b3d0bb46d0ea618c31ba32c7f1425773e62d09c3", "expectedKernelTypeSha256": "3e8000fd18d8836c1ae1ded6b3d0bb46d0ea618c31ba32c7f1425773e62d09c3", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.LockedNAND.strictLockedNANDPolynomialReduction_output", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "06df8e094590a5b0bf522a9daf5921a28667c7d44e73d447d9366ae201ca3ef4", "expectedKernelTypeSha256": "06df8e094590a5b0bf522a9daf5921a28667c7d44e73d447d9366ae201ca3ef4", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.LockedNAND.strictLockedNANDPolynomialReduction_correct", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "94955e4d9826f8364e0ecae478eb33cc92620403673db079c7f4a961c15b3739", "expectedKernelTypeSha256": "94955e4d9826f8364e0ecae478eb33cc92620403673db079c7f4a961c15b3739", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.LockedNAND.encodedNANDSAT_reducesTo_encodedLockedNANDThreshold", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "6c133d95b3eb1d8d04f89be467dfbe6405883cb352654eacc5b8a23b99857f4d", "expectedKernelTypeSha256": "6c133d95b3eb1d8d04f89be467dfbe6405883cb352654eacc5b8a23b99857f4d", "kernelTypeFingerprintMatches": true }, { "name": "PNP.Concrete.LockedNAND.strictLockedNANDPolynomialReduction_hasRawRefinement", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "9f0b15fadf98e634edafe5d6a2025d1d92424a6876a018802153a96b934b1061", "expectedKernelTypeSha256": "9f0b15fadf98e634edafe5d6a2025d1d92424a6876a018802153a96b934b1061", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-polynomial-reduction" }, { "classification": "formalized-semantic-boundary", "id": "concrete-cnf-to-nand-semantic-compiler", "nonClaim": "This milestone is the pure semantic and size-bound layer; the subsequent all-input milestone supplies the finite-machine, PolynomialTimeFunction, RawRefinement, and PolynomialReduction interfaces. Neither layer decides CNF-SAT, proves CNFSAT is in deterministic polynomial time, discharges the abstract report-level locked-NAND premise, completes ZeroSlack/PCCMin, or proves P = NP.", "requiredTheorems": [ "PNP.Concrete.CNFToNAND.encodeCNF_of_decodeEncodedCNF", "PNP.Concrete.CNFToNAND.compileFormula_inputCount", "PNP.Concrete.CNFToNAND.compileFormula_output_is_gate", "PNP.Concrete.CNFToNAND.compileFormula_wellFormed", "PNP.Concrete.CNFToNAND.decodeValidCircuit_encode_compileFormula", "PNP.Concrete.CNFToNAND.compiledFormulaCircuit_eval_eq_true_iff", "PNP.Concrete.CNFToNAND.compiledFormulaCircuit_satisfiable_iff", "PNP.Concrete.CNFToNAND.compileFormula_satisfiable_iff", "PNP.Concrete.CNFToNAND.formula_satisfiable_iff_encoded_compileFormula", "PNP.Concrete.CNFToNAND.compileFormula_gateCount_exact", "PNP.Concrete.CNFToNAND.compileFormula_gateCount_le", "PNP.Concrete.CNFToNAND.cnfToNANDOutputSizePolynomial_eval", "PNP.Concrete.CNFToNAND.compileEncodedCNFToNAND_of_decoded", "PNP.Concrete.CNFToNAND.compileEncodedCNFToNAND_of_malformed", "PNP.Concrete.CNFToNAND.compileEncodedCNFToNAND_size_le", "PNP.Concrete.CNFToNAND.empty_not_encodedNANDSAT", "PNP.Concrete.CNFToNAND.compileEncodedCNFToNAND_correct", "PNP.Concrete.CNFToNAND.buildLockedNANDFromCNF_correct" ], "scope": "A total answer-independent compiler transforms every strict canonical CNF formula into an intrinsically topological well-formed NAND circuit, preserves satisfiability exactly, proves the exact gate count and a quadratic serialized-output bound, fails closed on every malformed bitstring, and composes semantically with the concrete locked-NAND threshold builder.", "title": "Concrete CNF-to-NAND semantic compiler", "theoremRows": [ { "name": "PNP.Concrete.CNFToNAND.encodeCNF_of_decodeEncodedCNF", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "a56dccb2670f82612e631706445aed45fcb7870cec9ce8870554c815c5d4aa90", "expectedKernelTypeSha256": 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the concrete locked-NAND target to the abstract report-level threshold theorem, complete residual minimization or ZeroSlack/PCCMin, discharge any project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.Concrete.CNFSourceParser.allInput_exact", "PNP.Concrete.CNFSourceParser.compiledMachineOutput_eq_validatedCNFBytes", "PNP.Concrete.CNFSourceParser.compiledBoundedDecide_ne_timeout", "PNP.Concrete.CNFToNANDCarrierEncoder.canonical_exact", "PNP.Concrete.CNFToNANDCarrierEncoder.canonicalWorkSteps_polynomial_bound", "PNP.Concrete.CNFToNANDWorkspace.exact_execution_output", "PNP.Concrete.CNFToNANDController.rules_length_literal", "PNP.Concrete.CNFToNANDControllerTotalTrace.canonical_path", "PNP.Concrete.CNFToNANDControllerTotalTrace.canonical_bounded_exact", "PNP.Concrete.CNFToNANDCompilerMachine.rules_length_literal", "PNP.Concrete.CNFToNANDCompilerTotalTrace.malformed_bounded_exact", "PNP.Concrete.CNFToNANDCompilerTotalTrace.decoded_bounded_exact", "PNP.Concrete.CNFToNANDCompilerTotalTrace.allInput_bounded_exact", "PNP.Concrete.CNFToNANDCompilerPolynomialBound.allInputWorkTimePolynomial_eval", "PNP.Concrete.CNFToNANDCompilerPolynomialBound.compiledRawTimePolynomial_eval", "PNP.Concrete.CNFToNANDCompilerCompiled.compiledMachineOutput_eq_compileEncodedCNFToNAND", "PNP.Concrete.CNFToNANDCompilerCompiled.compiledBoundedDecide_accept_iff", "PNP.Concrete.CNFToNANDCompilerCompiled.compiledBoundedDecide_ne_timeout", "PNP.Concrete.CNFToNANDCompilerCompiled.cnfToNANDPolynomialTimeFunction_output", "PNP.Concrete.CNFToNAND.cnfToNANDPolynomialReduction_function", "PNP.Concrete.CNFToNAND.cnfToNANDPolynomialReduction_output", "PNP.Concrete.CNFToNAND.cnfToNANDPolynomialReduction_correct", "PNP.Concrete.CNFToNAND.cnfSAT_reducesTo_encodedNANDSAT", "PNP.Concrete.CNFToNAND.cnfToNANDPolynomialReduction_hasRawRefinement", "PNP.Concrete.CNFToNAND.cnfToLockedNANDPolynomialReduction_output", 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It does not derive global absence from a finite scan, generate a route, prove candidate-list or route completeness, construct the manuscript's ZeroSlack certificate, establish polynomial checking or PCCMin runtime, put SAT in P, discharge a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.referenceMinimumImplementation_gateCount_eq_referenceMinimum", "PNP.DirectWire.referenceMinimumImplementation_equivalent", "PNP.DirectWire.referenceMinimumImplementation_isSemanticallyMinimum", "PNP.DirectWire.referenceMinimumImplementation_residualSlack_eq_zero", "PNP.DirectWire.referenceMinimumImplementation_strictEquivalentGain_of_residualSlack_pos", "PNP.DirectWire.residualSlack_pos_iff_exists_strictEquivalentGain", "PNP.DirectWire.residualSlack_eq_zero_iff_forall_not_strictEquivalentGain", "PNP.DirectWire.isSemanticallyMinimum_iff_forall_not_strictEquivalentGain", "PNP.DirectWire.StrictGainChain.end_residualSlack_eq_zero_of_no_strictEquivalentGain", "PNP.DirectWire.strictGainChainBool_end_residualSlack_eq_zero_of_no_strictEquivalentGain" ], "scope": "For every finite direct-wire implementation, positive exhaustive-reference residual slack is equivalent to the existence of some strictly smaller semantically equivalent implementation; zero slack and semantic minimality are each equivalent to global absence of such an implementation. A verified chain endpoint with separately proved global no-gain evidence therefore has zero slack and packages an exact minimum result.", "title": "Global strict-gain stopping specification", "theoremRows": [ { "name": "PNP.DirectWire.referenceMinimumImplementation_gateCount_eq_referenceMinimum", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "d94a3b2887fb9c25feef33c7474419851d6516b8647e2638ac67e526c4512480", "expectedKernelTypeSha256": "d94a3b2887fb9c25feef33c7474419851d6516b8647e2638ac67e526c4512480", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.referenceMinimumImplementation_equivalent", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "dbbb97229e4cc25be761758a73728b6801d96cacc8c1b2f8c1dd2950c29bdf64", "expectedKernelTypeSha256": "dbbb97229e4cc25be761758a73728b6801d96cacc8c1b2f8c1dd2950c29bdf64", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.referenceMinimumImplementation_isSemanticallyMinimum", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "f3c4b31840dc29d0636cb49760894393c06801381abf74353dc57d3a0755affa", "expectedKernelTypeSha256": "f3c4b31840dc29d0636cb49760894393c06801381abf74353dc57d3a0755affa", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.referenceMinimumImplementation_residualSlack_eq_zero", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "20d2abe2e74494d70b5e7516f3727e8cf19893cb4449a0c78b2d6280489feca9", "expectedKernelTypeSha256": "20d2abe2e74494d70b5e7516f3727e8cf19893cb4449a0c78b2d6280489feca9", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.referenceMinimumImplementation_strictEquivalentGain_of_residualSlack_pos", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "2eab777e2469136148219b5883ecc42b8cc630d9d71c2ede182b46bddf642198", "expectedKernelTypeSha256": "2eab777e2469136148219b5883ecc42b8cc630d9d71c2ede182b46bddf642198", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.residualSlack_pos_iff_exists_strictEquivalentGain", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "fcdc9fe1d4bd4c75f9b08a2a2ced36009344d52c11cb14506eb325f516e40612", "expectedKernelTypeSha256": "fcdc9fe1d4bd4c75f9b08a2a2ced36009344d52c11cb14506eb325f516e40612", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.residualSlack_eq_zero_iff_forall_not_strictEquivalentGain", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "9b656ab2415d453f6db8fcf47f45d82e22d51383e6f7885cdb7fb78139409094", "expectedKernelTypeSha256": "9b656ab2415d453f6db8fcf47f45d82e22d51383e6f7885cdb7fb78139409094", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.isSemanticallyMinimum_iff_forall_not_strictEquivalentGain", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "c292c91ea97e54e8fa2d1ca5989b2ffd67d386e81ecfc138d66bcbe9635c7991", "expectedKernelTypeSha256": "c292c91ea97e54e8fa2d1ca5989b2ffd67d386e81ecfc138d66bcbe9635c7991", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.StrictGainChain.end_residualSlack_eq_zero_of_no_strictEquivalentGain", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "7e70d84ebd96fc91004f391498a0ba3869902abc56060ab5517dc38d0f0b06cd", "expectedKernelTypeSha256": "7e70d84ebd96fc91004f391498a0ba3869902abc56060ab5517dc38d0f0b06cd", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.strictGainChainBool_end_residualSlack_eq_zero_of_no_strictEquivalentGain", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "0f14074de58498b3edef73c065b3520fa9827229b32643a865338c16ae05f592", "expectedKernelTypeSha256": "0f14074de58498b3edef73c065b3520fa9827229b32643a865338c16ae05f592", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": true, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-semantic-stopping-only" }, { "classification": "formalized-terminal-full-mode-semantic-bridge", "id": "residual-terminal-full-carrier-bridge", "nonClaim": "This is the direct-wire terminal full-mode specialization of the manuscript bridge. It does not formalize the quotient carrier or quotient-to-full firewall, proper or governed supports, SaturatePositive, BCEL/BN2-BN6, packet or selector completeness, route generation, the ZeroSlack certificate, PCCMin, polynomial minimum search or checking, SAT in P, discharge a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalize_implementation", "PNP.DirectWire.terminalize_gateCount", "PNP.DirectWire.TerminalFullRealization.realize_equivalent", "PNP.DirectWire.TerminalFullRealization.realize_semantics", "PNP.DirectWire.referenceMinimumTerminalFullRealization_gateCount", "PNP.DirectWire.terminalFullMinimum_eq_referenceMinimum", "PNP.DirectWire.terminalFullMinimum_spec", "PNP.DirectWire.isTerminalFullMinimum_iff_eq_terminalFullMinimum", "PNP.DirectWire.isTerminalFullMinimum_iff_eq_referenceMinimum", "PNP.DirectWire.WholeSpanResidualWitness.strictResidualDescent", "PNP.DirectWire.residualSlack_pos_iff_exists_wholeSpanResidualWitness", "PNP.DirectWire.residualSlack_eq_zero_iff_no_wholeSpanResidualWitness", "PNP.DirectWire.StrictEquivalentGain.strictResidualDescent" ], "scope": "For every finite direct-wire implementation, terminalization preserves the exact whole implementation, gate count, and semantics at every input/output coordinate. An independently stated terminal minimum is attained, universally lower-bounds every complete terminal realization, and equals the exhaustive semantic reference minimum. Positive residual slack is equivalent to a cheaper whole-span full realization, every such realization gives strict residual descent, and zero slack is equivalent to absence of one.", "title": "Terminal full-carrier residual bridge", "theoremRows": [ { "name": "PNP.DirectWire.terminalize_implementation", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "f8b947cab5adb66f9a7314b3522f9e96a083ff752df2bb7f26e09bf8159a40c5", "expectedKernelTypeSha256": "f8b947cab5adb66f9a7314b3522f9e96a083ff752df2bb7f26e09bf8159a40c5", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalize_gateCount", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "05ffc10d0fcfd63531119a3a7ff69baf871d270412c93d345e45dc1167dc1aee", "expectedKernelTypeSha256": "05ffc10d0fcfd63531119a3a7ff69baf871d270412c93d345e45dc1167dc1aee", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalFullRealization.realize_equivalent", "present": 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"sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-terminal-full-mode-semantic-bridge" }, { "classification": "formalized-terminal-mode-firewall", "id": "residual-terminal-mode-firewall", "nonClaim": "This is a terminal comparison/lifting firewall only. It supplies no proper or governed supports, arbitrary quotient construction, support or projection-defect minimum, saturation, Package E, BCEL/BN2-BN6, packet or selector completeness, global residual route, ZeroSlack certificate, PCCMin exactness or polynomial runtime, SAT-in-P result, discharged project assumption, or proof that P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalFullCarrierRealization.project_realization", "PNP.DirectWire.TerminalFullCarrierRealization.project_implementation", "PNP.DirectWire.TerminalFullCarrierRealization.project_gateCount", "PNP.DirectWire.TerminalFullCarrierRealization.project_equivalent", "PNP.DirectWire.TerminalFullCarrierRealization.project_semantics", "PNP.DirectWire.TerminalCheckedFullLift.fullRealization_realization", "PNP.DirectWire.TerminalCheckedFullLift.fullRealization_profileEqual", "PNP.DirectWire.terminalCheckedFullLift_iff_fullProfileEqual", "PNP.DirectWire.TerminalQuotientComparison.checkedFullLift_of_keepsAll", "PNP.DirectWire.TerminalFullCarrierRealization.obligationsDischarged", "PNP.DirectWire.TerminalCheckedFullLift.obligationsDischarged", "PNP.DirectWire.terminalQuotientEqualityNotConstructive" ], "scope": "For every finite direct-wire implementation, a computed finite profile observer records the ten terminal carrier roles and an explicit forgetful projection selects the quotient coordinates. Projection retains the exact implementation, gate count, and complete multi-output Boolean semantics. A quotient comparison has a checked full lift exactly when every forgotten profile coordinate agrees, lossless projections lift directly, and obligation discharge transports across a checked lift.", "title": "Terminal quotient/full mode firewall", "theoremRows": [ { "name": "PNP.DirectWire.TerminalFullCarrierRealization.project_realization", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "e18ceb7862c428c17a83e7709f12149179b9618aecdb92de9885dec5a7a6cf8a", "expectedKernelTypeSha256": "e18ceb7862c428c17a83e7709f12149179b9618aecdb92de9885dec5a7a6cf8a", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalFullCarrierRealization.project_implementation", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "4adb38449d826339de7be4ef2f076ee3a96bf5da4b0fff319ef956391012acf5", "expectedKernelTypeSha256": "4adb38449d826339de7be4ef2f076ee3a96bf5da4b0fff319ef956391012acf5", "kernelTypeFingerprintMatches": true }, { "name": 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"residual-terminal-projection-minimum", "nonClaim": "These are exhaustive finite reference minima through the supplied implementation size. This milestone proves no polynomial runtime, proper or governed support construction, arbitrary manuscript quotient carrier, SaturatePositive, Package E, BCEL/BN2-BN6, complete residual routing, ZeroSlack certificate, PCCMin exactness, SAT-in-P result, discharged project assumption, or proof that P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalFullProfileMatchBool_complete", "PNP.DirectWire.terminalQuotientProfileMatchBool_complete", "PNP.DirectWire.terminalFullProfileMinimumRealization_gateCount", "PNP.DirectWire.terminalQuotientProfileMinimumComparison_gateCount", "PNP.DirectWire.terminalFullProfileMinimum_le", "PNP.DirectWire.terminalQuotientProfileMinimum_le", "PNP.DirectWire.terminalFullProfileMinimum_spec", "PNP.DirectWire.terminalQuotientProfileMinimum_spec", "PNP.DirectWire.terminalProjectionMinimum_mono", "PNP.DirectWire.terminalQuotientMinimum_add_projectionDefect", "PNP.DirectWire.terminalProjectionDefect_eq_zero_iff_minima_eq", "PNP.DirectWire.terminalProjectionDefect_eq_zero_iff_exists_checkedFullLiftAtMinimum", "PNP.DirectWire.terminalProfileMinima_eq_of_keepsAll", "PNP.DirectWire.terminalProjectionDefect_pos_no_checkedFullLiftAtMinimum" ], "scope": "For every finite direct-wire implementation, computed finite terminal-profile observer, and explicit forgetful projection, complete enumeration through the current gate count computes an attained full-profile minimum and an attained quotient-profile minimum. Both minima universally lower-bound every matching realization, forgetting coordinates cannot increase the minimum, the full minimum decomposes as the quotient minimum plus a nonnegative projection defect, and that defect is zero exactly when an attained quotient minimum has a checked full lift.", "title": "Terminal full/quotient projection minima", "theoremRows": [ { "name": "PNP.DirectWire.terminalFullProfileMatchBool_complete", "present": true, "kind": "theorem", "axioms": [ "propext" ], "actualKernelTypeSha256": "ce2551dced7c7b0f50749844f9113fcc9d9de4daf288a07f3932874ac5a9467f", "expectedKernelTypeSha256": "ce2551dced7c7b0f50749844f9113fcc9d9de4daf288a07f3932874ac5a9467f", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalQuotientProfileMatchBool_complete", "present": true, "kind": "theorem", "axioms": [ "propext" ], "actualKernelTypeSha256": "bbbe9447463d113efe13ac57c0fbadc387eda47aeb1eef6c08c06e70f885e4c2", "expectedKernelTypeSha256": 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It does not construct or certify a proper governed support square, prove SaturatePositive, discharge Package E or BCEL/BN2-BN6, generate a complete residual route, prove ZeroSlack or PCCMin, establish polynomial runtime, put SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalProjectionDefect_int", "PNP.DirectWire.TerminalProjectionFourCorners.transferIdentity", "PNP.DirectWire.TerminalProjectionFourCorners.constantCutEquation_of_defects", "PNP.DirectWire.TerminalProjectionFourCorners.projectionExcess_pos_of_constantCut" ], "scope": "For every finite direct-wire four-corner terminal-profile family sharing one computed observer and one explicit projection, signed full and quotient minimum deltas obey the exact Section 5.2 transfer identity. 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"formalized-terminal-projection-transfer" }, { "classification": "formalized-terminal-saturation-closure", "id": "residual-terminal-saturation-closure", "nonClaim": "This closure theorem does not derive the dependency relation from an arbitrary circuit, construct proper support, prove support completion or square legitimacy, instantiate a projection-compatible square, prove SaturatePositive or BCELReady, discharge Package E or BCEL/BN2-BN6, generate a complete residual route, prove ZeroSlack or PCCMin, establish polynomial runtime, put SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.mem_allTerminalPrimitiveRecords", "PNP.DirectWire.terminalSaturate_extensive", "PNP.DirectWire.terminalSaturate_closed", "PNP.DirectWire.terminalSaturate_least", "PNP.DirectWire.terminalSaturate_monotone", "PNP.DirectWire.terminalSaturate_idempotent", "PNP.DirectWire.terminalSaturate_fixed_iff_closed" ], "scope": "For every finite terminal primitive-record universe and every explicit Boolean dependency system tagged by the manuscript's ten closure mechanisms, the generated reflexive transitive closure contains the seed, is dependency-closed, is least among closed supersets, is monotone and idempotent, and has exactly the closed supports as fixed points.", "title": "Terminal saturation closure", "theoremRows": [ { "name": "PNP.DirectWire.mem_allTerminalPrimitiveRecords", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "06dc32945403e76a76eed4fdbe31b1962c878af2cd8c136103fa1243d304a9a6", "expectedKernelTypeSha256": "06dc32945403e76a76eed4fdbe31b1962c878af2cd8c136103fa1243d304a9a6", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalSaturate_extensive", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "80a5d112df19cb4fd03210ffdd06be067829549a28e2da0746230f79994761fc", "expectedKernelTypeSha256": "80a5d112df19cb4fd03210ffdd06be067829549a28e2da0746230f79994761fc", 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This milestone does not construct proper positive support, prove support completion in the manuscript's full sense or square legitimacy, instantiate the required projection square, prove SaturatePositive, discharge Package E or BCEL/BN2-BN6, generate a complete residual route, prove ZeroSlack or PCCMin, establish polynomial runtime, put SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.mem_allTerminalSaturationRuleKinds", "PNP.DirectWire.terminalSaturationEdge_eq_true_iff", "PNP.DirectWire.terminalSaturateRecords_extensive", "PNP.DirectWire.terminalSaturateRecords_sound", "PNP.DirectWire.terminalSaturateRecords_closed", "PNP.DirectWire.mem_terminalSaturateRecords_iff", "PNP.DirectWire.mem_allTerminalSupportWires", "PNP.DirectWire.mem_terminalBoundaryPorts_iff", "PNP.DirectWire.mem_terminalInterfacePorts_iff", "PNP.DirectWire.completeTerminalPhysicalSupport_incoming_complete", "PNP.DirectWire.completeTerminalPhysicalSupport_outgoing_complete", "PNP.DirectWire.completeTerminalPhysicalSupport_compatible", "PNP.DirectWire.completeSaturatedTerminalPhysicalSupport_records", "PNP.DirectWire.completeSaturatedTerminalPhysicalSupport_compatible" ], "scope": "For every finite direct-wire candidate, explicit terminal dependency system, and finite seed list, a deterministic finite work list computes exactly the inductive saturation, then the actual program computes canonically ordered incoming boundary and outgoing interface wires. 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This milestone does not construct a proper positive support, prove full governed support completion or square legitimacy, instantiate the required projection square, prove SaturatePositive, discharge Package E or BCEL/BN2-BN6, generate a complete residual route, prove ZeroSlack or PCCMin, establish polynomial runtime, put SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.mem_terminalSelectedGateIndices_iff", "PNP.DirectWire.mem_terminalSelectedGates_iff", "PNP.DirectWire.terminalSelectedGateIndices_nodup", "PNP.DirectWire.terminalSelectedGates_nodup", "PNP.DirectWire.extractTerminalSupport_records", "PNP.DirectWire.extractTerminalSupport_boundary", "PNP.DirectWire.extractTerminalSupport_selectedGates", "PNP.DirectWire.extractTerminalSupport_interface", "PNP.DirectWire.extractTerminalSupport_gateCount", "PNP.DirectWire.terminalOpenGateEvaluation_induced_selected", "PNP.DirectWire.terminalOpenSupportSemantics_induced", "PNP.DirectWire.extractTerminalSupport_semantics", "PNP.DirectWire.extractTerminalSupport_induced", "PNP.DirectWire.extractSaturatedTerminalSupport_records", "PNP.DirectWire.extractSaturatedTerminalSupport_gateCount", "PNP.DirectWire.extractSaturatedTerminalSupport_semantics", "PNP.DirectWire.extractSaturatedTerminalSupport_induced", "PNP.DirectWire.mem_terminalSaturateRecords_iff", "PNP.DirectWire.completeTerminalPhysicalSupport_incoming_complete", "PNP.DirectWire.completeTerminalPhysicalSupport_compatible", "PNP.DirectWire.completeSaturatedTerminalPhysicalSupport_compatible" ], "scope": "For every finite direct-wire candidate and finite terminal record list, including noncontiguous selections, the actual program is structurally extracted over its exact canonical incoming boundary and ordered outgoing interface. 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"actualKernelTypeSha256": "831f8ce697624f2e413d5da6a56fde592b2157793a1dd1f4f85060154faafc58", "expectedKernelTypeSha256": "831f8ce697624f2e413d5da6a56fde592b2157793a1dd1f4f85060154faafc58", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-terminal-support-extraction" }, { "classification": "formalized-governed-proper-positive-support-search", "id": "residual-terminal-proper-positive-support-search", "nonClaim": "The terminal dependency system remains explicit input rather than a profile frontier derived from the circuit, and the search is exhaustive reference computation rather than a polynomial algorithm. This milestone does not prove global gain completeness, full manuscript support completion or square legitimacy, instantiate a projection-compatible square, prove SaturatePositive or BCELReady, discharge Package E or BCEL/BN2-BN6, generate a complete residual route, prove ZeroSlack or PCCMin, put SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.canonicalTerminalSupportSeed_mem", "PNP.DirectWire.mem_canonicalTerminalSupportSeed_iff", "PNP.DirectWire.terminalProperPositiveSupportBool_eq_true_iff", "PNP.DirectWire.findTerminalProperPositiveSupport_sound", "PNP.DirectWire.findTerminalProperPositiveSupport_exists_of_seed", "PNP.DirectWire.findTerminalProperPositiveSupport_eq_none_iff", "PNP.DirectWire.findTerminalProperPositiveSupport_unique", "PNP.DirectWire.TerminalProperPositiveSupport.saturatedRecords_closed", "PNP.DirectWire.TerminalProperPositiveSupport.physically_compatible", "PNP.DirectWire.TerminalProperPositiveSupport.gateCount_bounds", "PNP.DirectWire.TerminalProperPositiveSupport.extracted_semantics", "PNP.DirectWire.TerminalProperPositiveSupport.extracted_induced", "PNP.DirectWire.TerminalProperPositiveSupport.minimumReplacement_equivalent", "PNP.DirectWire.TerminalProperPositiveSupport.referenceMinimum_lt_gateCount", "PNP.DirectWire.TerminalProperPositiveSupport.minimumReplacement_size_lt", "PNP.DirectWire.mem_terminalSaturateRecords_iff", "PNP.DirectWire.completeSaturatedTerminalPhysicalSupport_compatible", "PNP.DirectWire.extractSaturatedTerminalSupport_gateCount", "PNP.DirectWire.extractSaturatedTerminalSupport_semantics", "PNP.DirectWire.extractSaturatedTerminalSupport_induced", "PNP.DirectWire.Candidate.referenceMinimumReplacement_equivalent", "PNP.DirectWire.Candidate.referenceMinimumReplacement_size" ], "scope": "For every finite direct-wire candidate and explicit terminal dependency system, Lean enumerates the complete canonical finite universe of primitive-record seeds, saturates and physically completes each seed, extracts its exact open support, and computes exact local gain from the exhaustive semantic reference minimum. 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"formalized-terminal-saturated-support-square-closure", "id": "residual-terminal-saturated-support-square-closure", "nonClaim": "The terminal dependency system remains explicit input rather than a profile frontier derived from the circuit. This milestone proves finite closed-corner algebra and computed physical extraction, not the manuscript's obstruction routing, frontier pushout, projection-compatible square, side-tight four-corner minima, BN2 square legitimacy, SaturatePositive, Package E, BCEL/BN2-BN6, complete residual routing, ZeroSlack, PCCMin, polynomial runtime, SAT in P, removal of a project assumption, or P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalSaturatedSupportSquare.mem_meetRecords_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.leftRecords_closed", "PNP.DirectWire.TerminalSaturatedSupportSquare.rightRecords_closed", "PNP.DirectWire.TerminalSaturatedSupportSquare.meetRecords_closed", "PNP.DirectWire.TerminalSaturatedSupportSquare.mem_joinRecords_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.joinRecords_closed", "PNP.DirectWire.TerminalSaturatedSupportSquare.records_closed", "PNP.DirectWire.TerminalSaturatedSupportSquare.meetRecords_subset_left", "PNP.DirectWire.TerminalSaturatedSupportSquare.meetRecords_subset_right", "PNP.DirectWire.TerminalSaturatedSupportSquare.leftRecords_subset_join", "PNP.DirectWire.TerminalSaturatedSupportSquare.rightRecords_subset_join", "PNP.DirectWire.TerminalSaturatedSupportSquare.meetRecords_greatest", "PNP.DirectWire.TerminalSaturatedSupportSquare.joinRecords_least", "PNP.DirectWire.terminalSaturateRecords_mem_congr", "PNP.DirectWire.TerminalSaturatedSupportSquare.records_congr", "PNP.DirectWire.TerminalSaturatedSupportSquare.physically_compatible", "PNP.DirectWire.TerminalSaturatedSupportSquare.extracted_gateCount", "PNP.DirectWire.TerminalSaturatedSupportSquare.extracted_semantics", "PNP.DirectWire.TerminalSaturatedSupportSquare.extracted_induced", "PNP.DirectWire.mem_terminalSaturateRecords_iff", "PNP.DirectWire.completeTerminalPhysicalSupport_compatible", "PNP.DirectWire.extractTerminalSupport_semantics", "PNP.DirectWire.extractTerminalSupport_induced" ], "scope": "For every finite direct-wire candidate, every explicit terminal dependency system, and every pair of finite terminal seeds, Lean computes saturated left and right corners, their canonical closed meet, and their closed saturated-union join. 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This milestone computes a governed finite completion of each saturated support-square corner, not the manuscript's obstruction routing, frontier pushout, projection-compatible square, side-tight four-corner minima, BN2 square legitimacy, SaturatePositive, Package E, BCEL/BN2-BN6, complete residual routing, ZeroSlack, PCCMin, polynomial runtime, SAT in P, removal of a project assumption, or P = NP.", "requiredTheorems": [ "PNP.DirectWire.mem_allTerminalProfileRoles", "PNP.DirectWire.mem_terminalProfileCoordinatesForRole_iff", "PNP.DirectWire.terminalProfileCoordinatesForRole_nodup", "PNP.DirectWire.completeTerminalGovernedSupport_records", "PNP.DirectWire.TerminalGovernedCompletedSupport.frontier_boundary", "PNP.DirectWire.TerminalGovernedCompletedSupport.frontier_interface", "PNP.DirectWire.TerminalGovernedCompletedSupport.mem_profileCoordinates_iff", "PNP.DirectWire.TerminalGovernedCompletedSupport.profileCoordinates_nodup", "PNP.DirectWire.TerminalGovernedCompletedSupport.mem_own_profile_role_iff", "PNP.DirectWire.TerminalGovernedCompletedSupport.profile_role_unique", "PNP.DirectWire.TerminalGovernedCompletedSupport.profileCoordinates_disjoint", "PNP.DirectWire.TerminalGovernedCompletedSupport.profile_record_covered_iff", "PNP.DirectWire.TerminalGovernedCompletedSupport.required_mem", "PNP.DirectWire.TerminalGovernedCompletedSupport.required_profile_mem", "PNP.DirectWire.completeSaturatedTerminalGovernedSupport_records", "PNP.DirectWire.completeSaturatedTerminalGovernedSupport_compatible", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_records", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_compatible", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_profile_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_required_mem", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_required_profile_mem", "PNP.DirectWire.completeTerminalPhysicalSupport_compatible", "PNP.DirectWire.terminalSaturateRecords_closed", "PNP.DirectWire.mem_terminalSaturateRecords_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.records_closed", "PNP.DirectWire.TerminalSaturatedSupportSquare.physically_compatible" ], "scope": "For every finite direct-wire candidate, explicit terminal dependency system, finite seed list, and computed saturated support-square corner, Lean computes the exact physical boundary, ordered interface, and partition of selected profile coordinates among all ten terminal profile roles. 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"2a19ccf8594f3749b91d263b915e1d2156a90ad2681ca6ed527f124de4b564f3", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-terminal-governed-support-completion" }, { "classification": "formalized-terminal-governed-frontier-pushout", "id": "residual-terminal-governed-frontier-pushout", "nonClaim": "The terminal dependency system remains explicit input rather than a profile frontier derived from the circuit. This milestone proves exact frontier gluing for computed saturated support squares, not projection compatibility, side-tight four-corner minima, BN2 square legitimacy, SaturatePositive, Package E, BCEL/BN2-BN6, complete obstruction routing, ZeroSlack, PCCMin, polynomial runtime, SAT in P, removal of a project assumption, or P = NP.", "requiredTheorems": [ "PNP.DirectWire.allTerminalSupportWires_nodup", "PNP.DirectWire.TerminalGovernedFrontier.extensionality", "PNP.DirectWire.mem_terminalBoundaryFrontierPushout_iff", "PNP.DirectWire.mem_terminalInterfaceFrontierPushout_iff", "PNP.DirectWire.mem_terminalProfileFrontierPushout_iff", "PNP.DirectWire.terminalBoundaryFrontierPushout_nodup", "PNP.DirectWire.terminalInterfaceFrontierPushout_nodup", "PNP.DirectWire.terminalProfileFrontierPushout_nodup", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_join_boundary_eq_pushout", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_join_interface_eq_pushout", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_meet_profile_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_join_profile_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_join_profile_eq_pushout", "PNP.DirectWire.TerminalSaturatedSupportSquare.side_profile_mem_join", "PNP.DirectWire.TerminalSaturatedSupportSquare.left_boundary_disposition", "PNP.DirectWire.TerminalSaturatedSupportSquare.right_boundary_disposition", "PNP.DirectWire.TerminalSaturatedSupportSquare.side_interface_disposition", "PNP.DirectWire.TerminalSaturatedSupportSquare.governed_frontier_pushout", "PNP.DirectWire.terminalGateSelected_eq_true_iff", "PNP.DirectWire.terminalWireExternal_eq_true_iff", "PNP.DirectWire.terminalBoundaryWire_eq_true_iff", "PNP.DirectWire.terminalGateHasExternalConsumer_eq_true_iff", "PNP.DirectWire.terminalInterfaceGate_eq_true_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.mem_meetRecords_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.mem_joinRecords_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.records_closed", "PNP.DirectWire.TerminalSaturatedSupportSquare.physically_compatible", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_profile_iff" ], "scope": "For every finite direct-wire candidate, explicit terminal dependency system, and computed saturated support square, Lean constructs the governed boundary, interface, and role-preserving profile pushout from the two side completions alone. 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"allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-terminal-governed-frontier-pushout" }, { "classification": "formalized-terminal-governed-projection-square", "id": "residual-terminal-governed-projection-square", "nonClaim": "The terminal dependency system remains explicit input rather than a profile frontier derived from the circuit. This milestone proves structural projection commutation for computed saturated support squares, not side-tight four-corner minima, BN2 square legitimacy, SaturatePositive, Package E, BCEL/BN2-BN6, complete obstruction routing, ZeroSlack, PCCMin, polynomial runtime, SAT in P, removal of a project assumption, or P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalGovernedFrontier.project_boundary", "PNP.DirectWire.TerminalGovernedFrontier.project_interface", "PNP.DirectWire.TerminalGovernedFrontier.mem_project_profiles_iff", "PNP.DirectWire.TerminalGovernedFrontier.project_profiles_nodup", "PNP.DirectWire.TerminalGovernedFrontier.project_idempotent", "PNP.DirectWire.mem_terminalProjectedGovernedFrontierPushout_profiles_iff", "PNP.DirectWire.TerminalGovernedFrontier.project_pushout", "PNP.DirectWire.TerminalSaturatedSupportSquare.projectedFrontier_boundary", "PNP.DirectWire.TerminalSaturatedSupportSquare.projectedFrontier_interface", "PNP.DirectWire.TerminalSaturatedSupportSquare.mem_projectedFrontier_profiles_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.projectedFrontier_profiles_nodup", "PNP.DirectWire.TerminalSaturatedSupportSquare.forgotten_not_mem_projectedFrontier", "PNP.DirectWire.TerminalSaturatedSupportSquare.projected_meet_profile_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.projected_join_profile_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.projected_join_eq_pushout", "PNP.DirectWire.TerminalSaturatedSupportSquare.governed_projection_compatible", "PNP.DirectWire.TerminalGovernedFrontier.extensionality", "PNP.DirectWire.mem_terminalProfileFrontierPushout_iff", "PNP.DirectWire.terminalProfileFrontierPushout_nodup", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_profile_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_meet_profile_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_join_profile_iff", 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This milestone proves numerical arithmetic and fail-closed exactness, not construction of one coherent four-corner basis, coherent completion, maximization over a finite tight family, BN2 square legitimacy, SaturatePositive, Package E, BCELReady or BCEL/BN2-BN6, complete obstruction routing, ZeroSlack, PCCMin, polynomial runtime, SAT in P, removal of a project assumption, or P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalFourCornerSizes.componentwiseLE_refl", "PNP.DirectWire.TerminalFourCornerSizes.numericallySideTight_iff_eq", "PNP.DirectWire.TerminalFourCornerSizes.sideTightBool_eq_true_iff", "PNP.DirectWire.TerminalFourCornerSizes.tightValue?_eq_some_iff", "PNP.DirectWire.TerminalFourCornerSizes.tightValue?_sound", "PNP.DirectWire.TerminalFourCornerSizes.tightValue?_complete", "PNP.DirectWire.TerminalFourCornerSizes.incidenceValue_eq_minimum_add_slacks", "PNP.DirectWire.TerminalProjectionFourCorners.fullMinimumSizes_incidenceValue", "PNP.DirectWire.TerminalFullFourCornerBasis.minimum_componentwiseLE_sizes", "PNP.DirectWire.TerminalFullFourCornerBasis.incidenceValue_eq_fullDelta_add_slacks", "PNP.DirectWire.TerminalProjectionFourCorners.canonicalFullBasis_sizes", "PNP.DirectWire.TerminalProjectionFourCorners.canonicalFullBasis_numericallySideTight", "PNP.DirectWire.TerminalProjectionFourCorners.canonicalFullBasis_tightValue?", "PNP.DirectWire.TerminalProjectionFourCorners.quotientMinimumSizes_incidenceValue", "PNP.DirectWire.TerminalQuotientFourCornerBasis.minimum_componentwiseLE_sizes", "PNP.DirectWire.TerminalQuotientFourCornerBasis.incidenceValue_eq_quotientDelta_add_slacks", "PNP.DirectWire.TerminalProjectionFourCorners.canonicalQuotientBasis_sizes", "PNP.DirectWire.TerminalProjectionFourCorners.canonicalQuotientBasis_numericallySideTight", "PNP.DirectWire.TerminalProjectionFourCorners.canonicalQuotientBasis_tightValue?", "PNP.DirectWire.TerminalProjectionFourCorners.canonical_numericallySideTight_values", "PNP.DirectWire.terminalFullProfileMinimumRealization_gateCount", "PNP.DirectWire.terminalQuotientProfileMinimumComparison_gateCount", "PNP.DirectWire.terminalFullProfileMinimum_le", "PNP.DirectWire.terminalQuotientProfileMinimum_le" ], "scope": "For every finite terminal projection four-corner family and every independently attained typed full or quotient basis, Lean proves componentwise minimum bounds and the exact signed four-slack identity. 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It does not transport four optimum realizers, prove the full four-corner optimum carrier-compatibility obligation, construct a coherent four-corner optimum, prove side-tight completion or BN2 square legitimacy, establish SaturatePositive, Package E, BCELReady or BCEL/BN2-BN6, complete obstruction routing, prove ZeroSlack, PCCMin, polynomial runtime, SAT in P, removal of a project assumption, or P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalFourCornerCarrier.boundaryDisposition?_eq_some_iff", "PNP.DirectWire.TerminalFourCornerCarrier.interfaceDisposition?_eq_some_iff", "PNP.DirectWire.TerminalFourCornerCarrier.boundary_nodup", "PNP.DirectWire.TerminalFourCornerCarrier.interface_nodup", "PNP.DirectWire.TerminalFourCornerCarrier.profile_nodup", "PNP.DirectWire.TerminalFourCornerCarrier.extracted_boundary", "PNP.DirectWire.TerminalFourCornerCarrier.extracted_interface", "PNP.DirectWire.TerminalFourCornerCarrier.corner_compatible", "PNP.DirectWire.TerminalFourCornerCarrier.meet_profile_transport", "PNP.DirectWire.TerminalFourCornerCarrier.side_profile_transport", "PNP.DirectWire.TerminalFourCornerCarrier.join_profile_transport", "PNP.DirectWire.TerminalFourCornerCarrier.boundary_retained", "PNP.DirectWire.TerminalFourCornerCarrier.boundary_internalized", "PNP.DirectWire.TerminalFourCornerCarrier.interface_retained", "PNP.DirectWire.TerminalFourCornerCarrier.interface_internalized", "PNP.DirectWire.TerminalFourCornerCarrier.projection_compatible", "PNP.DirectWire.TerminalFourCornerCarrier.complete_transport", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_compatible", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_meet_profile_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.side_profile_mem_join", "PNP.DirectWire.TerminalSaturatedSupportSquare.governedCompleted_join_profile_iff", "PNP.DirectWire.TerminalSaturatedSupportSquare.left_boundary_disposition", 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It does not prove coherent transport along the square legs, construct a coherent four-corner optimum, prove sideTightCompletionExists or BN2 square legitimacy, derive the terminal dependency system, establish SaturatePositive, Package E, BCELReady or BCEL/BN2-BN6, complete obstruction routing, prove ZeroSlack, PCCMin, polynomial runtime, SAT in P, removal of a project assumption, or P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalSupportWireAt_ambientIndex", "PNP.DirectWire.TerminalSupportWire.ambientIndex_terminalSupportWireAt", "PNP.DirectWire.TerminalSupportWire.ambientIndex_injective", "PNP.DirectWire.TerminalFourCornerCarrier.boundaryIndex?_eq_some_iff", "PNP.DirectWire.TerminalFourCornerCarrier.interfaceIndex?_eq_some_iff", "PNP.DirectWire.TerminalFourCornerCarrier.boundaryIndex?_ambient_get", "PNP.DirectWire.TerminalFourCornerCarrier.interfaceIndex?_get", "PNP.DirectWire.TerminalFourCornerCarrier.boundaryAdapter_semantics_get", "PNP.DirectWire.TerminalFourCornerCarrier.ambientizeCandidate_semantics_present", "PNP.DirectWire.TerminalFourCornerCarrier.ambientizeCandidate_semantics_absent", "PNP.DirectWire.TerminalFourCornerCarrier.localizeCandidate_semantics", "PNP.DirectWire.TerminalFourCornerCarrier.localize_ambientize_semantics", "PNP.DirectWire.TerminalFourCornerCarrier.ambientizeCandidate_gateCount", "PNP.DirectWire.TerminalFourCornerCarrier.localizeCandidate_gateCount", "PNP.DirectWire.TerminalFourCornerCarrier.ambientizeCandidate_equivalent", "PNP.DirectWire.TerminalFourCornerCarrier.localizeCandidate_equivalent", "PNP.DirectWire.TerminalFourCornerCarrier.localize_ambientize_equivalent", "PNP.DirectWire.TerminalFourCornerCarrier.localizeImplementation_gateCount", "PNP.DirectWire.TerminalFourCornerCarrier.ambient_referenceMinimum_eq_corner", "PNP.DirectWire.TerminalFourCornerCarrier.optimizationCorners_at", "PNP.DirectWire.TerminalFourCornerCarrier.optimizationCorners_role", "PNP.DirectWire.TerminalFourCornerCarrier.optimizationCorners_projection", "PNP.DirectWire.TerminalFourCornerCarrier.localizeRealization_gateCount", "PNP.DirectWire.TerminalFourCornerCarrier.fourCornerOptimaCarrierCompatible", "PNP.DirectWire.TerminalFourCornerCarrier.complete_transport", "PNP.DirectWire.TerminalProjectionFourCorners.canonicalFullBasis_sizes", "PNP.DirectWire.TerminalProjectionFourCorners.canonicalQuotientBasis_sizes", "PNP.DirectWire.terminalFullProfileMinimumRealization_gateCount", "PNP.DirectWire.terminalQuotientProfileMinimumComparison_gateCount", "PNP.DirectWire.referenceMinimum_le_of_equivalent" ], "scope": "For every finite computed saturated terminal support square and every explicit observer, Lean embeds all four exact corner candidates into one common ambient carrier, proves reversible semantic and gate-count preservation, proves exact ambient and corner reference minima agree, and localizes canonical full and quotient optima from one shared observer and 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It does not prove that every square is coherent, construct the later no-outcome route, prove sideTightCompletionExists or BN2 square legitimacy, establish SaturatePositive, Package E, BCELReady or BCEL/BN2-BN6, complete obstruction routing, prove ZeroSlack, PCCMin, polynomial runtime, SAT in P, removal of a project assumption, or P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalOptimumLegTransport.recordsSubset", "PNP.DirectWire.TerminalOptimumLegTransport.profileTransport", "PNP.DirectWire.TerminalOptimumLegTransport.ambientCoordinate_exact", "PNP.DirectWire.TerminalOptimumLegTransport.retainedOutput?_eq_some_iff", "PNP.DirectWire.TerminalOptimumLegTransport.retained_or_internalized", "PNP.DirectWire.TerminalFourCornerCarrier.optimumTransportTheta", "PNP.DirectWire.TerminalFourCornerCarrier.firstOptimumCoherenceFailure?_sound", "PNP.DirectWire.TerminalFourCornerCarrier.firstOptimumModeMismatch?_sound", "PNP.DirectWire.TerminalFourCornerCarrier.noFailure_iff_coherentOptimumTuple", "PNP.DirectWire.TerminalFourCornerCarrier.classifyOptimumCoherence_exhaustive", "PNP.DirectWire.TerminalFourCornerCarrier.fourCornerOptimumCoherenceDichotomy", "PNP.DirectWire.TerminalFourCornerCarrier.fourCornerOptimaCarrierCompatible", "PNP.DirectWire.TerminalFourCornerCarrier.complete_transport", "PNP.DirectWire.TerminalFourCornerCarrier.meet_profile_transport", "PNP.DirectWire.TerminalFourCornerCarrier.side_profile_transport", "PNP.DirectWire.TerminalFourCornerCarrier.interfaceIndex?_eq_some_iff", "PNP.DirectWire.TerminalProjectionFourCorners.canonicalFullBasis_numericallySideTight", "PNP.DirectWire.TerminalProjectionFourCorners.canonicalQuotientBasis_numericallySideTight", "PNP.DirectWire.TerminalProjectionFourCorners.canonical_numericallySideTight_values" ], "scope": "For every finite computed terminal support square, every explicit observer, every terminal projection, and either full or quotient 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milestone closes the remaining local BN2 tight-basis maximum under computed local route silence. It does not prove universal route silence, connect a local obstruction to the complete global no-outcome route system, prove BN2 square legitimacy, derive the terminal dependency system, establish SaturatePositive, Package E, BCELReady or BCEL/BN2-BN6, complete obstruction routing, prove ZeroSlack, PCCMin, polynomial runtime, SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalOptimumCoherenceMode.minimumAt_le_current", "PNP.DirectWire.TerminalProjectionFourCorners.mem_minimumImplementationsAt_sound", "PNP.DirectWire.TerminalProjectionFourCorners.mem_minimumImplementationsAt_complete", "PNP.DirectWire.TerminalProjectionFourCorners.mem_minimumImplementationsAt_iff", "PNP.DirectWire.TerminalProjectionFourCorners.mem_minimumImplementationBases_iff", "PNP.DirectWire.TerminalFourCornerCarrier.tightBasisBool_eq_true_iff", "PNP.DirectWire.TerminalFourCornerCarrier.mem_tightBasisFamily_sound", "PNP.DirectWire.TerminalFourCornerCarrier.mem_tightBasisFamily_complete", "PNP.DirectWire.TerminalFourCornerCarrier.mem_tightBasisFamily_iff", "PNP.DirectWire.TerminalFourCornerCarrier.canonicalImplementationBasis_at", "PNP.DirectWire.TerminalFourCornerCarrier.canonicalImplementationBasis_sizes", "PNP.DirectWire.TerminalFourCornerCarrier.canonicalImplementationBasis_isTightCoherent", "PNP.DirectWire.TerminalFourCornerCarrier.canonicalImplementationBasis_mem_tightFamily", "PNP.DirectWire.TerminalFourCornerCarrier.tightBasis_incidenceValue_eq_delta", "PNP.DirectWire.TerminalFourCornerCarrier.mem_tightBasisValues_eq_delta", "PNP.DirectWire.TerminalFourCornerCarrier.tightBasisMaximum?_eq_delta", "PNP.DirectWire.TerminalFourCornerCarrier.tightBasisMaximum?_full", "PNP.DirectWire.TerminalFourCornerCarrier.tightBasisMaximum?_quotient", "PNP.DirectWire.TerminalFourCornerCarrier.firstBasisCoherenceFailure?_sound", "PNP.DirectWire.TerminalFourCornerCarrier.firstOptimumCoherenceFailure?_eq_basis", "PNP.DirectWire.mem_allBoundedCandidates", "PNP.DirectWire.terminalFullProfileMatchBool_complete", "PNP.DirectWire.terminalQuotientProfileMatchBool_complete", "PNP.DirectWire.terminalFullProfileMinimum_le", "PNP.DirectWire.terminalQuotientProfileMinimum_le", "PNP.DirectWire.TerminalFourCornerSizes.numericallySideTight_iff_eq", "PNP.DirectWire.TerminalFourCornerCarrier.firstOptimumCoherenceFailure?_sound", "PNP.DirectWire.TerminalFourCornerCarrier.sideTightCompletionExists" ], "scope": "For every finite computed terminal support square, every explicit observer, and either full or quotient mode, Lean enumerates the complete finite tight-basis family, retains every exact profile-constrained minimum implementation at each corner, filters the full Cartesian product with the arbitrary-family coherence query, and proves under exact local route silence that the signed maximum equals the selected delta.", 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true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-four-corner-complete-tight-basis-maximum" }, { "classification": "formalized-residual-terminal-computed-bn2-square-legitimacy", "id": "residual-terminal-computed-bn2-square-legitimacy", "nonClaim": "This milestone packages computed structural legitimacy and the exact local no-route conclusion. It does not derive the terminal dependency system from an arbitrary circuit, prove universal route silence, connect a local failure to the complete global no-outcome route system, identify a BCEL anchor square, establish SaturatePositive, Package E, BCELReady or later BCEL/BN2-BN6 conclusions, prove ZeroSlack, PCCMin, polynomial runtime, SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalComputedBN2SquareLegitimate.cornerCompatible", "PNP.DirectWire.TerminalComputedBN2SquareLegitimate.meetProfile", "PNP.DirectWire.TerminalComputedBN2SquareLegitimate.joinProfile", "PNP.DirectWire.TerminalComputedBN2SquareLegitimate.projectionCompatible", "PNP.DirectWire.TerminalFourCornerCarrier.computedBN2SquareLegitimate", "PNP.DirectWire.TerminalComputedBN2SquareQuantities.sharedRole", "PNP.DirectWire.TerminalComputedBN2SquareQuantities.sharedProjection", "PNP.DirectWire.TerminalComputedBN2SquareQuantities.referenceMinimumPreserved", "PNP.DirectWire.TerminalComputedBN2SquareQuantities.transferIdentity", "PNP.DirectWire.TerminalFourCornerCarrier.computedBN2SquareQuantities", "PNP.DirectWire.TerminalFourCornerCarrier.computedBN2LocalConclusion", "PNP.DirectWire.TerminalFourCornerCarrier.computedBN2LocalConclusionOrFirstRoute", "PNP.DirectWire.TerminalFourCornerCarrier.complete_transport", "PNP.DirectWire.TerminalSaturatedSupportSquare.governed_frontier_pushout", "PNP.DirectWire.TerminalFourCornerCarrier.fourCornerOptimaCarrierCompatible", "PNP.DirectWire.TerminalFourCornerCarrier.sideTightCompletionExistsEachMode", "PNP.DirectWire.TerminalFourCornerCarrier.tightBasisMaximum?_full", "PNP.DirectWire.TerminalFourCornerCarrier.tightBasisMaximum?_quotient", "PNP.DirectWire.TerminalFourCornerOptimumRoutedFailure.sound", "PNP.DirectWire.TerminalProjectionFourCorners.transferIdentity" ], "scope": "For every finite computed terminal support square built from two finite seeds under one explicit terminal dependency system, direct-wire candidate, observer, and forgetful projection, Lean constructs the exact compatible governed frontier and projection square, keeps full and quotient minimum quantities on the same carrier, and returns either the complete local conclusion under exact local route silence or the deterministic full-then-quotient proof-bearing first coherence route.", "title": "Computed terminal BN2 square legitimacy", "theoremRows": [ { "name": "PNP.DirectWire.TerminalComputedBN2SquareLegitimate.cornerCompatible", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "353ae46eab6aa34d13a35841d9645d0872fee705b8bad037d3e767e56a72b071", "expectedKernelTypeSha256": "353ae46eab6aa34d13a35841d9645d0872fee705b8bad037d3e767e56a72b071", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalComputedBN2SquareLegitimate.meetProfile", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": 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It does not derive either premise, identify manuscript activation or charge equivalence classes absent from the terminal model, connect a local failure to the complete global no-outcome route system, establish SaturatePositive, Package E, BCELReady or later BCEL/BN2-BN6 conclusions, prove ZeroSlack, PCCMin, polynomial runtime, SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalBCELAnchorProblem.mem_anchorRecords_iff", "PNP.DirectWire.TerminalBCELAnchorProblem.anchorRecords_nodup", "PNP.DirectWire.TerminalBCELAnchorProblem.anchorRecords_mem_allAnchorSubfamilies", "PNP.DirectWire.findTerminalPositiveAnchorNucleus_sound", "PNP.DirectWire.findTerminalPositiveAnchorNucleus_eq_none_iff", "PNP.DirectWire.findTerminalPositiveAnchorNucleus_exists_of_whole_positive", "PNP.DirectWire.findTerminalPositiveAnchorNucleus_unique", "PNP.DirectWire.TerminalBCELAnchorAlgebraCheck.disagrees_eq_true_iff", "PNP.DirectWire.terminalBCELAnchorAlgebraCheck_mem", "PNP.DirectWire.mem_allTerminalBCELAnchorAlgebraChecks_governed", "PNP.DirectWire.firstTerminalBCELAnchorAlgebraMismatch?_sound", "PNP.DirectWire.firstTerminalBCELAnchorAlgebraMismatch?_eq_none_iff", "PNP.DirectWire.terminalBCELProperCutSeedBool_eq_true_iff", "PNP.DirectWire.mem_allTerminalBCELProperCutSeeds_iff", "PNP.DirectWire.TerminalBCELCutDefectCheck.disagrees_eq_true_iff", "PNP.DirectWire.terminalBCELCutDefectCheck_mem", "PNP.DirectWire.mem_allTerminalBCELCutDefectChecks_proper", "PNP.DirectWire.firstTerminalBCELCutDefectMismatch?_sound", "PNP.DirectWire.firstTerminalBCELCutDefectMismatch?_eq_none_all", "PNP.DirectWire.firstTerminalBCELCutRoute?_sound", "PNP.DirectWire.firstTerminalBCELCutRoute?_eq_none_noRoutes", "PNP.DirectWire.computedBCELCutConclusionOfNoFailures", "PNP.DirectWire.TerminalComputedBCELAnchorNucleus.strictSubfamily_defect_zero", "PNP.DirectWire.TerminalComputedBCELAnchorNucleus.anchorSizeAtLeastTwo", "PNP.DirectWire.TerminalComputedBCELAnchorNucleus.properCutConstantEquation", "PNP.DirectWire.TerminalComputedBCELAnchorNucleus.properCutLocalConclusion", "PNP.DirectWire.classifyTerminalBCELAnchorNucleus_exhaustive", "PNP.DirectWire.allTerminalPrimitiveRecords_nodup", "PNP.DirectWire.filter_mem_terminalListSubsets", "PNP.DirectWire.mem_allTerminalPrimitiveRecords", "PNP.DirectWire.TerminalFourCornerCarrier.firstOptimumRoute?_sound", "PNP.DirectWire.TerminalFourCornerOptimumRoutedFailure.sound", "PNP.DirectWire.TerminalFourCornerCarrier.computedBN2SquareLegitimate", "PNP.DirectWire.TerminalFourCornerCarrier.computedBN2LocalConclusion", "PNP.DirectWire.TerminalProjectionFourCorners.constantCutEquation_of_defects", "PNP.DirectWire.TerminalProjectionFourCorners.projectionExcess_pos_of_constantCut" ], "scope": "For every finite direct-wire candidate, explicit terminal dependency system, computed governed proper-positive support, forgetful projection, executable ambient observer, and positive whole-support projection defect, Lean computes the canonical minimum-cardinality positive anchor nucleus and returns either an insufficient nucleus, the exact first anchor-algebra mismatch, the exact first proper-cut defect mismatch, the proof-bearing first full-before-quotient local route, or exact constant-cut and local BN2 conclusions for every proper cut.", "title": "Computed terminal BCEL anchor nucleus and cut-square dichotomy", "theoremRows": [ { "name": "PNP.DirectWire.TerminalBCELAnchorProblem.mem_anchorRecords_iff", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "e5e4b4afb6377b6691fb406fdcc8221ff33b65f4a0df4fa5d4ad1b737c3b7cea", "expectedKernelTypeSha256": "e5e4b4afb6377b6691fb406fdcc8221ff33b65f4a0df4fa5d4ad1b737c3b7cea", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalBCELAnchorProblem.anchorRecords_nodup", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], 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It assumes an explicit terminal dependency system and an already computed governed proper-positive support. 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The executable observer and forgetful projection remain explicit model inputs, and a nontransparent event is recorded rather than routed. It does not discharge interfaceExposureRoutesToE or originKernelObligationClosureRouted; establish full SaturatePositive, Package E, BCELReady or later BCEL/BN2-BN6 conclusions; prove ZeroSlack, PCCMin, polynomial runtime, SAT in P; remove a project assumption; or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalSaturateTrace_eventsLinked", "PNP.DirectWire.terminalSaturateTrace_records", "PNP.DirectWire.terminalCandidateSaturationSystem_profileSystem", "PNP.DirectWire.TerminalTransparentSaturationStep.uniqueMaterializerOwner", "PNP.DirectWire.TerminalTransparentSaturationStep.supportCostBalanced", "PNP.DirectWire.TerminalTransparentSaturationStep.fullCostBalanced", "PNP.DirectWire.TerminalTransparentSaturationStep.quotientCostBounded", "PNP.DirectWire.TerminalTransparentSaturationStep.fullSlack_preserved", "PNP.DirectWire.TerminalTransparentSaturationStep.projectionDefect_mono", "PNP.DirectWire.TerminalTransparentSaturationStep.fullPositive_preserved", 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"formalized-residual-terminal-interface-exposure-routing", "id": "residual-terminal-interface-exposure-routing", "nonClaim": "This closes only the finite local form of interfaceExposureRoutesToE. The proof-bearing local E-route is an exposure-obligation coordinate, not a full Package E VerifyDW acceptance, a verified global gain, or global route completeness. The executable observer and forgetful projection remain explicit model inputs. It does not discharge originKernelObligationClosureRouted; establish full SaturatePositive, Package E, BCELReady or later BCEL/BN2-BN6 conclusions; prove ZeroSlack, PCCMin, polynomial runtime, SAT in P; remove a project assumption; or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalInterfaceExposureCoordinate?_sound", "PNP.DirectWire.terminalCandidateInterfaceExposureCoordinate?_shape", "PNP.DirectWire.terminalCandidateInterfaceExposureCoordinate?_edge", "PNP.DirectWire.terminalInterfaceOutgoingCoordinate_eventCost_zero", "PNP.DirectWire.TerminalInterfaceExposureERoute.sound", "PNP.DirectWire.TerminalInterfaceExposureZeroCostRetract.eventCost_zero", "PNP.DirectWire.TerminalInterfaceExposureZeroCostRetract.fullSlack_preserved", "PNP.DirectWire.terminalInterfaceExposure_transparent_or_eRoute", "PNP.DirectWire.TerminalFirstInterfaceExposureRoute.sound", "PNP.DirectWire.classifyTerminalSaturationInterfaceRouting_exhaustive" ], "scope": "For every finite direct-wire candidate, executable ambient observer, forgetful projection, and finite terminal seed, Lean recognizes only an exact candidate-derived interface-consumer edge. Each recognized event is transparently cost-balanced or produces a proof-bearing local E-route; the production trace result records the exact first nontransparent event and complete transparent prefix, while non-interface first failures remain fail-closed.", "title": "Finite terminal interface-exposure routing", "theoremRows": [ { "name": "PNP.DirectWire.terminalInterfaceExposureCoordinate?_sound", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "fd588e1aab5d670afcc23cadd40cc3dcf03214f8f4d222b955116e80a5c8f125", "expectedKernelTypeSha256": "fd588e1aab5d670afcc23cadd40cc3dcf03214f8f4d222b955116e80a5c8f125", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalCandidateInterfaceExposureCoordinate?_shape", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "0f6ea712760c01fdb762902ad020dcf4a9f3c45dba0e2a91790b6bfd327cace7", "expectedKernelTypeSha256": "0f6ea712760c01fdb762902ad020dcf4a9f3c45dba0e2a91790b6bfd327cace7", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalCandidateInterfaceExposureCoordinate?_edge", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "d168a3362f7b52afd52451d87eff1eedb6302016786c4c763e0965ebcea08880", "expectedKernelTypeSha256": "d168a3362f7b52afd52451d87eff1eedb6302016786c4c763e0965ebcea08880", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalInterfaceOutgoingCoordinate_eventCost_zero", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "74c702f16d6714a455bad571c647ae4903581a174b7c31b1e37b426b10a78160", "expectedKernelTypeSha256": "74c702f16d6714a455bad571c647ae4903581a174b7c31b1e37b426b10a78160", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalInterfaceExposureERoute.sound", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "57ae2f434a17a9c5c9a91cd17844f55fa7d72086827598e9bcf6ce7d2441803e", "expectedKernelTypeSha256": "57ae2f434a17a9c5c9a91cd17844f55fa7d72086827598e9bcf6ce7d2441803e", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalInterfaceExposureZeroCostRetract.eventCost_zero", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "c098e9bfd76067bbc7a2d4223c0107dae920d3378e47a002f58e71f24ed0404b", "expectedKernelTypeSha256": "c098e9bfd76067bbc7a2d4223c0107dae920d3378e47a002f58e71f24ed0404b", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalInterfaceExposureZeroCostRetract.fullSlack_preserved", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "e47bd2a49bb380282d5c6b7b950390dbfea623a55c0c6a5b93a9875215cff293", "expectedKernelTypeSha256": "e47bd2a49bb380282d5c6b7b950390dbfea623a55c0c6a5b93a9875215cff293", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalInterfaceExposure_transparent_or_eRoute", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "71207620ad161eb833ad413ed00fccd4d14af5c5720f812c7ecff00eca892336", "expectedKernelTypeSha256": "71207620ad161eb833ad413ed00fccd4d14af5c5720f812c7ecff00eca892336", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalFirstInterfaceExposureRoute.sound", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "cef4ceb9d4650c0847c740c6126132dc9fa78100d83502b1c887cd6d3d31b8d0", "expectedKernelTypeSha256": "cef4ceb9d4650c0847c740c6126132dc9fa78100d83502b1c887cd6d3d31b8d0", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.classifyTerminalSaturationInterfaceRouting_exhaustive", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "3f2cf5f03389537c02418758fd531dfa427be40d6773ddfa5fd6104c3a798c38", "expectedKernelTypeSha256": "3f2cf5f03389537c02418758fd531dfa427be40d6773ddfa5fd6104c3a798c38", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-interface-exposure-routing" }, { "classification": "formalized-residual-terminal-finite-saturate-positive-composition", "id": "residual-terminal-finite-saturate-positive-composition", "nonClaim": "This closes the finite local form of originKernelObligationClosureRouted and composes the five reconstructed terminal sub-obligations only for an explicit proof-bearing problem. A local route is not a complete global outcome, Package E VerifyDW acceptance, verified gain, or global route-completeness result. The positive initial full-slack premise remains explicit. It does not establish manuscript-wide SaturatePositive, BCELReady, RankWF, ZeroSlack, PCCMin, polynomial runtime, SAT in P; remove a project assumption; or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalOriginKernelObligationCoordinate?_sound", "PNP.DirectWire.terminalCandidateOriginKernelObligationCoordinate?_shape", "PNP.DirectWire.terminalCandidateOriginKernelObligationCoordinate?_edge", "PNP.DirectWire.TerminalOriginKernelObligationClosureRoute.sound", "PNP.DirectWire.terminalOriginKernelObligation_safe_or_route", "PNP.DirectWire.TerminalSaturationClosureSafeStep.transparent", "PNP.DirectWire.classifyTerminalSaturationClosureRouting_exhaustive", "PNP.DirectWire.TerminalFiniteSaturatePositiveOutcome.sound", "PNP.DirectWire.classifyTerminalFiniteSaturatePositive_exhaustive" ], "scope": "For every finite direct-wire candidate, executable ambient observer, forgetful projection, and proof-bearing candidate BCEL anchor problem whose normalized seed has positive full slack, Lean recognizes exact candidate-derived origin, kernel, and obligation closures in both gate/profile orientations; checks cost transparency, obligation discharge, and forgotten-profile stability; preserves positive full slack across an all-safe trace into the checked-lift or BCEL firewall; or returns the exact first interface, closure, or other fail-closed nontransparent route with its complete safe prefix.", "title": "Finite terminal SaturatePositive composition", "theoremRows": [ { "name": "PNP.DirectWire.terminalOriginKernelObligationCoordinate?_sound", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "fe2fafdfcb67fecf4c1b2b544645b0f6af7ec9a5b5761ee8d6a23ba2ae140a60", "expectedKernelTypeSha256": "fe2fafdfcb67fecf4c1b2b544645b0f6af7ec9a5b5761ee8d6a23ba2ae140a60", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalCandidateOriginKernelObligationCoordinate?_shape", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "5473c0dae862ab3face4f83003143ffb94887a88cdac6edba739f2470fe9b0eb", "expectedKernelTypeSha256": "5473c0dae862ab3face4f83003143ffb94887a88cdac6edba739f2470fe9b0eb", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalCandidateOriginKernelObligationCoordinate?_edge", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "12fcaaad403c4bf2f0ed3e218fac247139db20db8b103bd35c751664ebfad8be", "expectedKernelTypeSha256": "12fcaaad403c4bf2f0ed3e218fac247139db20db8b103bd35c751664ebfad8be", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalOriginKernelObligationClosureRoute.sound", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "a6ccf72ac02146fb0d00f77f1bd68f5d10e95bd3a9575881976593dbca19ab17", "expectedKernelTypeSha256": "a6ccf72ac02146fb0d00f77f1bd68f5d10e95bd3a9575881976593dbca19ab17", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalOriginKernelObligation_safe_or_route", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "996fad4b79e3d3e5b0cce63e094b597d9ee731556df3c73f38bbb9761343330f", "expectedKernelTypeSha256": "996fad4b79e3d3e5b0cce63e094b597d9ee731556df3c73f38bbb9761343330f", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalSaturationClosureSafeStep.transparent", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "cb47374327a74960b19f7f9bb46af7bd0ad6b766608187d755f10edddd74d4bd", "expectedKernelTypeSha256": "cb47374327a74960b19f7f9bb46af7bd0ad6b766608187d755f10edddd74d4bd", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.classifyTerminalSaturationClosureRouting_exhaustive", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "bff24fddcdc569451d7963003194317f0d018bdd4906314ff791d8effd8fac92", "expectedKernelTypeSha256": "bff24fddcdc569451d7963003194317f0d018bdd4906314ff791d8effd8fac92", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalFiniteSaturatePositiveOutcome.sound", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "5acc708881d1450e3e86abe5c1e0152ee6000d58ed0b5999c249d6efafcab69b", "expectedKernelTypeSha256": "5acc708881d1450e3e86abe5c1e0152ee6000d58ed0b5999c249d6efafcab69b", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.classifyTerminalFiniteSaturatePositive_exhaustive", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "2f114bbe06ba71cfb038dcc5ff686516e7af0461d9a85e6dbda0e1104a814655", "expectedKernelTypeSha256": "2f114bbe06ba71cfb038dcc5ff686516e7af0461d9a85e6dbda0e1104a814655", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-finite-saturate-positive-composition" }, { "classification": "formalized-residual-terminal-rank-wf", "id": "residual-terminal-rank-wf", "nonClaim": "This establishes the fixed residual rank domain and RankWF only. It does not map the current finite terminal routes into the manuscript's complete global outcome system, prove that any existing route strictly decreases the rank, establish route completeness or Package E, remove the explicit positive premise from the finite composition, establish full manuscript-wide SaturatePositive or BCELReady, prove ZeroSlack, PCCMin, polynomial runtime, SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalResidualRank.coordinates_mk", "PNP.DirectWire.TerminalResidualRank.coordinates_length", "PNP.DirectWire.terminalResidualRankLTBool_eq_true_iff", "PNP.DirectWire.terminalResidualRankLTBool_eq_false_iff", "PNP.DirectWire.terminalResidualRankLexLT_wellFounded", "PNP.DirectWire.terminalResidualRank_accessible", "PNP.DirectWire.terminalResidualRank_induction", "PNP.DirectWire.terminalResidualRank_witnessType_lt", "PNP.DirectWire.terminalResidualRank_spanType_lt", "PNP.DirectWire.terminalResidualRank_mode_lt", "PNP.DirectWire.terminalResidualRank_frontierDefect_lt", "PNP.DirectWire.terminalResidualRank_projectionDefect_lt", "PNP.DirectWire.terminalResidualRank_saturationDefect_lt", "PNP.DirectWire.terminalResidualRank_anchorCount_lt", "PNP.DirectWire.terminalResidualRank_chargeSize_lt", "PNP.DirectWire.terminalResidualRank_profileSize_lt", "PNP.DirectWire.terminalResidualRank_canonicalCode_lt", "PNP.DirectWire.TerminalResidualRankDescent.sound" ], "scope": "For the fixed manuscript residual rank of exactly ten natural coordinates in the stated witness-type, span-type, mode, frontier-defect, projection-defect, saturation-defect, anchor-count, charge-size, profile-size, canonical-code priority order, Lean provides the exact lexicographic proposition, an equivalent executable comparison, all ten priority witnesses, proof-bearing descent, accessibility, induction, and kernel-checked well-foundedness.", "title": "Residual terminal RankWF", "theoremRows": [ { "name": 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"axioms": [], "actualKernelTypeSha256": "bba66c1a0a7249c35dd7ecc9cf53a7303ce0187058cb0fd25342e6d7154e11ba", "expectedKernelTypeSha256": "bba66c1a0a7249c35dd7ecc9cf53a7303ce0187058cb0fd25342e6d7154e11ba", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalResidualRank_profileSize_lt", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "42228d10e71af76dd08669f00511a1b43bebb0085ccff8b5d14b848e5903a07f", "expectedKernelTypeSha256": "42228d10e71af76dd08669f00511a1b43bebb0085ccff8b5d14b848e5903a07f", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalResidualRank_canonicalCode_lt", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "b3783268f850ea30f5e444e8a78a09ed278d1224990bbe725a89681dbd59d10e", "expectedKernelTypeSha256": "b3783268f850ea30f5e444e8a78a09ed278d1224990bbe725a89681dbd59d10e", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalResidualRankDescent.sound", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "4c4bf650e6b68d7ec6d654c4fe37ad523959cb254bb5cb871cda50c79515ada5", "expectedKernelTypeSha256": "4c4bf650e6b68d7ec6d654c4fe37ad523959cb254bb5cb871cda50c79515ada5", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-rank-wf" }, { "classification": "formalized-residual-terminal-bn3-request-envelope", "id": "residual-terminal-bn3-request-envelope", "nonClaim": "This establishes one exact candidate-derived finite BN3 envelope only after the existing computed BCEL anchor-nucleus classifier succeeds. Proper cuts are enumerated through all subsets, so the construction is exponential reference computation rather than a polynomial algorithm. It does not derive the terminal dependency system, map local routes into the manuscript's complete global outcome system, construct BN4-BN6, prove selector or realizer completeness, establish global ZeroSlack or PCCMin, prove SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalListSubsets_sublist", "PNP.DirectWire.TerminalComputedBCELAnchorNucleus.requestAtoms_nodup", "PNP.DirectWire.terminalBN3RequestPredicateBool_eq_true_iff", "PNP.DirectWire.terminalBN3RequestPredicate_monotone", "PNP.DirectWire.terminalBN3RequestPredicate_stable", "PNP.DirectWire.terminalBN3MinimalConsumer_exact", "PNP.DirectWire.TerminalComputedBCELAnchorNucleus.mem_activeRequestAtoms_iff_properCut", "PNP.DirectWire.TerminalComputedBCELAnchorNucleus.activeRequestAtoms_nodup", "PNP.DirectWire.TerminalComputedBCELAnchorNucleus.canonicalRequestBasis_jointlySideTight", "PNP.DirectWire.TerminalComputedBCELAnchorNucleus.computedBN3RequestEnvelope", "PNP.DirectWire.classifyTerminalBN3RequestEnvelope_exhaustive" ], "scope": "From every successful computed finite BCEL anchor nucleus, Lean uses one canonical duplicate-free primitive-record identity list across every proper cut; gives exact executable monotone request membership stable under extensional transport and exact singleton minimal consumers; accounts active incidences without duplicates; selects one canonical full or quotient side-tight coherent basis for every proper cut; and preserves all upstream proof-bearing classifier failures in a total outcome.", "title": "Candidate-derived finite BN3 request envelope", "theoremRows": [ { "name": "PNP.DirectWire.terminalListSubsets_sublist", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "ea0fefb3d241ebd81df124fbdb49a8409c1e6e77a111af3b6d4755e78c0c4d75", "expectedKernelTypeSha256": "ea0fefb3d241ebd81df124fbdb49a8409c1e6e77a111af3b6d4755e78c0c4d75", "kernelTypeFingerprintMatches": 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"status": "formalized-residual-terminal-bn3-request-envelope" }, { "classification": "formalized-residual-terminal-bn4-activation-cancellation", "id": "residual-terminal-bn4-activation-cancellation", "nonClaim": "This is a finite cancellation kernel over an explicit typed cell ledger. It does not derive cells, semantic signatures, or transport types from four-corner bases and is not the full historical BN4 theorem. It supplies no polynomial construction or size bound; does not construct PkgC or BN6; does not complete global routes, selectors, or realizers; does not establish ZeroSlack or PCCMin; does not put SAT in P; does not remove a project assumption; and does not prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalBN4ActivationCode_active_iff", "PNP.DirectWire.terminalBN4ActivationCode_eq_iff_activation", "PNP.DirectWire.terminalBN4ActivationKey_eq_iff", "PNP.DirectWire.terminalBN4IntegerMassLedger_exact", "PNP.DirectWire.TerminalBN4KeyCancellation.residual_key_eq", "PNP.DirectWire.TerminalBN4KeyCancellation.residual_mass_positive", "PNP.DirectWire.TerminalBN4KeyCancellation.no_opposite_sign_residual", "PNP.DirectWire.TerminalBN4KeyCancellation.residual_signedContribution_exact", "PNP.DirectWire.terminalBN4CancelAtKey_signedContribution_exact", "PNP.DirectWire.terminalBN4CanonicalKeys_nodup", "PNP.DirectWire.terminalBN4CellsUseCanonicalAtoms_iff", "PNP.DirectWire.TerminalComputedBCELAnchorNucleus.computedBN4ActivationCancellation", "PNP.DirectWire.classifyTerminalBN4ActivationCancellation_exhaustive" ], "scope": "After a successful computed finite BN3 envelope, Lean gives every request atom a canonical singleton activation code; proves activation-code equality exactly equivalent to equality of activation functions without enumerating cuts; checks equality of a complete typed key containing the atom, explicit semantic signature, and explicit transport type; totals positive and negative natural mass only at that same complete key; classifies a canonical balanced, positive, or negative residual; proves exact integer mass conservation, complete-key preservation, positive residual mass, and absence of opposite-sign residual pairs; computes duplicate-free ledger keys; and preserves all upstream failure branches while rejecting foreign request atoms.", "title": "Finite BN4 activation-exact cancellation kernel", "theoremRows": [ { 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"axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-bn4-activation-cancellation" }, { "classification": "formalized-residual-terminal-bn5-full-shadow-localization", "id": "residual-terminal-bn5-full-shadow-localization", "nonClaim": "This kernel starts from explicit finite inputs: one complete BN4 key, a negative cancellation result, a payload list, a cut, and a quotient-shadow coordinate list. It does not derive payloads or shadows from four-corner bases, connect complete matching back to a BN4 contradiction, or prove the full CritC/Q/E/L/X2/X3/X4 diagnosis, so it is not the full historical BN5 theorem. It does not construct PkgC or BN6; complete global routes, selectors, or realizers; establish polynomial generation or runtime, ZeroSlack, or PCCMin; put SAT in P; remove a project assumption; or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalBN5ShadowCoordinate_eq_iff", "PNP.DirectWire.terminalBN5FullUnits_length", "PNP.DirectWire.terminalBN5FullUnits_key_eq", "PNP.DirectWire.TerminalBN5HallDeficit.neighbor_card_lt_full_card", "PNP.DirectWire.TerminalBN5HallDeficit.fullSubset_coordinate_eq", "PNP.DirectWire.TerminalBN5HallDeficit.neighbor_coordinate_eq", "PNP.DirectWire.classifyTerminalBN5ShadowMatching_exhaustive", "PNP.DirectWire.TerminalBN4KeyCancellation.negativeResidualMass?_positive", "PNP.DirectWire.TerminalBN5HallDeficit.namedLocalRoute_eq_x1Hall", "PNP.DirectWire.classifyTerminalBN5FullShadowLocalization_active", "PNP.DirectWire.TerminalBN5HallDeficit.unmatchedShadowNotSilent", "PNP.DirectWire.classifyTerminalBN5FullShadowLocalization_exhaustive" ], "scope": "For every explicit finite negative-unit refinement and quotient-shadow ledger, Lean preserves the complete exact-coordinate data, validates the negative mass refinement, computes whether the cut is silent, and otherwise returns either complete multiplicity coverage or a strict Hall deficit with a literal smaller shadow-neighbor fibre, complete-coordinate preservation, and a proof-bearing local X1 route that prevents active unmatched units from disappearing silently.", "title": "Finite BN5 full-shadow localization kernel", "theoremRows": [ { "name": "PNP.DirectWire.terminalBN5ShadowCoordinate_eq_iff", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "8bff617398ee48e2e0b703c252fe76e54989a81857c112937a71e1544db6ce92", "expectedKernelTypeSha256": "8bff617398ee48e2e0b703c252fe76e54989a81857c112937a71e1544db6ce92", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalBN5FullUnits_length", "present": true, "kind": "theorem", "axioms": [ 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"daa39d3dc5252bc6eafa09474a9f1f91642c40957251fe7d69b1db28bbd35e29", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalBN4KeyCancellation.negativeResidualMass?_positive", "present": true, "kind": "theorem", "axioms": [ "propext" ], "actualKernelTypeSha256": "dd8fcb3c7caa6eefd822407110d5621906a6a3f812f651d52ff2d0a5d6c07c9d", "expectedKernelTypeSha256": "dd8fcb3c7caa6eefd822407110d5621906a6a3f812f651d52ff2d0a5d6c07c9d", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalBN5HallDeficit.namedLocalRoute_eq_x1Hall", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "76b66e4be0893605464be5e7498b300a796a6913b5194e09577eae4b00b17e04", "expectedKernelTypeSha256": "76b66e4be0893605464be5e7498b300a796a6913b5194e09577eae4b00b17e04", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.classifyTerminalBN5FullShadowLocalization_active", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "022b89b6ed7a65c81d886a4467cea34ee98e37c50fae199996774070ca7b52d4", "expectedKernelTypeSha256": "022b89b6ed7a65c81d886a4467cea34ee98e37c50fae199996774070ca7b52d4", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalBN5HallDeficit.unmatchedShadowNotSilent", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "1472e6b4b8992222aac9a8a93b52fd0cce8b696f13bc5d2403ff321462be823f", "expectedKernelTypeSha256": "1472e6b4b8992222aac9a8a93b52fd0cce8b696f13bc5d2403ff321462be823f", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.classifyTerminalBN5FullShadowLocalization_exhaustive", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "be1af480ae6864a553e86c154bcbc67ac7b72b01df96b5580c9b9eadeebb4cc3", "expectedKernelTypeSha256": "be1af480ae6864a553e86c154bcbc67ac7b72b01df96b5580c9b9eadeebb4cc3", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-bn5-full-shadow-localization" }, { "classification": "formalized-residual-terminal-pkgc-separating-consumers", "id": "residual-terminal-pkgc-separating-consumers", "nonClaim": "The restoration coordinate universe remains explicit. This theorem does not derive consumers or restorations from a terminal candidate, connect complete coverage back to a BN4 or BN5 contradiction, embed the Hall route into the complete global outcome system, prove global route silence, or establish the full historical PkgC theorem. It does not prove full BN6 or Packet selector-realizer completeness, polynomial generation or runtime, ZeroSlack or PCCMin, SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalPkgCPairNeedsRestoration_eq_true_iff", "PNP.DirectWire.firstTerminalPkgCSeparatingPair?_sound", "PNP.DirectWire.firstTerminalPkgCSeparatingPair?_eq_none_iff", "PNP.DirectWire.TerminalPkgCSeparatingPair.quotientUnits_length", "PNP.DirectWire.TerminalPkgCSeparatingPair.quotientUnits_nonempty", "PNP.DirectWire.terminalPkgC_restorationEdge_preservesCoordinate", "PNP.DirectWire.TerminalBN5HallDeficit.pkgCRestorationNotSilent", "PNP.DirectWire.terminalPkgC_separatingConsumers_restorationDichotomy", "PNP.DirectWire.classifyTerminalPkgCSeparatingConsumers_exhaustive" ], "scope": "For an arbitrary finite explicit minimal-consumer antichain, Lean canonically scans for the first disjoint pair that is not singleton-singleton. Absence proves exactly V54's singletonization premise. A found pair's atoms are canonically indexed into exact-coordinate quotient units and an explicit full-restoration universe is classified into complete multiplicity coverage or a strict Hall deficit with a deterministic local Q route.", "title": "Finite PkgC separating-consumer restoration dichotomy", "theoremRows": [ { "name": "PNP.DirectWire.terminalPkgCPairNeedsRestoration_eq_true_iff", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "184ddf69a99d22a191c773492f5544b457d72987222ef40fc00edadd977a95e2", "expectedKernelTypeSha256": "184ddf69a99d22a191c773492f5544b457d72987222ef40fc00edadd977a95e2", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.firstTerminalPkgCSeparatingPair?_sound", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "d13683190a71031aa6df3b201dc1507fe83dd005867b9c5d8a4f8738ad3fd6bf", "expectedKernelTypeSha256": "d13683190a71031aa6df3b201dc1507fe83dd005867b9c5d8a4f8738ad3fd6bf", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.firstTerminalPkgCSeparatingPair?_eq_none_iff", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "4b671f164446da2714ecf89d93203f53cf07380f60cfd5d70b7301f6c831cbdc", "expectedKernelTypeSha256": "4b671f164446da2714ecf89d93203f53cf07380f60cfd5d70b7301f6c831cbdc", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalPkgCSeparatingPair.quotientUnits_length", "present": true, "kind": "theorem", "axioms": [ "propext" ], "actualKernelTypeSha256": "207c3a318d1e4bbde2a33bc01afd96f674b6973a0cf5d595e3739af8f3b0e6c9", "expectedKernelTypeSha256": "207c3a318d1e4bbde2a33bc01afd96f674b6973a0cf5d595e3739af8f3b0e6c9", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalPkgCSeparatingPair.quotientUnits_nonempty", "present": true, "kind": "theorem", "axioms": [ "propext" ], "actualKernelTypeSha256": "1be63df6d57723d5183ecc39a11d3ed346430ecaf57c118399b2358a5862bfbc", "expectedKernelTypeSha256": "1be63df6d57723d5183ecc39a11d3ed346430ecaf57c118399b2358a5862bfbc", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalPkgC_restorationEdge_preservesCoordinate", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "614c064c69d3e39441036d8b1e70a6f2f5f15b2e6523b1ae094ed34b7bae1e8d", "expectedKernelTypeSha256": "614c064c69d3e39441036d8b1e70a6f2f5f15b2e6523b1ae094ed34b7bae1e8d", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalBN5HallDeficit.pkgCRestorationNotSilent", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "bec861192dedeca58ccdb1b9764e428715730d5b66c2dd5bdd3f688957ff7b27", "expectedKernelTypeSha256": "bec861192dedeca58ccdb1b9764e428715730d5b66c2dd5bdd3f688957ff7b27", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalPkgC_separatingConsumers_restorationDichotomy", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "690a8f82885a284547a453cbfd6858ef8d8d7c4201e710f79a9eca9faa92df81", "expectedKernelTypeSha256": "690a8f82885a284547a453cbfd6858ef8d8d7c4201e710f79a9eca9faa92df81", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.classifyTerminalPkgCSeparatingConsumers_exhaustive", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "a05d8069e347cf7f63b59aef148e10f5587461d09bd0fff8f400b7cc7b4f1d2f", "expectedKernelTypeSha256": "a05d8069e347cf7f63b59aef148e10f5587461d09bd0fff8f400b7cc7b4f1d2f", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-pkgc-separating-consumers" }, { "classification": "formalized-residual-terminal-pkgc-typed-restoration", "id": "residual-terminal-pkgc-typed-restoration", "nonClaim": "The typed restoration operation remains explicit caller data. This milestone does not construct it from a terminal candidate or prove its full semantic adequacy. It does not connect complete restoration to a BN4 or BN5 contradiction, embed local routes into the complete global outcome system, prove global PkgC route silence or the full historical PkgC theorem, establish full BN6 or Packet selector-realizer completeness, polynomial generation or runtime, ZeroSlack or PCCMin, put SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalPkgCSeparatingPair.fullRestorationCandidates_length", "PNP.DirectWire.TerminalPkgCSeparatingPair.fullRestorationCandidates_coordinates", "PNP.DirectWire.TerminalPkgCTypedRestorer.coordinateUniverse_coordinates", "PNP.DirectWire.terminalBN5FullMultiplicity_indexed_eq", "PNP.DirectWire.terminalBN5ShadowMultiplicity_indexed_eq", "PNP.DirectWire.TerminalPkgCSeparatingPair.typedRestoration_exactCoverage", "PNP.DirectWire.terminalBN5CompleteMultiplicityMatching_not_hallDeficit", "PNP.DirectWire.terminalPkgC_typedRestoration_realization", "PNP.DirectWire.classifyTerminalPkgCTypedRestoration_exhaustive" ], "scope": "For an arbitrary finite explicit minimal-consumer antichain and a typed coordinate-preserving restoration operation, Lean materializes typed full-restoration candidates for every atom of the canonical first disjoint nonsingleton pair, proves exact candidate count and positional coordinate preservation, derives complete equality-fibre multiplicity coverage, excludes a strict Hall deficit for that graph, and otherwise proves V54 singletonization.", "title": "Finite PkgC typed restoration realization", "theoremRows": [ { "name": "PNP.DirectWire.TerminalPkgCSeparatingPair.fullRestorationCandidates_length", "present": true, "kind": "theorem", "axioms": [ "propext" ], "actualKernelTypeSha256": "e8ebfbe79ccfd79fd39fa8fb08f4acc15b21ac3db9921fa8d5b52e0e2a5a79fc", "expectedKernelTypeSha256": "e8ebfbe79ccfd79fd39fa8fb08f4acc15b21ac3db9921fa8d5b52e0e2a5a79fc", "kernelTypeFingerprintMatches": true }, { "name": 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"theorem", "axioms": [], "actualKernelTypeSha256": "ccad95b996ac06b5c602110f765022263fab1d9db31c979f0fd7b46f9c615929", "expectedKernelTypeSha256": "ccad95b996ac06b5c602110f765022263fab1d9db31c979f0fd7b46f9c615929", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalPkgC_typedRestoration_realization", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "0b26c740c02a95570364ba233ce3671bb5250dae0d99237e06e0374e017b5301", "expectedKernelTypeSha256": "0b26c740c02a95570364ba233ce3671bb5250dae0d99237e06e0374e017b5301", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.classifyTerminalPkgCTypedRestoration_exhaustive", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "8a161f64784fb6fc510c7cf8820556b4e774432191713bd1369944b99db13583", "expectedKernelTypeSha256": "8a161f64784fb6fc510c7cf8820556b4e774432191713bd1369944b99db13583", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-pkgc-typed-restoration" }, { "classification": "formalized-residual-terminal-pkgc-same-key-cancellation", "id": "residual-terminal-pkgc-same-key-cancellation", "nonClaim": "The typed restoration operation and its complete coordinate maps remain explicit inputs, and the generated opposite-sign cells are not yet proved to be the cells of the terminal candidate's ambient BN4 ledger. This milestone does not construct semantic restorations from a terminal candidate, embed cancellation or Hall outcomes into the complete global route system, prove global route silence or the full historical PkgC theorem, establish full BN6 or Packet selector-realizer completeness, polynomial generation or runtime, ZeroSlack or PCCMin, put SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalPkgCRestorationCancellationCellsForAtom_key_eq", "PNP.DirectWire.terminalPkgCRestorationCancellationCellsForAtom_balanced", "PNP.DirectWire.terminalPkgCRestorationCancellationCellsForAtoms_length", "PNP.DirectWire.terminalPkgCRestorationCancellationCellsForAtoms_balanced", "PNP.DirectWire.TerminalPkgCSeparatingPair.restorationCancellationCells_length", "PNP.DirectWire.TerminalPkgCSeparatingPair.restorationCancellation_balanced", "PNP.DirectWire.TerminalPkgCSeparatingPair.restorationCancellation_residualCells_empty", "PNP.DirectWire.TerminalPkgCSeparatingPair.restorationCancellation_signedMass_zero", "PNP.DirectWire.terminalPkgC_typedRestoration_sameKeyCancellation", "PNP.DirectWire.terminalPkgC_sameKeyCancellation_silence_singletonizes", "PNP.DirectWire.classifyTerminalPkgCSameKeyCancellation_exhaustive" ], "scope": "For an arbitrary finite explicit minimal-consumer antichain and typed exact-BN5-coordinate restoration operation, Lean mechanically pairs every quotient atom with its restored full candidate as opposite-sign unit cells, proves the complete BN5 coordinate gives the same nested BN4 key, proves exact cell count and positive/negative multiplicity equality at every BN4 key, computes an empty canonical residual and zero signed mass at every key, and derives V54 singletonization from exact absence of every such proof-bearing cancellation outcome.", "title": "Finite PkgC typed restoration same-key cancellation", "theoremRows": [ { "name": 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"0524782f70301d75c278d5bda7865945e932e2fce0f4a2eff65a284118312a48", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalPkgC_typedRestoration_sameKeyCancellation", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "b8acb10618219c2980c026a7bdfa92bddd5da3810f449e8249bd820f31abb1e7", "expectedKernelTypeSha256": "b8acb10618219c2980c026a7bdfa92bddd5da3810f449e8249bd820f31abb1e7", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalPkgC_sameKeyCancellation_silence_singletonizes", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "c0feb56a30d17811ddf2d13bb7a645ac55d8e99af90d1ccd9cad99409745c229", "expectedKernelTypeSha256": "c0feb56a30d17811ddf2d13bb7a645ac55d8e99af90d1ccd9cad99409745c229", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.classifyTerminalPkgCSameKeyCancellation_exhaustive", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "c8c63bd0d8f1c967d9e4809d87f62692c9532dd52ea3f895b372baf3aacdfe95", "expectedKernelTypeSha256": "c8c63bd0d8f1c967d9e4809d87f62692c9532dd52ea3f895b372baf3aacdfe95", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-pkgc-same-key-cancellation" }, { "classification": "formalized-residual-terminal-pkgc-ambient-bn4-ledger", "id": "residual-terminal-pkgc-ambient-bn4-ledger", "nonClaim": "The ambient ledger, typed restoration operation, exact permutation certificate or canonical serialization, and successful candidate-derived BN4 kernel remain explicit proof-bearing inputs. This milestone does not derive the ambient ledger or restorer from a terminal candidate, prove the restorer's semantic adequacy, embed local cancellation or Hall outcomes into the complete global route system, prove global PkgC route silence or the full historical PkgC theorem, establish full BN6 or Packet selector-realizer completeness, polynomial generation or runtime, ZeroSlack or PCCMin, put SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalBN4PositiveMass_perm", "PNP.DirectWire.terminalBN4NegativeMass_perm", "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.generatedCell_mem_ambient", "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.cellMultiplicity", "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.length_eq", "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.positiveMass_decomposition", "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.negativeMass_decomposition", "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.signedMass_eq_remainder", "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.residualSignedContribution_eq_remainder", "PNP.DirectWire.classifyTerminalPkgCAmbientBN4LedgerBinding_exhaustive", "PNP.DirectWire.TerminalPkgCComputedAmbientBN4Cancellation.generatedCell_usesCanonicalAtom", "PNP.DirectWire.terminalPkgC_computedAmbientBN4_silence_singletonizes" ], "scope": "For arbitrary finite explicit BN4 cell ledgers, Lean proves that a proof-bearing exact multiset embedding identifies the generated PkgC opposite-sign cancellation ledger with an ambient subledger and preserves every duplicate. Positive and negative mass decompose at every complete key; removing the balanced generated subledger leaves the ambient signed mass and executable residual signed contribution exactly equal to an explicit remainder. A successful candidate-derived BN4 kernel additionally proves every embedded generated cell uses its canonical request-atom space, and complete bindings plus exact absence of every computed bridge imply V54 singletonization.", "title": "Finite PkgC ambient BN4 ledger embedding", "theoremRows": [ { "name": "PNP.DirectWire.terminalBN4PositiveMass_perm", "present": true, "kind": "theorem", "axioms": [ "propext" ], "actualKernelTypeSha256": "85bd3e3748cd1d2bb7a6a749fccbe75077fe4106abdb04f09f33a565d1490e1a", "expectedKernelTypeSha256": "85bd3e3748cd1d2bb7a6a749fccbe75077fe4106abdb04f09f33a565d1490e1a", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalBN4NegativeMass_perm", "present": true, "kind": "theorem", "axioms": [ "propext" ], "actualKernelTypeSha256": "a435189ebdd2cadb72ffdc828902e5742a70454f3ac0ea4e5c3662a77ec0617a", "expectedKernelTypeSha256": "a435189ebdd2cadb72ffdc828902e5742a70454f3ac0ea4e5c3662a77ec0617a", "kernelTypeFingerprintMatches": true }, { 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"allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-pkgc-ambient-bn4-ledger" }, { "classification": "formalized-residual-terminal-pkgc-ambient-bn4-residual-reduction", "id": "residual-terminal-pkgc-ambient-bn4-residual-reduction", "nonClaim": "The ambient ledger, typed restoration operation, exact embedding, and explicit remainder remain proof-bearing inputs. This milestone does not derive those inputs from an arbitrary terminal candidate, prove that the remainder is empty or route-producing, establish restoration semantic adequacy or complete global route silence, reconstruct the full historical PkgC/BN6/Packet path, prove polynomial generation or runtime, establish ZeroSlack or PCCMin, put SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.terminalBN4ResidualCells_add_common", "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.residualCells_eq_remainder", "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.residualLedgerOver_eq_remainder", "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.remainderKey_mem_ambientCanonicalKeys", "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.canonicalResidualLedger_eq_remainder", "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.canonicalResidualLedger_empty_of_remainder_empty", "PNP.DirectWire.classifyTerminalPkgCAmbientBN4ResidualReduction_exhaustive", "PNP.DirectWire.TerminalPkgCComputedAmbientBN4Cancellation.residualLedger_empty_of_remainder_empty" ], "scope": "For arbitrary finite explicit ambient BN4 ledgers, Lean proves that removing an exactly embedded balanced PkgC generated subledger preserves the executable residual cell at every complete key and the complete canonical executable residual ledger over the ambient key universe. Every remainder key occurs in that universe; a fail-closed canonical classifier constructs the exact reduction without caller-provided proof bits; and an empty remainder yields an empty ambient residual ledger, including for the existing candidate-derived computed bridge.", "title": "Finite PkgC ambient BN4 residual reduction", "theoremRows": [ { "name": "PNP.DirectWire.terminalBN4ResidualCells_add_common", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "1a6f5d332c00b6d10c7d0e1b0b703f0199bf1f72024e963754cbf1b32e45bb48", "expectedKernelTypeSha256": "1a6f5d332c00b6d10c7d0e1b0b703f0199bf1f72024e963754cbf1b32e45bb48", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.residualCells_eq_remainder", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "136ed6fe2e3bf2ebe118aa9241c6ed9be9716aa60c38a96f242154bd79a65245", "expectedKernelTypeSha256": "136ed6fe2e3bf2ebe118aa9241c6ed9be9716aa60c38a96f242154bd79a65245", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.residualLedgerOver_eq_remainder", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "b89662618db4ee812256d3088c53e8e1b910ace8d65db0a5f8a6a99571955f85", "expectedKernelTypeSha256": "b89662618db4ee812256d3088c53e8e1b910ace8d65db0a5f8a6a99571955f85", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.remainderKey_mem_ambientCanonicalKeys", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "917b2e93bf46b18681e610599182e5c55086feb311b348585a0527ebfa588cc5", "expectedKernelTypeSha256": "917b2e93bf46b18681e610599182e5c55086feb311b348585a0527ebfa588cc5", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.canonicalResidualLedger_eq_remainder", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "b36d0d118451d5de98066c9fab1ebe4beec820c718e5760bbe752184df2f479c", "expectedKernelTypeSha256": "b36d0d118451d5de98066c9fab1ebe4beec820c718e5760bbe752184df2f479c", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalPkgCAmbientBN4LedgerEmbedding.canonicalResidualLedger_empty_of_remainder_empty", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "b7071dc1bfedd34a95972ddb2306d4acadd1fc2908aa7bca12145846d3e50df3", "expectedKernelTypeSha256": "b7071dc1bfedd34a95972ddb2306d4acadd1fc2908aa7bca12145846d3e50df3", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.classifyTerminalPkgCAmbientBN4ResidualReduction_exhaustive", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "a9a7ae5b7dfdf9f1d7e459535bf090da559792ed829a3272cc70f1fa0730451d", "expectedKernelTypeSha256": "a9a7ae5b7dfdf9f1d7e459535bf090da559792ed829a3272cc70f1fa0730451d", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalPkgCComputedAmbientBN4Cancellation.residualLedger_empty_of_remainder_empty", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "88dbcbe83183f60d9fb391aeca7bd156d853637353339d899f5185ffa7c3fdf4", "expectedKernelTypeSha256": "88dbcbe83183f60d9fb391aeca7bd156d853637353339d899f5185ffa7c3fdf4", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-pkgc-ambient-bn4-residual-reduction" }, { "classification": "formalized-residual-terminal-v54-consumer-antichain-normal-form", "id": "residual-terminal-consumer-antichain-normal-form", "nonClaim": "This finite kernel starts from an explicit minimal-consumer antichain and an explicit proof that every disjoint consumer pair is singletonized. It does not construct PkgC, derive that singletonization premise, or connect the footprint back to the full BN6 proof. It does not construct complete global routes, selectors, or realizers; establish polynomial generation or runtime, ZeroSlack, or PCCMin; put SAT in P; remove a project assumption; or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalV54ConsumerSystem.requestActive_monotone", "PNP.DirectWire.TerminalV54ConsumerSystem.requestActive_empty_false", "PNP.DirectWire.TerminalV54ConsumerSystem.consumer_is_minimal", "PNP.DirectWire.TerminalV54ConsumerSystem.cutActive_has_disjoint_consumers", "PNP.DirectWire.terminalV54_cutActivation_nonzero_iff_disjoint_consumers", "PNP.DirectWire.terminalV54_consumerAntichain_normal_form_iff", "PNP.DirectWire.terminalV54_consumerAntichain_normal_form" ], "scope": "For an arbitrary finite carrier and its explicit minimal-consumer antichain, Lean proves monotonicity and empty-request inactivity, proves that nonzero two-sided cut activation is equivalent to the existence of a disjoint consumer pair, and under the exact singletonized-disjoint-pair premise proves literal equality with the corresponding footprint cut indicator.", "title": "Finite V54 consumer-antichain normal form", "theoremRows": [ { "name": "PNP.DirectWire.TerminalV54ConsumerSystem.requestActive_monotone", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "2b4b2abc251a6f227f312189bef5f8af6ad139173a5b56bd041b5c0490c56009", "expectedKernelTypeSha256": "2b4b2abc251a6f227f312189bef5f8af6ad139173a5b56bd041b5c0490c56009", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalV54ConsumerSystem.requestActive_empty_false", "present": true, "kind": "theorem", "axioms": [ "propext" ], "actualKernelTypeSha256": "6a1188e91a30e61ffbabab94b6419a4e0d9980de8723c4bf3e717c82735e95c9", "expectedKernelTypeSha256": "6a1188e91a30e61ffbabab94b6419a4e0d9980de8723c4bf3e717c82735e95c9", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalV54ConsumerSystem.consumer_is_minimal", "present": true, "kind": "theorem", "axioms": [], "actualKernelTypeSha256": "962b718805829299b8f72bb3623e64290a082100336661f4bf8c35c7e7b2ad01", "expectedKernelTypeSha256": "962b718805829299b8f72bb3623e64290a082100336661f4bf8c35c7e7b2ad01", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalV54ConsumerSystem.cutActive_has_disjoint_consumers", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "85e9ce925cd90213f31797bde659b24a409759777dd994270114a8b1d4476bc6", "expectedKernelTypeSha256": "85e9ce925cd90213f31797bde659b24a409759777dd994270114a8b1d4476bc6", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalV54_cutActivation_nonzero_iff_disjoint_consumers", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "49b42196b9207ded24b4c2c90443a4e704e7ac89eb72742ad2b08d9e492c0b9e", "expectedKernelTypeSha256": "49b42196b9207ded24b4c2c90443a4e704e7ac89eb72742ad2b08d9e492c0b9e", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalV54_consumerAntichain_normal_form_iff", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "8551d1d8aaab04e96fe846f653c9f7327144623374ab9d7d18a401fb448c0de6", "expectedKernelTypeSha256": "8551d1d8aaab04e96fe846f653c9f7327144623374ab9d7d18a401fb448c0de6", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalV54_consumerAntichain_normal_form", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "2f11e9bb5314944cd0ee20579865210e16792d95a20e754c72d18f401c0bdbf9", "expectedKernelTypeSha256": "2f11e9bb5314944cd0ee20579865210e16792d95a20e754c72d18f401c0bdbf9", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-v54-consumer-antichain-normal-form" }, { "classification": "formalized-residual-terminal-v53-constant-cut-hypergraph-rigidity", "id": "residual-terminal-constant-cut-hypergraph-rigidity", "nonClaim": "This theorem consumes an explicit sparse positive hypergraph and an explicit proof that every nonempty proper cut has the same positive value. It does not construct PkgC, derive the hypergraph from a terminal candidate or the V54 consumer system, build BN6 cells or payloads, complete global routes, selectors, or realizers, establish polynomial generation or runtime, prove ZeroSlack or PCCMin, put SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalV53Hypergraph.cell_partition", "PNP.DirectWire.TerminalV53Hypergraph.cut_partition", "PNP.DirectWire.TerminalV53Hypergraph.pair_complement_identity", "PNP.DirectWire.TerminalV53Hypergraph.pairWeights_equal_of_shared", "PNP.DirectWire.TerminalV53Hypergraph.pairWeight_eq_zero_of_four", "PNP.DirectWire.TerminalV53Hypergraph.properFootprintWeight_eq_zero_of_four", "PNP.DirectWire.TerminalV53Hypergraph.twoAnchor_fullWeight", "PNP.DirectWire.TerminalV53Hypergraph.threeAnchor_rigidity", "PNP.DirectWire.TerminalV53Hypergraph.fourAnchor_rigidity", "PNP.DirectWire.terminalV53_constantCut_hypergraph_rigidity" ], "scope": "For an arbitrary finite duplicate-free carrier and sparse nonnegative weighted hypergraph with positive listed cells, exact equality of every nonempty proper cut proves the complete V53 q=2, q=3, and q>=4 classification: full-span weight D; one common pair weight p with w_A + 2p = D; or zero weight on every proper footprint with full-span weight D.", "title": "Finite V53 constant-cut hypergraph rigidity", "theoremRows": [ { "name": "PNP.DirectWire.TerminalV53Hypergraph.cell_partition", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "d16379065824e05c2fb07404dc6102c1e715431af0412428fe58762e7c443163", "expectedKernelTypeSha256": "d16379065824e05c2fb07404dc6102c1e715431af0412428fe58762e7c443163", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalV53Hypergraph.cut_partition", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "e2a434fd77b4beb8c1165f6bedd35df5dbe529b00362ce5915acddd46c44239e", "expectedKernelTypeSha256": "e2a434fd77b4beb8c1165f6bedd35df5dbe529b00362ce5915acddd46c44239e", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalV53Hypergraph.pair_complement_identity", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "5fbeb36fa268a1891851320dda5716ced89a17d9234e1b39380faded0ca97b5c", "expectedKernelTypeSha256": "5fbeb36fa268a1891851320dda5716ced89a17d9234e1b39380faded0ca97b5c", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalV53Hypergraph.pairWeights_equal_of_shared", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "aaf54c3a5054d667fb42dc481a8c11bfabe9fd097491bca8e218b5e1257c8b9d", "expectedKernelTypeSha256": "aaf54c3a5054d667fb42dc481a8c11bfabe9fd097491bca8e218b5e1257c8b9d", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalV53Hypergraph.pairWeight_eq_zero_of_four", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "5763916217506b3ec2ffbfb4ac6beeac75a09978651daa42ce1e586084d893b2", "expectedKernelTypeSha256": "5763916217506b3ec2ffbfb4ac6beeac75a09978651daa42ce1e586084d893b2", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalV53Hypergraph.properFootprintWeight_eq_zero_of_four", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "5875a9ea0aaab8e4d1598a79262ccae893e25e64f8e280c86145e3e5f2353d82", "expectedKernelTypeSha256": "5875a9ea0aaab8e4d1598a79262ccae893e25e64f8e280c86145e3e5f2353d82", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalV53Hypergraph.twoAnchor_fullWeight", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "f69d38ce0fc3179c4b64601fc7bc972e5b4d40becc39af61ca811b442f1ec692", "expectedKernelTypeSha256": "f69d38ce0fc3179c4b64601fc7bc972e5b4d40becc39af61ca811b442f1ec692", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalV53Hypergraph.threeAnchor_rigidity", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "7042a91984504430f872f5496b3492f33cdf368f7e6cee2d6f21689ce411e574", "expectedKernelTypeSha256": "7042a91984504430f872f5496b3492f33cdf368f7e6cee2d6f21689ce411e574", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalV53Hypergraph.fourAnchor_rigidity", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "046214923b999ee22b3da8460b12de05b47f0f15631fb0f76478b282d6dac585", "expectedKernelTypeSha256": "046214923b999ee22b3da8460b12de05b47f0f15631fb0f76478b282d6dac585", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalV53_constantCut_hypergraph_rigidity", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "5b5dd2a2dc158dcc4e91ca9f08de3c94a134e530eafc96a096279a6ab48c8b07", "expectedKernelTypeSha256": "5b5dd2a2dc158dcc4e91ca9f08de3c94a134e530eafc96a096279a6ab48c8b07", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-v53-constant-cut-hypergraph-rigidity" }, { "classification": "formalized-residual-terminal-bn6-hypergraph-packet", "id": "residual-terminal-bn6-hypergraph-packet", "nonClaim": "This finite bridge consumes explicit exact footprint grouping, PkgC singletonization proofs, positive atom ledgers, payload data, and the BCEL constant-cut equation. It does not construct PkgC, derive or group survivors from a terminal candidate, establish full historical BN6 or Packet selector/realizer completeness, complete global routes, prove polynomial generation or runtime, ZeroSlack or PCCMin, put SAT in P, remove a project assumption, or prove P = NP.", "requiredTheorems": [ "PNP.DirectWire.TerminalBN6GroupedCell.massPositive", "PNP.DirectWire.TerminalBN6GroupedCell.crosses_iff_footprintCrosses", "PNP.DirectWire.TerminalBN6GroupedCell.crossesBool_eq_cutActivationBool", "PNP.DirectWire.TerminalBN6GroupedFamily.cutWeight_eq_activationWeight", "PNP.DirectWire.TerminalBN6GroupedFamily.constantProperCuts", "PNP.DirectWire.TerminalBN6GroupedFamily.footprintWeight_eq_groupedMass", "PNP.DirectWire.TerminalBN6GroupedFamily.hasPayloadAt_of_footprintWeight_positive", "PNP.DirectWire.terminalBN6_hypergraph_packet" ], "scope": "For an arbitrary finite duplicate-free anchor carrier and explicit already-grouped family of positive payload-bearing survivor cells, V54 activation is transported exactly into the constructed V53 hypergraph cut sum. A positive BCEL constant-cut premise then yields the complete pair, mixed three-anchor balanced-triple/full-span, or four-or-more-anchor full-span classification with original payload witnesses.", "title": "Finite BN6 grouped hypergraph packet bridge", "theoremRows": [ { "name": "PNP.DirectWire.TerminalBN6GroupedCell.massPositive", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "52dcb7f7bc53bd8efa1f550ea96f76596b5a267dc1570fc2c7d516bc9177018d", "expectedKernelTypeSha256": "52dcb7f7bc53bd8efa1f550ea96f76596b5a267dc1570fc2c7d516bc9177018d", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalBN6GroupedCell.crosses_iff_footprintCrosses", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "046aa9fa7795d8b905a24e3812d93bfcd64a461a5a0de1ffa175e7016e91851b", "expectedKernelTypeSha256": "046aa9fa7795d8b905a24e3812d93bfcd64a461a5a0de1ffa175e7016e91851b", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalBN6GroupedCell.crossesBool_eq_cutActivationBool", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "ae3cb49dfb2c70dde2e96d5b48a86a5a1f5787715d4e2480f26dde6a5ed521dc", "expectedKernelTypeSha256": "ae3cb49dfb2c70dde2e96d5b48a86a5a1f5787715d4e2480f26dde6a5ed521dc", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalBN6GroupedFamily.cutWeight_eq_activationWeight", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "938b6617e45e2d77005a52cb207b3049a852a9bb27647ba5cbc44bcf7da8b15c", "expectedKernelTypeSha256": "938b6617e45e2d77005a52cb207b3049a852a9bb27647ba5cbc44bcf7da8b15c", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalBN6GroupedFamily.constantProperCuts", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "34b06191ae4d2b810e5073bffd5a361d9ce5a8c0a64a0bf88bc561e61cbe4f3e", "expectedKernelTypeSha256": "34b06191ae4d2b810e5073bffd5a361d9ce5a8c0a64a0bf88bc561e61cbe4f3e", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalBN6GroupedFamily.footprintWeight_eq_groupedMass", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "f97902c2a9421081048eae6a85ffa03a1ecc2eda72ba6aacb43bbbed125652c0", "expectedKernelTypeSha256": "f97902c2a9421081048eae6a85ffa03a1ecc2eda72ba6aacb43bbbed125652c0", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.TerminalBN6GroupedFamily.hasPayloadAt_of_footprintWeight_positive", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "c9f5cfab81884d8636e41bff0dd401fb27e8a4a99c5918ce41027a55e26291f4", "expectedKernelTypeSha256": "c9f5cfab81884d8636e41bff0dd401fb27e8a4a99c5918ce41027a55e26291f4", "kernelTypeFingerprintMatches": true }, { "name": "PNP.DirectWire.terminalBN6_hypergraph_packet", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "1275e898a6b23670bfbc97c3684c2da75a6b413059e4d2d9c0f0c1cd967a11ba", "expectedKernelTypeSha256": "1275e898a6b23670bfbc97c3684c2da75a6b413059e4d2d9c0f0c1cd967a11ba", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-residual-terminal-bn6-hypergraph-packet" }, { "classification": "formalized-concrete-locked-nand-threshold", "id": "global-locked-nand-threshold", "nonClaim": "This closes the uniform all-bitstring CNFSAT-to-concrete-locked-threshold builder and report-facing linkage in the finite charged-pipeline model. It does not put the concrete locked threshold language in P, discharge residual-band minimization, ZeroSlack or PCCMin, prove concrete CNFSAT NP-hardness, activate the legacy string-handle bridge, or prove P = NP.", "requiredTheorems": [ "PNP.Main.locked_nand_threshold" ], "scope": "A uniform encoded polynomial-time SAT instance builder and the report-level locked-NAND threshold theorem linked to that builder.", "title": "Global locked-NAND construction and threshold", "theoremRows": [ { "name": "PNP.Main.locked_nand_threshold", "present": true, "kind": "theorem", "axioms": [ "Quot.sound", "propext" ], "actualKernelTypeSha256": "951ec63c09e9a096aacc26332a97607dade4a1f412229f9185aff5c7f36aa591", "expectedKernelTypeSha256": "951ec63c09e9a096aacc26332a97607dade4a1f412229f9185aff5c7f36aa591", "kernelTypeFingerprintMatches": true } ], "allPresent": true, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": true, "allKernelTypesMatch": true, "sourceClosureFingerprintMatches": true, "earned": true, "status": "formalized-concrete-locked-nand-threshold" }, { "classification": "not-formalized", "id": "global-zeroslack-pccmin", "nonClaim": "The finite candidate-derived BN3 envelope supplies stable request identities and one jointly side-tight canonical basis family; the finite BN4 kernel supplies activation-exact same-key integer cancellation over an explicit typed cell ledger; and the finite BN5 kernel localizes explicit full/shadow multiplicity failure to a strict Hall deficit and local X1 route. The construction still does not derive the BN4 ledger or BN5 payload/shadow universe from the bases, connect matching back to a contradiction, establish the full historical BN4 or BN5 theorems, construct PkgC or BN6, map all residual routes into a decreasing complete global outcome system, or provide selector/realizer and polynomial-runtime completeness; global ZeroSlack and polynomial PCCMin therefore remain unformalized.", "requiredTheorems": [ "PNP.Main.pccmin_polynomial_exact", "PNP.Main.zero_slack_complete" ], "scope": "Complete residual routing, global ZeroSlack contradiction, exact minimization, and polynomial bounds.", "title": "Global ZeroSlack, PCCMin, and polynomial runtime", "theoremRows": [ { "name": "PNP.Main.pccmin_polynomial_exact", "present": false, "kind": null, "axioms": null, "actualKernelTypeSha256": null, "expectedKernelTypeSha256": null, "kernelTypeFingerprintMatches": false }, { "name": "PNP.Main.zero_slack_complete", "present": false, "kind": null, "axioms": null, "actualKernelTypeSha256": null, "expectedKernelTypeSha256": null, "kernelTypeFingerprintMatches": false } ], "allPresent": false, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": false, "allKernelTypesMatch": false, "sourceClosureFingerprintMatches": true, "earned": false, "status": "not-formalized" }, { "classification": "not-formalized", "id": "concrete-publication-root", "nonClaim": "The concrete target definition remains inactive: CNF-SAT now has a raw-machine verifier, NP-membership proof, and an all-input finite-machine polynomial reduction through NAND to the concrete locked target, but a deterministic polynomial-time CNF-SAT decider, the concrete-to-abstract threshold link, SAT NP-hardness and CNFSAT NP-completeness transport, and PNP.Main.p_eq_np remain absent; the legacy string-handle bridge is ineligible.", "requiredTheorems": [ "PNP.Main.p_eq_np" ], "scope": "Raw-machine-linked complexity classes, concrete SAT completeness and a SAT decider, plus the publication root theorem.", "title": "Concrete standard P-versus-NP target and root theorem", "theoremRows": [ { "name": "PNP.Main.p_eq_np", "present": false, "kind": null, "axioms": null, "actualKernelTypeSha256": null, "expectedKernelTypeSha256": null, "kernelTypeFingerprintMatches": false } ], "allPresent": false, "allAssumptionFree": false, "axiomClosureUsesOnlyLeanStandardAllowlist": false, "allKernelTypesMatch": false, "sourceClosureFingerprintMatches": true, "earned": false, "status": "not-formalized" } ], "canonicalReportCoordinate": "PNP-CANONICAL-FORMAL-RECONSTRUCTION-REPORT-2026-08-12-134", "canonicalReportSource": "canonical_proof_report.tex", "canonicalReportPdf": "canonical_proof_report.pdf", "canonicalReportDerivedFromLeanInventory": true }