https://pnplabs.com.au/updates.html Plain-language updates with source-bound technical details for newly earned formal milestones. 2026-08-12T09:30:00Z PNP Labs https://pnplabs.com.au/updates.html#2026-08-12-residual-terminal-pkgc-ambient-bn4-ledger 2026-08-12T09:30:00Z 2026-08-12T09:30:00Z

For an arbitrary finite explicit BN4 cell ledger, Lean now accepts a proof-bearing exact multiset embedding of the generated PkgC opposite-sign cancellation cells. The embedding preserves every duplicate and proves that the ambient ledger is exactly the generated balanced subledger followed by an explicit remainder. At every complete BN4 key, positive and negative mass split across the generated cells and remainder, so both ambient signed mass and the executable residual signed contribution equal the remainder. A successful candidate-derived BN4 kernel also ties every generated cell to its canonical request-atom space, and complete bindings plus exact absence of every computed bridge imply V54 singletonization. The ambient ledger, typed restorer, exact permutation certificate or canonical serialization, and successful kernel remain proof-bearing inputs. Lean has not derived those inputs from a terminal candidate, proved global PkgC route silence or the full historical PkgC theorem, completed global routing, BN6 or Packet selector-realizer completeness, polynomial runtime, ZeroSlack, or PCCMin, put SAT in P, removed a project assumption, or proved P = NP. The 94% figure is a revisable editorial estimate of known reconstruction work, separate from the 109 of 111 scoped formal-publication rows and not a probability that the claim is correct. Editorial progress estimate at publication: 94 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>For an arbitrary finite explicit BN4 cell ledger, Lean now accepts a proof-bearing exact multiset embedding of the generated PkgC opposite-sign cancellation cells. The embedding preserves every duplicate and proves that the ambient ledger is exactly the generated balanced subledger followed by an explicit remainder.</p><p>At every complete BN4 key, positive and negative mass split across the generated cells and remainder, so both ambient signed mass and the executable residual signed contribution equal the remainder. A successful candidate-derived BN4 kernel also ties every generated cell to its canonical request-atom space, and complete bindings plus exact absence of every computed bridge imply V54 singletonization. The ambient ledger, typed restorer, exact permutation certificate or canonical serialization, and successful kernel remain proof-bearing inputs. Lean has not derived those inputs from a terminal candidate, proved global PkgC route silence or the full historical PkgC theorem, completed global routing, BN6 or Packet selector-realizer completeness, polynomial runtime, ZeroSlack, or PCCMin, put SAT in P, removed a project assumption, or proved P = NP. The 94% figure is a revisable editorial estimate of known reconstruction work, separate from the 109 of 111 scoped formal-publication rows and not a probability that the claim is correct.</p><p data-progress-estimate-percent="94">Editorial progress estimate at publication: 94%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-12-residual-terminal-pkgc-ambient-bn4-ledger">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-12-residual-terminal-pkgc-same-key-cancellation 2026-08-12T04:11:09Z 2026-08-12T04:11:09Z

For every atom of the canonical first disjoint nonsingleton pair, Lean now constructs one positive unit cell for the quotient atom and one negative unit cell for its typed restored candidate. Exact preservation of the complete BN5 coordinate proves that each pair has the same nested BN4 key. Lean proves the exact cell count, equal positive and negative multiplicity at every BN4 key, an empty computed residual, and zero signed mass. Its total classifier returns either V54 singletonization or a proof-bearing cancellation realization, and exact absence of every cancellation realization forces singletonization. The typed restorer and coordinate maps remain explicit, and the generated cells are not yet identified with a terminal candidate's ambient BN4 ledger. Lean has not completed global route integration or silence, the full historical PkgC result, BN6 or Packet selector-realizer completeness, polynomial runtime, ZeroSlack, or PCCMin, put SAT in P, removed a project assumption, or proved P = NP. The 93% figure is a revisable editorial estimate of known reconstruction work, separate from the 108 of 110 scoped formal-publication rows and not a probability that the claim is correct. Editorial progress estimate at publication: 93 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>For every atom of the canonical first disjoint nonsingleton pair, Lean now constructs one positive unit cell for the quotient atom and one negative unit cell for its typed restored candidate. Exact preservation of the complete BN5 coordinate proves that each pair has the same nested BN4 key.</p><p>Lean proves the exact cell count, equal positive and negative multiplicity at every BN4 key, an empty computed residual, and zero signed mass. Its total classifier returns either V54 singletonization or a proof-bearing cancellation realization, and exact absence of every cancellation realization forces singletonization. The typed restorer and coordinate maps remain explicit, and the generated cells are not yet identified with a terminal candidate&#39;s ambient BN4 ledger. Lean has not completed global route integration or silence, the full historical PkgC result, BN6 or Packet selector-realizer completeness, polynomial runtime, ZeroSlack, or PCCMin, put SAT in P, removed a project assumption, or proved P = NP. The 93% figure is a revisable editorial estimate of known reconstruction work, separate from the 108 of 110 scoped formal-publication rows and not a probability that the claim is correct.</p><p data-progress-estimate-percent="93">Editorial progress estimate at publication: 93%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-12-residual-terminal-pkgc-same-key-cancellation">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-12-residual-terminal-pkgc-typed-restoration 2026-08-12T00:57:40Z 2026-08-12T00:57:40Z

For an arbitrary finite explicitly supplied minimal-consumer antichain and an explicitly supplied typed coordinate-preserving restoration operation, Lean now materializes one full-restoration candidate for every atom of the canonical first disjoint nonsingleton pair. It proves the exact candidate count and that every position keeps the source atom's complete coordinate. The resulting equality-fibre graph has exact full and shadow multiplicities, gives complete coverage, and cannot satisfy a strict Hall deficit; when no nonsingleton pair exists, the classifier instead proves the V54 singletonization premise. The typed restoration operation remains caller-supplied, and Lean has not constructed it from a terminal candidate, proved its full semantic adequacy, connected restoration coverage to a BN4 or BN5 contradiction, completed PkgC route silence or global routing, established polynomial runtime, ZeroSlack, or PCCMin, put SAT in P, removed a project assumption, or proved P = NP. The 92% figure is a revisable editorial estimate of known reconstruction work, separate from the 107 of 109 scoped formal-publication rows and not a probability that the claim is correct. Editorial progress estimate at publication: 92 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>For an arbitrary finite explicitly supplied minimal-consumer antichain and an explicitly supplied typed coordinate-preserving restoration operation, Lean now materializes one full-restoration candidate for every atom of the canonical first disjoint nonsingleton pair. It proves the exact candidate count and that every position keeps the source atom&#39;s complete coordinate.</p><p>The resulting equality-fibre graph has exact full and shadow multiplicities, gives complete coverage, and cannot satisfy a strict Hall deficit; when no nonsingleton pair exists, the classifier instead proves the V54 singletonization premise. The typed restoration operation remains caller-supplied, and Lean has not constructed it from a terminal candidate, proved its full semantic adequacy, connected restoration coverage to a BN4 or BN5 contradiction, completed PkgC route silence or global routing, established polynomial runtime, ZeroSlack, or PCCMin, put SAT in P, removed a project assumption, or proved P = NP. The 92% figure is a revisable editorial estimate of known reconstruction work, separate from the 107 of 109 scoped formal-publication rows and not a probability that the claim is correct.</p><p data-progress-estimate-percent="92">Editorial progress estimate at publication: 92%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-12-residual-terminal-pkgc-typed-restoration">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-12-residual-terminal-pkgc-separating-consumers 2026-08-11T19:58:23Z 2026-08-11T19:58:23Z

For an arbitrary finite explicitly supplied minimal-consumer antichain, Lean now scans in a deterministic order for the first disjoint pair that is not singleton-singleton. If no such pair exists, it proves exactly the singletonization premise used by the V54 normal form. If a pair is found, Lean canonically indexes its atoms as exact-coordinate quotient units and compares them with an explicit finite full-restoration universe. The classifier returns complete multiplicity coverage or a strict Hall deficit with a deterministic local Q route, and every restoration edge preserves the full coordinate. The consumer antichain and restoration universe remain explicit inputs. Lean has not derived them from terminal candidates, connected complete coverage back to a BN4 or BN5 contradiction, embedded the local route into the complete global outcome system, proved global route silence or the full historical PkgC result, completed BN6 or Packet selectors and realizers, established polynomial runtime, ZeroSlack, or PCCMin, put SAT in P, removed a project assumption, or proved P = NP. The 91% figure is a revisable editorial estimate of known reconstruction work, separate from the 106 of 108 scoped formal-publication rows and not a probability that the claim is correct. Editorial progress estimate at publication: 91 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>For an arbitrary finite explicitly supplied minimal-consumer antichain, Lean now scans in a deterministic order for the first disjoint pair that is not singleton-singleton. If no such pair exists, it proves exactly the singletonization premise used by the V54 normal form. If a pair is found, Lean canonically indexes its atoms as exact-coordinate quotient units and compares them with an explicit finite full-restoration universe.</p><p>The classifier returns complete multiplicity coverage or a strict Hall deficit with a deterministic local Q route, and every restoration edge preserves the full coordinate. The consumer antichain and restoration universe remain explicit inputs. Lean has not derived them from terminal candidates, connected complete coverage back to a BN4 or BN5 contradiction, embedded the local route into the complete global outcome system, proved global route silence or the full historical PkgC result, completed BN6 or Packet selectors and realizers, established polynomial runtime, ZeroSlack, or PCCMin, put SAT in P, removed a project assumption, or proved P = NP. The 91% figure is a revisable editorial estimate of known reconstruction work, separate from the 106 of 108 scoped formal-publication rows and not a probability that the claim is correct.</p><p data-progress-estimate-percent="91">Editorial progress estimate at publication: 91%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-12-residual-terminal-pkgc-separating-consumers">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-12-residual-terminal-bn6-hypergraph-packet 2026-08-11T16:31:22Z 2026-08-11T16:31:22Z

For an arbitrary finite set of anchors and an explicitly supplied, already-grouped family of positive survivor cells carrying payloads, Lean now converts their V54 cut activation into the exact V53 hypergraph cut sum. Given one common positive value on every nonempty proper cut, it returns the two-anchor pair case, the three-anchor mixed balanced-triple or full-span case, or the four-or-more-anchor full-span case, together with witnesses back to the original payloads. The survivor family, grouping, payloads, PkgC singletonization proofs, and constant-cut equation remain explicit inputs. Lean has not constructed PkgC, derived or grouped the survivors from terminal candidates, established the full historical BN6 or Packet selector and realizer results, completed global routes, ZeroSlack, PCCMin, or polynomial runtime, put SAT in P, removed a project assumption, or proved P = NP. The 90% figure is a revisable editorial estimate of known reconstruction work, separate from the 105 of 107 scoped formal-publication rows and not a probability that the claim is correct. Editorial progress estimate at publication: 90 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>For an arbitrary finite set of anchors and an explicitly supplied, already-grouped family of positive survivor cells carrying payloads, Lean now converts their V54 cut activation into the exact V53 hypergraph cut sum. Given one common positive value on every nonempty proper cut, it returns the two-anchor pair case, the three-anchor mixed balanced-triple or full-span case, or the four-or-more-anchor full-span case, together with witnesses back to the original payloads.</p><p>The survivor family, grouping, payloads, PkgC singletonization proofs, and constant-cut equation remain explicit inputs. Lean has not constructed PkgC, derived or grouped the survivors from terminal candidates, established the full historical BN6 or Packet selector and realizer results, completed global routes, ZeroSlack, PCCMin, or polynomial runtime, put SAT in P, removed a project assumption, or proved P = NP. The 90% figure is a revisable editorial estimate of known reconstruction work, separate from the 105 of 107 scoped formal-publication rows and not a probability that the claim is correct.</p><p data-progress-estimate-percent="90">Editorial progress estimate at publication: 90%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-12-residual-terminal-bn6-hypergraph-packet">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-11-residual-terminal-constant-cut-hypergraph-rigidity 2026-08-11T11:31:58Z 2026-08-11T11:31:58Z

For an arbitrary finite duplicate-free carrier and an explicitly supplied sparse positive weighted hypergraph, Lean now classifies every case in which all nonempty proper cuts have the same positive value. With two anchors the full-span weight is that value. With three anchors all pair weights agree and the full-span weight plus twice the common pair weight is that value. With four or more anchors every proper footprint has zero weight and the full-span weight is that value. The hypergraph and constant-cut proof remain explicit inputs. Lean has not constructed PkgC, derived the hypergraph from terminal candidates or the V54 consumer system, built BN6 cells or payloads, completed global routes, ZeroSlack, PCCMin, or polynomial runtime, put SAT in P, removed a project assumption, or proved P = NP. The 89% figure is a revisable editorial estimate of known reconstruction work, separate from the 104 of 106 scoped formal-publication rows and not a probability that the claim is correct. Editorial progress estimate at publication: 89 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>For an arbitrary finite duplicate-free carrier and an explicitly supplied sparse positive weighted hypergraph, Lean now classifies every case in which all nonempty proper cuts have the same positive value. With two anchors the full-span weight is that value. With three anchors all pair weights agree and the full-span weight plus twice the common pair weight is that value. With four or more anchors every proper footprint has zero weight and the full-span weight is that value.</p><p>The hypergraph and constant-cut proof remain explicit inputs. Lean has not constructed PkgC, derived the hypergraph from terminal candidates or the V54 consumer system, built BN6 cells or payloads, completed global routes, ZeroSlack, PCCMin, or polynomial runtime, put SAT in P, removed a project assumption, or proved P = NP. The 89% figure is a revisable editorial estimate of known reconstruction work, separate from the 104 of 106 scoped formal-publication rows and not a probability that the claim is correct.</p><p data-progress-estimate-percent="89">Editorial progress estimate at publication: 89%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-11-residual-terminal-constant-cut-hypergraph-rigidity">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-11-residual-terminal-consumer-antichain-normal-form 2026-08-11T07:40:05Z 2026-08-11T07:40:05Z

For an arbitrary finite carrier and an explicitly supplied antichain of minimal consumers, Lean now proves that requests are monotone, the empty request is inactive, and two-sided cut activation is nonzero exactly when two consumers are disjoint. If every disjoint consumer pair is singletonized, the exact premise required by the manuscript's PkgC stage, Lean proves literal equality with the cut indicator of the singleton footprint. The minimal-consumer antichain and singletonization proof remain explicit inputs. Lean has not constructed PkgC or route silence, derived these inputs from terminal candidates, completed V53 or BN6, established complete global routes, ZeroSlack, PCCMin, or polynomial runtime, put SAT in P, removed a project assumption, or proved P = NP. The 88% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 88 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>For an arbitrary finite carrier and an explicitly supplied antichain of minimal consumers, Lean now proves that requests are monotone, the empty request is inactive, and two-sided cut activation is nonzero exactly when two consumers are disjoint. If every disjoint consumer pair is singletonized, the exact premise required by the manuscript&#39;s PkgC stage, Lean proves literal equality with the cut indicator of the singleton footprint.</p><p>The minimal-consumer antichain and singletonization proof remain explicit inputs. Lean has not constructed PkgC or route silence, derived these inputs from terminal candidates, completed V53 or BN6, established complete global routes, ZeroSlack, PCCMin, or polynomial runtime, put SAT in P, removed a project assumption, or proved P = NP. The 88% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="88">Editorial progress estimate at publication: 88%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-11-residual-terminal-consumer-antichain-normal-form">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-11-residual-terminal-bn5-full-shadow-localization 2026-08-11T04:17:43Z 2026-08-11T04:17:43Z

Starting from one verified negative BN4 cancellation result, Lean now expands its mass into explicit unit records and compares those records with an explicit finite list of quotient-shadow coordinates. It checks whether the chosen cut is silent. When it is active, the classifier either returns complete multiplicity coverage or proves a strict Hall deficit: a specific group of full records has fewer distinct shadow neighbours, and the resulting local route cannot disappear silently. The payloads, cut, and shadow universe remain explicit inputs rather than constructions from the four-corner bases. Complete matching is not yet connected back to a BN4 contradiction, and the full historical BN5 diagnosis, later package and BN6 stages, complete global routing, polynomial runtime, ZeroSlack, PCCMin, SAT in P, removal of a project assumption, and P = NP remain unproved. The 87% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 87 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>Starting from one verified negative BN4 cancellation result, Lean now expands its mass into explicit unit records and compares those records with an explicit finite list of quotient-shadow coordinates. It checks whether the chosen cut is silent. When it is active, the classifier either returns complete multiplicity coverage or proves a strict Hall deficit: a specific group of full records has fewer distinct shadow neighbours, and the resulting local route cannot disappear silently.</p><p>The payloads, cut, and shadow universe remain explicit inputs rather than constructions from the four-corner bases. Complete matching is not yet connected back to a BN4 contradiction, and the full historical BN5 diagnosis, later package and BN6 stages, complete global routing, polynomial runtime, ZeroSlack, PCCMin, SAT in P, removal of a project assumption, and P = NP remain unproved. The 87% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="87">Editorial progress estimate at publication: 87%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-11-residual-terminal-bn5-full-shadow-localization">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-11-residual-terminal-bn4-activation-cancellation 2026-08-11T00:32:18Z 2026-08-11T00:32:18Z

After the existing finite request-envelope classifier succeeds, Lean now gives each request identity one exact singleton activation code and compares complete typed cancellation keys without enumerating cuts. For an explicit ledger of positive and negative cells, it totals mass only at the same complete key, returns a canonical balanced, positive, or negative residual, and proves exact integer mass conservation, preserved key identity, positive residual mass, and absence of opposite-sign residual pairs. The ledger, semantic signatures, and transport types are supplied explicitly rather than derived from the four-corner bases, so this is a finite cancellation kernel rather than the full historical BN4 theorem. It has no polynomial construction or size bound and does not construct BN5, PkgC, or BN6; complete global routes, selectors, or realizers; ZeroSlack or PCCMin; SAT in P; removal of a project assumption; or P = NP. The 86% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 86 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>After the existing finite request-envelope classifier succeeds, Lean now gives each request identity one exact singleton activation code and compares complete typed cancellation keys without enumerating cuts. For an explicit ledger of positive and negative cells, it totals mass only at the same complete key, returns a canonical balanced, positive, or negative residual, and proves exact integer mass conservation, preserved key identity, positive residual mass, and absence of opposite-sign residual pairs.</p><p>The ledger, semantic signatures, and transport types are supplied explicitly rather than derived from the four-corner bases, so this is a finite cancellation kernel rather than the full historical BN4 theorem. It has no polynomial construction or size bound and does not construct BN5, PkgC, or BN6; complete global routes, selectors, or realizers; ZeroSlack or PCCMin; SAT in P; removal of a project assumption; or P = NP. The 86% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="86">Editorial progress estimate at publication: 86%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-11-residual-terminal-bn4-activation-cancellation">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-10-residual-terminal-bn3-request-envelope 2026-08-10T20:51:36Z 2026-08-10T20:51:36Z

After the existing finite anchor classifier succeeds, Lean now constructs one canonical duplicate-free list of request identities across every proper cut. It proves that executable request membership is exact, monotone, and stable, computes exact singleton minimal consumers, accounts for each active incidence once, and selects one shared full-or-quotient side-tight basis family for all cuts while preserving every earlier proof-bearing failure. This is an exact finite reference construction, but it checks every subset of the anchor family and can therefore take exponential time. It does not construct the later BN4 through BN6 stages, prove complete selector or realizer coverage, establish global ZeroSlack or polynomial PCCMin, put SAT in P, remove a project assumption, or prove P = NP. The 85% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 85 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>After the existing finite anchor classifier succeeds, Lean now constructs one canonical duplicate-free list of request identities across every proper cut. It proves that executable request membership is exact, monotone, and stable, computes exact singleton minimal consumers, accounts for each active incidence once, and selects one shared full-or-quotient side-tight basis family for all cuts while preserving every earlier proof-bearing failure.</p><p>This is an exact finite reference construction, but it checks every subset of the anchor family and can therefore take exponential time. It does not construct the later BN4 through BN6 stages, prove complete selector or realizer coverage, establish global ZeroSlack or polynomial PCCMin, put SAT in P, remove a project assumption, or prove P = NP. The 85% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="85">Editorial progress estimate at publication: 85%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-10-residual-terminal-bn3-request-envelope">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-10-concrete-locked-nand-threshold-publication 2026-08-10T14:25:00Z 2026-08-10T14:25:00Z

Lean now proves one uniform polynomial-time transformation from encoded CNF satisfiability instances to the concrete locked-NAND threshold language. The witness is the fixed parser, circuit compiler, and locked-NAND emitter pipeline, and the theorem applies to every input bitstring with fail-closed malformed-input behavior. The theorem depends only on standard Lean logical principles, not on the legacy project assumption with a similar name. It does not put the target language in P, prove concrete CNF-SAT NP-hardness, discharge residual-band minimization, ZeroSlack or PCCMin, establish polynomial certificate checking for the final route, remove the remaining project assumptions, or prove P = NP. The 84% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 84 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>Lean now proves one uniform polynomial-time transformation from encoded CNF satisfiability instances to the concrete locked-NAND threshold language. The witness is the fixed parser, circuit compiler, and locked-NAND emitter pipeline, and the theorem applies to every input bitstring with fail-closed malformed-input behavior.</p><p>The theorem depends only on standard Lean logical principles, not on the legacy project assumption with a similar name. It does not put the target language in P, prove concrete CNF-SAT NP-hardness, discharge residual-band minimization, ZeroSlack or PCCMin, establish polynomial certificate checking for the final route, remove the remaining project assumptions, or prove P = NP. The 84% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="84">Editorial progress estimate at publication: 84%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-10-concrete-locked-nand-threshold-publication">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-10-residual-terminal-rank-wf 2026-08-10T11:35:00Z 2026-08-10T11:35:00Z

Lean now defines the manuscript's residual terminal rank as exactly ten natural-number coordinates in the stated priority order. It proves that the executable Boolean comparison agrees with the lexicographic proposition, supplies a proof for each of the ten possible first-decreasing coordinates, packages proof-bearing descent, and proves accessibility, induction, and kernel-checked well-foundedness for the fixed rank. This establishes the 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 current route strictly decreases the rank, establish route completeness or Package E, remove the explicit positive premise from the finite composition, establish manuscript-wide SaturatePositive or BCELReady, prove ZeroSlack or PCCMin, establish polynomial runtime, put SAT in P, remove a project assumption, or prove P = NP. The 83% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 83 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>Lean now defines the manuscript&#39;s residual terminal rank as exactly ten natural-number coordinates in the stated priority order. It proves that the executable Boolean comparison agrees with the lexicographic proposition, supplies a proof for each of the ten possible first-decreasing coordinates, packages proof-bearing descent, and proves accessibility, induction, and kernel-checked well-foundedness for the fixed rank.</p><p>This establishes the rank domain and RankWF only. It does not map the current finite terminal routes into the manuscript&#39;s complete global outcome system, prove that any current route strictly decreases the rank, establish route completeness or Package E, remove the explicit positive premise from the finite composition, establish manuscript-wide SaturatePositive or BCELReady, prove ZeroSlack or PCCMin, establish polynomial runtime, put SAT in P, remove a project assumption, or prove P = NP. The 83% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="83">Editorial progress estimate at publication: 83%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-10-residual-terminal-rank-wf">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-10-residual-terminal-finite-saturate-positive-composition 2026-08-10T04:52:48Z 2026-08-10T04:52:48Z

For every finite direct-wire candidate, executable observer, forgetful projection, and proof-bearing terminal anchor problem whose normalized starting point has positive full slack, Lean now checks the candidate-derived origin, kernel, and obligation closures in both gate and profile orientations. It verifies that each safe step is cost transparent, discharges its closure obligation, preserves the forgotten profile, and preserves positive full slack through the complete safe prefix into either the checked-lift or BCEL firewall. Otherwise it returns the exact first local route or nontransparent event together with that complete safe prefix. This composes the five reconstructed finite terminal obligations only for an explicit proof-bearing problem and closes the finite local form of originKernelObligationClosureRouted. A returned local route is not a complete global outcome, full Package E acceptance, verified gain, or proof of global route completeness, and the initial positive full-slack premise remains explicit. Lean has not established manuscript-wide SaturatePositive, BCELReady or RankWF, proved ZeroSlack or PCCMin, established polynomial runtime, put SAT in P, removed a project assumption, or proved P = NP. The 82% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 82 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>For every finite direct-wire candidate, executable observer, forgetful projection, and proof-bearing terminal anchor problem whose normalized starting point has positive full slack, Lean now checks the candidate-derived origin, kernel, and obligation closures in both gate and profile orientations. It verifies that each safe step is cost transparent, discharges its closure obligation, preserves the forgotten profile, and preserves positive full slack through the complete safe prefix into either the checked-lift or BCEL firewall. Otherwise it returns the exact first local route or nontransparent event together with that complete safe prefix.</p><p>This composes the five reconstructed finite terminal obligations only for an explicit proof-bearing problem and closes the finite local form of originKernelObligationClosureRouted. A returned local route is not a complete global outcome, full Package E acceptance, verified gain, or proof of global route completeness, and the initial positive full-slack premise remains explicit. Lean has not established manuscript-wide SaturatePositive, BCELReady or RankWF, proved ZeroSlack or PCCMin, established polynomial runtime, put SAT in P, removed a project assumption, or proved P = NP. The 82% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="82">Editorial progress estimate at publication: 82%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-10-residual-terminal-finite-saturate-positive-composition">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-10-residual-terminal-interface-exposure-routing 2026-08-09T23:52:23Z 2026-08-09T23:52:23Z

For every finite direct-wire candidate, executable ambient observer, forgetful projection, and finite terminal seed, Lean now recognizes only an exact interface-consumer edge derived from that candidate. Each recognized event is either proved transparently cost-balanced or turned into a proof-bearing local E-route. Across a saturation trace, Lean records the exact first interface-exposure event and the complete transparent prefix. This closes only the finite local form of interfaceExposureRoutesToE. The local E-route identifies an exposure obligation; it is not a full Package E acceptance, verified global gain, or proof that every global route has been found. The observer and projection remain explicit inputs, and Lean has not discharged originKernelObligationClosureRouted, established full SaturatePositive, Package E or BCELReady, proved ZeroSlack or PCCMin, established polynomial runtime, put SAT in P, removed a project assumption, or proved P = NP. The 81% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 81 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>For every finite direct-wire candidate, executable ambient observer, forgetful projection, and finite terminal seed, Lean now recognizes only an exact interface-consumer edge derived from that candidate. Each recognized event is either proved transparently cost-balanced or turned into a proof-bearing local E-route. Across a saturation trace, Lean records the exact first interface-exposure event and the complete transparent prefix.</p><p>This closes only the finite local form of interfaceExposureRoutesToE. The local E-route identifies an exposure obligation; it is not a full Package E acceptance, verified global gain, or proof that every global route has been found. The observer and projection remain explicit inputs, and Lean has not discharged originKernelObligationClosureRouted, established full SaturatePositive, Package E or BCELReady, proved ZeroSlack or PCCMin, established polynomial runtime, put SAT in P, removed a project assumption, or proved P = NP. The 81% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="81">Editorial progress estimate at publication: 81%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-10-residual-terminal-interface-exposure-routing">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-09-residual-terminal-candidate-saturation-cost-balance 2026-08-09T15:28:29Z 2026-08-09T15:28:29Z

For every finite direct-wire candidate, executable ambient observer, forgetful projection, and finite terminal seed, Lean now derives the physical and context-sensitive dependency system from the candidate itself and computes a deterministic rule-labelled saturation trace. If every event is transparent, Lean proves exact support and full-circuit cost balance while preserving full slack, positivity, and a nondecreasing projection defect across the linked history. If an event is not transparent, Lean records the exact first event, its typed reason, and the complete transparent prefix instead of silently continuing. This closes only the finite terminal forms of transparentSaturationCostBalanced and firstNontransparentStepRecorded. The observer and projection remain explicit inputs, and Lean has not routed a nontransparent event, discharged interfaceExposureRoutesToE or originKernelObligationClosureRouted, established full SaturatePositive, Package E or BCELReady, proved ZeroSlack or PCCMin, established polynomial runtime, put SAT in P, removed a project assumption, or proved P = NP. The 80% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 80 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>For every finite direct-wire candidate, executable ambient observer, forgetful projection, and finite terminal seed, Lean now derives the physical and context-sensitive dependency system from the candidate itself and computes a deterministic rule-labelled saturation trace. If every event is transparent, Lean proves exact support and full-circuit cost balance while preserving full slack, positivity, and a nondecreasing projection defect across the linked history.</p><p>If an event is not transparent, Lean records the exact first event, its typed reason, and the complete transparent prefix instead of silently continuing. This closes only the finite terminal forms of transparentSaturationCostBalanced and firstNontransparentStepRecorded. The observer and projection remain explicit inputs, and Lean has not routed a nontransparent event, discharged interfaceExposureRoutesToE or originKernelObligationClosureRouted, established full SaturatePositive, Package E or BCELReady, proved ZeroSlack or PCCMin, established polynomial runtime, put SAT in P, removed a project assumption, or proved P = NP. The 80% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="80">Editorial progress estimate at publication: 80%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-09-residual-terminal-candidate-saturation-cost-balance">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-09-residual-terminal-saturation-positivity-firewall 2026-08-09T10:51:52Z 2026-08-09T10:51:52Z

For an explicit finite direct-wire candidate, terminal dependency system, already computed governed proper-positive support, forgetful projection, and executable observer, Lean now computes the whole-support projection defect and handles both possible cases. If the defect is zero, it returns an attained reduced minimum together with a checked full lift. If the defect is positive, it delegates exactly to the existing fail-closed BCEL anchor-nucleus classifier. This closes only the named projectionPositivityNotLostSilently obligation in the current finite terminal model. It does not derive the dependency system or support, discharge the other four SaturatePositive obligations, establish full SaturatePositive, Package E or BCELReady, prove ZeroSlack or PCCMin, establish polynomial runtime, put SAT in P, or prove P = NP. The 79% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 79 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>For an explicit finite direct-wire candidate, terminal dependency system, already computed governed proper-positive support, forgetful projection, and executable observer, Lean now computes the whole-support projection defect and handles both possible cases. If the defect is zero, it returns an attained reduced minimum together with a checked full lift. If the defect is positive, it delegates exactly to the existing fail-closed BCEL anchor-nucleus classifier.</p><p>This closes only the named projectionPositivityNotLostSilently obligation in the current finite terminal model. It does not derive the dependency system or support, discharge the other four SaturatePositive obligations, establish full SaturatePositive, Package E or BCELReady, prove ZeroSlack or PCCMin, establish polynomial runtime, put SAT in P, or prove P = NP. The 79% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="79">Editorial progress estimate at publication: 79%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-09-residual-terminal-saturation-positivity-firewall">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-09-residual-terminal-computed-bcel-anchor-nucleus 2026-08-09T07:32:49Z 2026-08-09T07:32:49Z

For an explicit finite direct-wire candidate, terminal dependency system, governed proper-positive support, forgetful projection, and executable observer, Lean now enumerates every candidate anchor subfamily and selects the unique canonical minimum-cardinality one with positive projection defect. It then checks the anchor algebra and every proper cut in a deterministic order. The classifier is fail-closed: it returns an insufficient nucleus, the exact first anchor-algebra mismatch, the exact first proper-cut defect mismatch, the first proof-bearing full-before-reduced local route, or exact constant-cut and local BN2 conclusions for every proper cut. This milestone assumes both the terminal dependency system and a positive whole-support projection defect. It does not derive either premise, identify the manuscript's activation or charge classes, connect every local failure to the complete global route system, or establish SaturatePositive, Package E, BCELReady, ZeroSlack, PCCMin, polynomial runtime, SAT in P, or P = NP. The 78% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 78 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>For an explicit finite direct-wire candidate, terminal dependency system, governed proper-positive support, forgetful projection, and executable observer, Lean now enumerates every candidate anchor subfamily and selects the unique canonical minimum-cardinality one with positive projection defect. It then checks the anchor algebra and every proper cut in a deterministic order.</p><p>The classifier is fail-closed: it returns an insufficient nucleus, the exact first anchor-algebra mismatch, the exact first proper-cut defect mismatch, the first proof-bearing full-before-reduced local route, or exact constant-cut and local BN2 conclusions for every proper cut. This milestone assumes both the terminal dependency system and a positive whole-support projection defect. It does not derive either premise, identify the manuscript&#39;s activation or charge classes, connect every local failure to the complete global route system, or establish SaturatePositive, Package E, BCELReady, ZeroSlack, PCCMin, polynomial runtime, SAT in P, or P = NP. The 78% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="78">Editorial progress estimate at publication: 78%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-09-residual-terminal-computed-bcel-anchor-nucleus">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-09-residual-terminal-computed-bn2-square-legitimacy 2026-08-09T03:20:33Z 2026-08-09T03:20:33Z

Given two finite seed lists and one explicit set of terminal dependency rules, Lean now computes the four related terminal-support views, completes the governed frontier at every corner, and proves that the frontier and its reduced projection form the exact compatible square required by this local model. It also keeps the full and reduced minimum quantities on one checked carrier, so the comparison does not silently change what is being measured. The final checker is fail-closed: when the exact local route queries are silent, Lean returns the complete local square conclusion; otherwise it returns the first full-then-reduced mismatch together with a proof that the route is real. This is computed BN2 square legitimacy for the explicit finite data, not the manuscript's unrestricted conclusion. Lean has not derived the dependency rules from an arbitrary circuit, proved universal route silence, connected every local failure to the complete global no-outcome route system, identified a BCEL anchor square, or completed SaturatePositive, ZeroSlack, PCCMin, polynomial runtime, SAT in P, or P = NP. The 77% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 77 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>Given two finite seed lists and one explicit set of terminal dependency rules, Lean now computes the four related terminal-support views, completes the governed frontier at every corner, and proves that the frontier and its reduced projection form the exact compatible square required by this local model. It also keeps the full and reduced minimum quantities on one checked carrier, so the comparison does not silently change what is being measured.</p><p>The final checker is fail-closed: when the exact local route queries are silent, Lean returns the complete local square conclusion; otherwise it returns the first full-then-reduced mismatch together with a proof that the route is real. This is computed BN2 square legitimacy for the explicit finite data, not the manuscript&#39;s unrestricted conclusion. Lean has not derived the dependency rules from an arbitrary circuit, proved universal route silence, connected every local failure to the complete global no-outcome route system, identified a BCEL anchor square, or completed SaturatePositive, ZeroSlack, PCCMin, polynomial runtime, SAT in P, or P = NP. The 77% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="77">Editorial progress estimate at publication: 77%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-09-residual-terminal-computed-bn2-square-legitimacy">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-08-residual-terminal-four-corner-tight-basis-maximum 2026-08-08T11:58:13Z 2026-08-08T11:58:13Z

A computer circuit is a network of simple yes-or-no operations. This part of the proposed proof compares four related views of one circuit. Each view can have several different designs tied for the smallest size. Lean now lists every smallest design for all four views, tries every possible four-way combination, and keeps exactly the combinations that agree where the views overlap. When the existing local mismatch check finds no problem, Lean proves that at least one matching combination remains and that every retained combination gives the same signed number. It therefore calculates the exact largest value without incorrectly replacing a negative answer with zero. This works in both the full and reduced comparison modes. Lean has not proved that every square passes the local check, established the manuscript's BN2 square-legitimacy theorem, or completed SaturatePositive, ZeroSlack, PCCMin, polynomial runtime, SAT in P, or P = NP. The 76% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 76 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A computer circuit is a network of simple yes-or-no operations. This part of the proposed proof compares four related views of one circuit. Each view can have several different designs tied for the smallest size. Lean now lists every smallest design for all four views, tries every possible four-way combination, and keeps exactly the combinations that agree where the views overlap.</p><p>When the existing local mismatch check finds no problem, Lean proves that at least one matching combination remains and that every retained combination gives the same signed number. It therefore calculates the exact largest value without incorrectly replacing a negative answer with zero. This works in both the full and reduced comparison modes. Lean has not proved that every square passes the local check, established the manuscript&#39;s BN2 square-legitimacy theorem, or completed SaturatePositive, ZeroSlack, PCCMin, polynomial runtime, SAT in P, or P = NP. The 76% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="76">Editorial progress estimate at publication: 76%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-08-residual-terminal-four-corner-tight-basis-maximum">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-08-residual-terminal-four-corner-side-tight-completion 2026-08-08T07:38:25Z 2026-08-08T07:38:25Z

This section of the proposed proof compares four related smallest circuit views. The previous update gave Lean a fixed checklist for finding the first local mismatch between them. Lean can now finish the other side of that checklist: when the relevant checks report no mismatch, it constructs one verified package showing that the four selected views fit together in the chosen comparison mode and reach the exact required size balance. This result applies to every finite square covered by the formal model and keeps the full and reduced comparison modes separate. It is conditional on the local checklist finding no problem. Lean has not proved that this happens for every square, connected every local problem to the manuscript's complete obstruction route, or proved the manuscript's BN2 square-legitimacy theorem. Later SaturatePositive, ZeroSlack, PCCMin, polynomial runtime, SAT in P, and P = NP obligations remain open. The 75% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 75 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>This section of the proposed proof compares four related smallest circuit views. The previous update gave Lean a fixed checklist for finding the first local mismatch between them. Lean can now finish the other side of that checklist: when the relevant checks report no mismatch, it constructs one verified package showing that the four selected views fit together in the chosen comparison mode and reach the exact required size balance.</p><p>This result applies to every finite square covered by the formal model and keeps the full and reduced comparison modes separate. It is conditional on the local checklist finding no problem. Lean has not proved that this happens for every square, connected every local problem to the manuscript&#39;s complete obstruction route, or proved the manuscript&#39;s BN2 square-legitimacy theorem. Later SaturatePositive, ZeroSlack, PCCMin, polynomial runtime, SAT in P, and P = NP obligations remain open. The 75% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="75">Editorial progress estimate at publication: 75%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-08-residual-terminal-four-corner-side-tight-completion">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-08-residual-terminal-four-corner-optimum-coherence 2026-08-08T03:05:06Z 2026-08-08T03:05:06Z

This part of the proposed proof compares four related views of a computer circuit, each represented by a smallest known circuit for that view. Lean can now check whether those four choices agree where the views overlap. It checks the four connections in a fixed order, so the result does not depend on guesswork or on which comparison happens to be tried first. If every check passes, Lean returns one package showing exactly how the four choices fit together and retains the size and boundary facts needed by later work. If a check fails, Lean identifies the first exact reason, such as a missing connection, a changed output, a mismatched bookkeeping pattern, or use of the wrong comparison mode. This works for every finite square covered by the model. It does not prove that every square passes, complete the manuscript's square-legitimacy step, or establish P = NP. The 74% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 74 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>This part of the proposed proof compares four related views of a computer circuit, each represented by a smallest known circuit for that view. Lean can now check whether those four choices agree where the views overlap. It checks the four connections in a fixed order, so the result does not depend on guesswork or on which comparison happens to be tried first.</p><p>If every check passes, Lean returns one package showing exactly how the four choices fit together and retains the size and boundary facts needed by later work. If a check fails, Lean identifies the first exact reason, such as a missing connection, a changed output, a mismatched bookkeeping pattern, or use of the wrong comparison mode. This works for every finite square covered by the model. It does not prove that every square passes, complete the manuscript&#39;s square-legitimacy step, or establish P = NP. The 74% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="74">Editorial progress estimate at publication: 74%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-08-residual-terminal-four-corner-optimum-coherence">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-08-residual-terminal-four-corner-optimum-carrier-compatibility 2026-08-07T20:22:49Z 2026-08-07T20:22:49Z

This part of the proposed proof compares four related views of a computer circuit. For each view, earlier checked work identifies a circuit with the fewest gates. Those four smallest circuits could originally use different maps of their positions, which made a direct comparison unsafe. Lean now places all four on one finite common map without changing what any circuit does or how many gates it has. Lean also proves that every real position can be translated to the common map and back exactly, while a missing position is rejected instead of being invented. This works for every finite computed square covered by the model and uses one shared way of observing all four circuit views. It still does not prove that the four smallest circuits fit together consistently along the square's sides, establish the manuscript's square-legitimacy step, or complete the later ZeroSlack, PCCMin, efficient-runtime, SAT-in-P, or P = NP obligations. The 73% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 73 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>This part of the proposed proof compares four related views of a computer circuit. For each view, earlier checked work identifies a circuit with the fewest gates. Those four smallest circuits could originally use different maps of their positions, which made a direct comparison unsafe. Lean now places all four on one finite common map without changing what any circuit does or how many gates it has.</p><p>Lean also proves that every real position can be translated to the common map and back exactly, while a missing position is rejected instead of being invented. This works for every finite computed square covered by the model and uses one shared way of observing all four circuit views. It still does not prove that the four smallest circuits fit together consistently along the square&#39;s sides, establish the manuscript&#39;s square-legitimacy step, or complete the later ZeroSlack, PCCMin, efficient-runtime, SAT-in-P, or P = NP obligations. The 73% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="73">Editorial progress estimate at publication: 73%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-08-residual-terminal-four-corner-optimum-carrier-compatibility">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-07-residual-terminal-four-corner-carrier-transport 2026-08-07T08:01:41Z 2026-08-07T08:01:41Z

This part of the proposed proof compares four related views of a computer circuit: the part two alternatives share, each alternative on its own, and the result of combining them. Lean now gives those views one common map of positions. It proves that every boundary and connection is listed exactly and that no position is accidentally counted twice, so the same wire or bookkeeping item keeps the same identity in all four views. Lean also checks every position that appears on either side. It proves that the position either remains visible after the sides are combined or becomes internal for a verified reason, and its lookup rejects positions that are not actually present. This works for every finite computed square covered by the model, not one selected example. It still does not construct four compatible optimum circuits, one coherent four-corner minimum, or the manuscript's square-legitimacy result. Later minimization, efficient runtime, ZeroSlack, PCCMin, and P = NP remain open. The 72% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 72 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>This part of the proposed proof compares four related views of a computer circuit: the part two alternatives share, each alternative on its own, and the result of combining them. Lean now gives those views one common map of positions. It proves that every boundary and connection is listed exactly and that no position is accidentally counted twice, so the same wire or bookkeeping item keeps the same identity in all four views.</p><p>Lean also checks every position that appears on either side. It proves that the position either remains visible after the sides are combined or becomes internal for a verified reason, and its lookup rejects positions that are not actually present. This works for every finite computed square covered by the model, not one selected example. It still does not construct four compatible optimum circuits, one coherent four-corner minimum, or the manuscript&#39;s square-legitimacy result. Later minimization, efficient runtime, ZeroSlack, PCCMin, and P = NP remain open. The 72% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="72">Editorial progress estimate at publication: 72%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-07-residual-terminal-four-corner-carrier-transport">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-07-residual-terminal-side-tight-minimum-arithmetic 2026-08-07T02:05:20Z 2026-08-07T02:05:20Z

A later part of the proposed proof compares four related views of a computer circuit: the part they share, two alternatives, and their combination. Each view has a smallest possible size under the formal rules. Lean can now check whether all four sizes are exactly minimal and, when they are, calculate their combined difference without losing track of any excess size. The result applies to every finite four-view family covered by this model, not one chosen circuit. The checker fails closed if even one corner is not exactly minimal. An important limitation remains: the four minima may come from four different constructions, so Lean has not yet built one coherent four-corner object that reaches all of them together. Square legitimacy, efficient runtime, ZeroSlack, PCCMin, and P = NP remain open. The 72% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 72 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A later part of the proposed proof compares four related views of a computer circuit: the part they share, two alternatives, and their combination. Each view has a smallest possible size under the formal rules. Lean can now check whether all four sizes are exactly minimal and, when they are, calculate their combined difference without losing track of any excess size.</p><p>The result applies to every finite four-view family covered by this model, not one chosen circuit. The checker fails closed if even one corner is not exactly minimal. An important limitation remains: the four minima may come from four different constructions, so Lean has not yet built one coherent four-corner object that reaches all of them together. Square legitimacy, efficient runtime, ZeroSlack, PCCMin, and P = NP remain open. The 72% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="72">Editorial progress estimate at publication: 72%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-07-residual-terminal-side-tight-minimum-arithmetic">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-06-residual-terminal-governed-projection-square 2026-08-06T14:36:05Z 2026-08-06T14:36:05Z

A computer circuit can be split into overlapping parts. The previous milestone proved that Lean could combine the descriptions of two completed parts. This milestone adds a controlled way to hide selected bookkeeping details and proves that the visible result still has the same physical boundary and shared information. It does this for every finite circuit covered by the formal model and every allowed choice of details to hide, rather than for a fixed example. Combining first and hiding later gives the same visible structure as hiding each side first and then combining them. The dependency rulebook is still supplied, and important mathematical and efficiency steps remain open. This does not prove that P equals NP. The 71% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 71 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A computer circuit can be split into overlapping parts. The previous milestone proved that Lean could combine the descriptions of two completed parts. This milestone adds a controlled way to hide selected bookkeeping details and proves that the visible result still has the same physical boundary and shared information.</p><p>It does this for every finite circuit covered by the formal model and every allowed choice of details to hide, rather than for a fixed example. Combining first and hiding later gives the same visible structure as hiding each side first and then combining them. The dependency rulebook is still supplied, and important mathematical and efficiency steps remain open. This does not prove that P equals NP. The 71% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="71">Editorial progress estimate at publication: 71%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-06-residual-terminal-governed-projection-square">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-06-residual-terminal-governed-frontier-pushout 2026-08-06T10:59:14Z 2026-08-06T10:59:14Z

A circuit is a collection of simple yes-or-no operations connected by wires. Lean could already complete two chosen parts and describe each one's boundary. It can now combine those two descriptions to build the boundary of everything covered by either part, while keeping exactly the information the parts share. Lean proves that this combined description matches a separate calculation made from the completed whole. Each item at either side's boundary is accounted for: it remains exposed or becomes internal to the combined part. The dependency rulebook is still supplied rather than derived from the circuit. The required projection-compatible square, obstruction routing, efficient runtime, ZeroSlack, PCCMin, and P = NP remain open. The 70% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 70 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a collection of simple yes-or-no operations connected by wires. Lean could already complete two chosen parts and describe each one&#39;s boundary. It can now combine those two descriptions to build the boundary of everything covered by either part, while keeping exactly the information the parts share.</p><p>Lean proves that this combined description matches a separate calculation made from the completed whole. Each item at either side&#39;s boundary is accounted for: it remains exposed or becomes internal to the combined part. The dependency rulebook is still supplied rather than derived from the circuit. The required projection-compatible square, obstruction routing, efficient runtime, ZeroSlack, PCCMin, and P = NP remain open. The 70% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="70">Editorial progress estimate at publication: 70%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-06-residual-terminal-governed-frontier-pushout">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-06-residual-terminal-governed-support-completion 2026-08-06T04:20:09Z 2026-08-06T04:20:09Z

A circuit is a collection of simple yes-or-no operations connected by wires. Lean can now take any finite chosen part that has been completed under a supplied dependency rulebook, identify exactly which wires enter and leave it, and sort its selected bookkeeping positions into ten distinct roles. The result is calculated by the formal construction rather than supplied as a separate certificate. Lean proves that every selected record is covered, no profile position is duplicated or assigned to two roles, every required dependency remains present, and the completed part is physically compatible. The same guarantees now apply to all four related parts from the previous milestone. The dependency rulebook is still supplied rather than derived from the circuit, and obstruction routing, the required projection square, efficient runtime, ZeroSlack, PCCMin, and P = NP remain open. The 69% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 69 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a collection of simple yes-or-no operations connected by wires. Lean can now take any finite chosen part that has been completed under a supplied dependency rulebook, identify exactly which wires enter and leave it, and sort its selected bookkeeping positions into ten distinct roles. The result is calculated by the formal construction rather than supplied as a separate certificate.</p><p>Lean proves that every selected record is covered, no profile position is duplicated or assigned to two roles, every required dependency remains present, and the completed part is physically compatible. The same guarantees now apply to all four related parts from the previous milestone. The dependency rulebook is still supplied rather than derived from the circuit, and obstruction routing, the required projection square, efficient runtime, ZeroSlack, PCCMin, and P = NP remain open. The 69% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="69">Editorial progress estimate at publication: 69%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-06-residual-terminal-governed-support-completion">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-06-residual-terminal-saturated-support-square-closure 2026-08-05T23:29:23Z 2026-08-05T23:29:23Z

A circuit is a collection of simple yes-or-no operations connected by wires. Some later proof steps need to compare two chosen parts at once. Lean can now start from any two finite lists under a supplied dependency rulebook, complete both lists, and calculate four related parts: what they share, the completed left and right parts, and everything covered by either side. Lean proves all four parts obey every supplied dependency, the shared part is the largest one contained in both sides, and the combined part is the smallest one containing both. It also rebuilds each part as an induced circuit with the expected inputs and outputs. The rulebook is still supplied rather than derived automatically, and the harder manuscript requirements for a projection-compatible square, routing obstructions, efficient runtime, ZeroSlack, PCCMin, and the final P versus NP theorem remain open. The 68% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 68 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a collection of simple yes-or-no operations connected by wires. Some later proof steps need to compare two chosen parts at once. Lean can now start from any two finite lists under a supplied dependency rulebook, complete both lists, and calculate four related parts: what they share, the completed left and right parts, and everything covered by either side.</p><p>Lean proves all four parts obey every supplied dependency, the shared part is the largest one contained in both sides, and the combined part is the smallest one containing both. It also rebuilds each part as an induced circuit with the expected inputs and outputs. The rulebook is still supplied rather than derived automatically, and the harder manuscript requirements for a projection-compatible square, routing obstructions, efficient runtime, ZeroSlack, PCCMin, and the final P versus NP theorem remain open. The 68% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="68">Editorial progress estimate at publication: 68%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-06-residual-terminal-saturated-support-square-closure">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-05-residual-terminal-proper-positive-support-search 2026-08-05T14:30:00Z 2026-08-05T14:30:00Z

A circuit is a list of simple yes-or-no operations connected by wires. Lean can now examine every possible starting selection from a supplied finite dependency system, complete all recorded dependencies, and test whether the resulting part is nonempty, genuinely smaller than the whole circuit, and improves the exact size comparison. Lean proves that this finite search finds a qualifying part whenever one exists in that governed search space, and that the extracted smaller circuit preserves the intended boundary behaviour. The dependency system is still supplied rather than derived from the circuit, and the search is exhaustive rather than efficient. It does not complete the manuscript's full support or square arguments, finish ZeroSlack or PCCMin, provide a polynomial-time algorithm, or prove P = NP. The 67% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 67 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a list of simple yes-or-no operations connected by wires. Lean can now examine every possible starting selection from a supplied finite dependency system, complete all recorded dependencies, and test whether the resulting part is nonempty, genuinely smaller than the whole circuit, and improves the exact size comparison.</p><p>Lean proves that this finite search finds a qualifying part whenever one exists in that governed search space, and that the extracted smaller circuit preserves the intended boundary behaviour. The dependency system is still supplied rather than derived from the circuit, and the search is exhaustive rather than efficient. It does not complete the manuscript&#39;s full support or square arguments, finish ZeroSlack or PCCMin, provide a polynomial-time algorithm, or prove P = NP. The 67% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="67">Editorial progress estimate at publication: 67%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-05-residual-terminal-proper-positive-support-search">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-05-residual-terminal-support-extraction 2026-08-05T06:43:58Z 2026-08-05T06:43:58Z

A circuit is a collection of simple yes-or-no operations connected by wires. Lean can now select any collection of operations, including a scattered collection, and turn that selection into a smaller circuit of its own. It identifies every value entering the selection, rebuilds the selected operations in their original order, and exposes every selected result that the rest of the circuit uses. Lean proves the rebuilt circuit has exactly the selected number of operations and gives the intended outgoing values for every possible set of incoming values. When those incoming values come from the original whole circuit, the extracted circuit reproduces the original results, including after the dependency checklist is completed. The record list and dependency links are still supplied rather than derived automatically, and this does not construct the manuscript’s proper positive support or required square, finish ZeroSlack or PCCMin, provide a polynomial-time algorithm, or prove P = NP. The 66% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 66 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a collection of simple yes-or-no operations connected by wires. Lean can now select any collection of operations, including a scattered collection, and turn that selection into a smaller circuit of its own. It identifies every value entering the selection, rebuilds the selected operations in their original order, and exposes every selected result that the rest of the circuit uses.</p><p>Lean proves the rebuilt circuit has exactly the selected number of operations and gives the intended outgoing values for every possible set of incoming values. When those incoming values come from the original whole circuit, the extracted circuit reproduces the original results, including after the dependency checklist is completed. The record list and dependency links are still supplied rather than derived automatically, and this does not construct the manuscript’s proper positive support or required square, finish ZeroSlack or PCCMin, provide a polynomial-time algorithm, or prove P = NP. The 66% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="66">Editorial progress estimate at publication: 66%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-05-residual-terminal-support-extraction">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-05-residual-terminal-physical-support-completion 2026-08-05T01:37:59Z 2026-08-05T01:37:59Z

A circuit is a collection of simple yes-or-no operations connected by wires. Lean can now take any chosen group of operations in any finite circuit, finish its recorded dependency checklist, and identify every wire that brings a value into the group or carries a result out. The result is put in one consistent order, even if the starting checklist was duplicated or scrambled. Lean proves that no crossing wire is missed and no unrelated wire is added: constants stay inside, wires between chosen operations stay internal, and wires connecting to the rest of the circuit appear on the correct side. This still does not complete the extra bookkeeping records required by the manuscript, construct the positive witness, prove the required four-part square, finish ZeroSlack or PCCMin, provide a polynomial-time algorithm, or prove P = NP. The 65% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 65 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a collection of simple yes-or-no operations connected by wires. Lean can now take any chosen group of operations in any finite circuit, finish its recorded dependency checklist, and identify every wire that brings a value into the group or carries a result out. The result is put in one consistent order, even if the starting checklist was duplicated or scrambled.</p><p>Lean proves that no crossing wire is missed and no unrelated wire is added: constants stay inside, wires between chosen operations stay internal, and wires connecting to the rest of the circuit appear on the correct side. This still does not complete the extra bookkeeping records required by the manuscript, construct the positive witness, prove the required four-part square, finish ZeroSlack or PCCMin, provide a polynomial-time algorithm, or prove P = NP. The 65% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="65">Editorial progress estimate at publication: 65%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-05-residual-terminal-physical-support-completion">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-04-residual-terminal-saturation-closure 2026-08-04T15:42:54Z 2026-08-04T15:42:54Z

A later part of the proposed proof needs to start with a set of facts about a circuit and repeatedly add every other fact those facts depend on. Lean now defines that process for any finite checklist and any explicitly supplied dependency links. It proves the process includes everything initially selected and continues until no recorded dependency is missing. Lean also proves this completed checklist is the smallest one with those properties: starting with more facts cannot produce fewer, running the process again changes nothing, and a checklist is unchanged exactly when it was already complete. This does not yet create the correct dependency links from an arbitrary circuit or build the manuscript’s required support square. It does not complete ZeroSlack or PCCMin, provide a polynomial-time algorithm, or prove P = NP. The 64% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 64 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A later part of the proposed proof needs to start with a set of facts about a circuit and repeatedly add every other fact those facts depend on. Lean now defines that process for any finite checklist and any explicitly supplied dependency links. It proves the process includes everything initially selected and continues until no recorded dependency is missing.</p><p>Lean also proves this completed checklist is the smallest one with those properties: starting with more facts cannot produce fewer, running the process again changes nothing, and a checklist is unchanged exactly when it was already complete. This does not yet create the correct dependency links from an arbitrary circuit or build the manuscript’s required support square. It does not complete ZeroSlack or PCCMin, provide a polynomial-time algorithm, or prove P = NP. The 64% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="64">Editorial progress estimate at publication: 64%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-04-residual-terminal-saturation-closure">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-04-residual-terminal-projection-transfer 2026-08-04T09:07:29Z 2026-08-04T09:07:29Z

Sometimes this proof compares circuits using a complete checklist of their behaviour, and sometimes it uses a shorter checklist that deliberately ignores selected details. This update considers four related cases: the common part, two alternatives, and their combined case. It proves an exact accounting rule for how the shorter checklist changes the measured size differences. The rule applies only when all four cases use the same checklist and the same way of omitting details. Under the stated conditions, if no information is lost in the common part or either alternative, but the combined case loses D circuit operations, then the accounting difference is exactly D; if D is greater than zero, the difference is too. Lean checks increases and decreases correctly. It does not construct the four cases, prove that the manuscript’s required support and saturation objects exist, complete ZeroSlack or PCCMin, provide a fast algorithm, or prove P = NP. The 63% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 63 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>Sometimes this proof compares circuits using a complete checklist of their behaviour, and sometimes it uses a shorter checklist that deliberately ignores selected details. This update considers four related cases: the common part, two alternatives, and their combined case. It proves an exact accounting rule for how the shorter checklist changes the measured size differences. The rule applies only when all four cases use the same checklist and the same way of omitting details.</p><p>Under the stated conditions, if no information is lost in the common part or either alternative, but the combined case loses D circuit operations, then the accounting difference is exactly D; if D is greater than zero, the difference is too. Lean checks increases and decreases correctly. It does not construct the four cases, prove that the manuscript’s required support and saturation objects exist, complete ZeroSlack or PCCMin, provide a fast algorithm, or prove P = NP. The 63% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="63">Editorial progress estimate at publication: 63%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-04-residual-terminal-projection-transfer">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-04-residual-terminal-projection-minimum 2026-08-04T01:52:25Z 2026-08-04T01:52:25Z

A circuit is a finite collection of simple yes-or-no operations. Two circuits can be compared using either a full checklist of their behaviour or a shorter checklist that deliberately ignores some details. Lean now exhaustively finds the smallest matching circuit under each checklist, proves that a circuit of each reported size really exists, and proves that no matching circuit can be smaller. Ignoring details can never make the smallest matching circuit larger. Lean now measures the exact size gap between the two comparisons and proves that the gap is zero exactly when a smallest partial match also passes every omitted check. This exhaustive reference search is not an efficient algorithm and does not complete the remaining support, saturation, ZeroSlack, PCCMin, or complexity-theory work, so it does not prove P = NP. The 62% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 62 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a finite collection of simple yes-or-no operations. Two circuits can be compared using either a full checklist of their behaviour or a shorter checklist that deliberately ignores some details. Lean now exhaustively finds the smallest matching circuit under each checklist, proves that a circuit of each reported size really exists, and proves that no matching circuit can be smaller.</p><p>Ignoring details can never make the smallest matching circuit larger. Lean now measures the exact size gap between the two comparisons and proves that the gap is zero exactly when a smallest partial match also passes every omitted check. This exhaustive reference search is not an efficient algorithm and does not complete the remaining support, saturation, ZeroSlack, PCCMin, or complexity-theory work, so it does not prove P = NP. The 62% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="62">Editorial progress estimate at publication: 62%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-04-residual-terminal-projection-minimum">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-04-residual-terminal-mode-firewall 2026-08-03T17:25:02Z 2026-08-03T17:25:02Z

A circuit is a finite collection of simple yes-or-no operations. Some proof steps compare only a selected checklist of facts about two circuits, while later steps need every relevant fact to match. Lean now records exactly which facts were kept, preserves the circuit, its number of operations, and all input-and-output behaviour, and prevents a partial comparison from being treated as a complete one without the missing checks. A partial comparison can now be lifted to a complete comparison exactly when every omitted fact also agrees; keeping the full checklist makes that lift immediate. This is a safety boundary for future minimisation work, not a method for finding smaller circuits, completing the remaining support and saturation arguments, running the final algorithm efficiently, or proving P = NP. The 61% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 61 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a finite collection of simple yes-or-no operations. Some proof steps compare only a selected checklist of facts about two circuits, while later steps need every relevant fact to match. Lean now records exactly which facts were kept, preserves the circuit, its number of operations, and all input-and-output behaviour, and prevents a partial comparison from being treated as a complete one without the missing checks.</p><p>A partial comparison can now be lifted to a complete comparison exactly when every omitted fact also agrees; keeping the full checklist makes that lift immediate. This is a safety boundary for future minimisation work, not a method for finding smaller circuits, completing the remaining support and saturation arguments, running the final algorithm efficiently, or proving P = NP. The 61% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="61">Editorial progress estimate at publication: 61%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-04-residual-terminal-mode-firewall">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-03-residual-terminal-full-carrier-bridge 2026-08-03T12:39:26Z 2026-08-03T12:39:26Z

A circuit is a collection of simple yes-or-no operations. Lean now checks the full comparison for any finite circuit: another circuit counts as equivalent only when it gives the same answer for every possible input and every output. The wrapper used for that comparison also preserves the circuit and its exact number of operations. Lean also checks that any cheaper complete equivalent circuit is genuine progress toward the smallest one, and that no such circuit exists exactly when the remaining size gap is zero. It does not provide an efficient way to find that circuit or finish the manuscript’s remaining minimisation and complexity-theory steps, and it does not prove P = NP. The 60% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 60 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a collection of simple yes-or-no operations. Lean now checks the full comparison for any finite circuit: another circuit counts as equivalent only when it gives the same answer for every possible input and every output. The wrapper used for that comparison also preserves the circuit and its exact number of operations.</p><p>Lean also checks that any cheaper complete equivalent circuit is genuine progress toward the smallest one, and that no such circuit exists exactly when the remaining size gap is zero. It does not provide an efficient way to find that circuit or finish the manuscript’s remaining minimisation and complexity-theory steps, and it does not prove P = NP. The 60% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="60">Editorial progress estimate at publication: 60%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-03-residual-terminal-full-carrier-bridge">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-03-residual-gain-stopping-specification 2026-08-03T06:33:39Z 2026-08-03T06:33:39Z

A circuit is a finite list of simple yes-or-no operations. Two circuits are equivalent when they give exactly the same outputs for every possible input. Lean now proves an exact rule for when an equivalent circuit cannot be made smaller: there is no smaller equivalent circuit anywhere exactly when the current circuit is already as small as possible. This lets a previously checked shrinking sequence end with an exact minimum result, but only when a separate proof rules out every smaller equivalent circuit, not merely the circuits in a searched list. The milestone does not supply that global proof, an efficient search, the manuscript’s remaining ZeroSlack/PCCMin construction, or P = NP. The 59% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 59 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a finite list of simple yes-or-no operations. Two circuits are equivalent when they give exactly the same outputs for every possible input. Lean now proves an exact rule for when an equivalent circuit cannot be made smaller: there is no smaller equivalent circuit anywhere exactly when the current circuit is already as small as possible.</p><p>This lets a previously checked shrinking sequence end with an exact minimum result, but only when a separate proof rules out every smaller equivalent circuit, not merely the circuits in a searched list. The milestone does not supply that global proof, an efficient search, the manuscript’s remaining ZeroSlack/PCCMin construction, or P = NP. The 59% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="59">Editorial progress estimate at publication: 59%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-03-residual-gain-stopping-specification">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-08-03-residual-gain-chain-bound 2026-08-03T01:28:40Z 2026-08-03T01:28:40Z

A circuit is a collection of simple yes-or-no operations. Sometimes one circuit can be replaced by a smaller circuit that gives exactly the same answers. Lean now checks a general rule for any finite sequence of these replacements: when every step is independently checked to keep all answers the same and to use fewer operations, the sequence cannot continue for longer than the starting gap between the current circuit and the smallest equivalent circuit. For the locked comparison circuits already constructed in this project, that starting gap is at most four, so any such verified shrinking sequence has at most four steps. This rules out endless repetition, but it does not find the next smaller circuit, prove that every possible improvement will be found, or show that stopping early means the smallest circuit has been reached. Those search, completeness, runtime, and final P = NP steps remain open. The 58% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 58 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a collection of simple yes-or-no operations. Sometimes one circuit can be replaced by a smaller circuit that gives exactly the same answers. Lean now checks a general rule for any finite sequence of these replacements: when every step is independently checked to keep all answers the same and to use fewer operations, the sequence cannot continue for longer than the starting gap between the current circuit and the smallest equivalent circuit.</p><p>For the locked comparison circuits already constructed in this project, that starting gap is at most four, so any such verified shrinking sequence has at most four steps. This rules out endless repetition, but it does not find the next smaller circuit, prove that every possible improvement will be found, or show that stopping early means the smallest circuit has been reached. Those search, completeness, runtime, and final P = NP steps remain open. The 58% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="58">Editorial progress estimate at publication: 58%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-08-03-residual-gain-chain-bound">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-31-concrete-cnf-to-nand-polynomial-reduction 2026-07-31T17:22:27Z 2026-07-31T17:22:27Z

The previous update proved that a CNF formula and its NAND translation have the same yes-or-no answer. This update adds one fixed, inspectable machine that performs that translation for every possible input. Lean checks that valid formulas produce exactly the intended circuit, while malformed inputs stop safely and produce no result. Lean also proves that the work grows at a polynomial rate with the input size, which is the standard efficiency requirement for this kind of translation. The checked machine is now packaged as a formal reduction from CNF satisfiability to NAND-circuit satisfiability and then connected to the existing locked-circuit conversion. It still does not decide CNF satisfiability efficiently, remove the remaining locked-circuit assumption, or prove P = NP. The 57% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 57 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>The previous update proved that a CNF formula and its NAND translation have the same yes-or-no answer. This update adds one fixed, inspectable machine that performs that translation for every possible input. Lean checks that valid formulas produce exactly the intended circuit, while malformed inputs stop safely and produce no result.</p><p>Lean also proves that the work grows at a polynomial rate with the input size, which is the standard efficiency requirement for this kind of translation. The checked machine is now packaged as a formal reduction from CNF satisfiability to NAND-circuit satisfiability and then connected to the existing locked-circuit conversion. It still does not decide CNF satisfiability efficiently, remove the remaining locked-circuit assumption, or prove P = NP. The 57% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="57">Editorial progress estimate at publication: 57%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-31-concrete-cnf-to-nand-polynomial-reduction">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-31-concrete-cnf-to-nand-semantic-compiler 2026-07-30T23:06:02Z 2026-07-30T23:06:02Z

A CNF formula is a list of yes-or-no requirements, while a NAND circuit is a network built from one simple universal logic operation. Lean now checks a general translation between these two forms and proves that a solution exists before the translation exactly when one exists afterward. The proof covers empty formulas, empty clauses, invalid encodings, out-of-range variables, the exact number of added gates, and a polynomial limit on the output size. This is a semantic and size result, not yet the finite-machine polynomial-time reduction needed for the full complexity-theory bridge, and it does not prove P = NP. The 56% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 56 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A CNF formula is a list of yes-or-no requirements, while a NAND circuit is a network built from one simple universal logic operation. Lean now checks a general translation between these two forms and proves that a solution exists before the translation exactly when one exists afterward.</p><p>The proof covers empty formulas, empty clauses, invalid encodings, out-of-range variables, the exact number of added gates, and a polynomial limit on the output size. This is a semantic and size result, not yet the finite-machine polynomial-time reduction needed for the full complexity-theory bridge, and it does not prove P = NP. The 56% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="56">Editorial progress estimate at publication: 56%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-31-concrete-cnf-to-nand-semantic-compiler">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-30-concrete-locked-nand-polynomial-reduction 2026-07-30T04:25:46Z 2026-07-30T04:25:46Z

A reduction is a reliable translation from one yes-or-no problem into another. The project already had checked machines that validate a circuit description and build the corresponding comparison object. Lean now packages those machines as one formal translation and proves, for every possible input, that the original circuit has a successful input exactly when the translated comparison passes its threshold. Lean also records that this translation runs within an explicit polynomial limit, so the construction does not hide an impractical exhaustive search. This is an important complexity-theory bridge, but it does not yet connect ordinary CNF-SAT to this exact source format, prove that the target comparison can be solved efficiently, discharge the remaining assumptions, or prove P = NP. The 55% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 55 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A reduction is a reliable translation from one yes-or-no problem into another. The project already had checked machines that validate a circuit description and build the corresponding comparison object. Lean now packages those machines as one formal translation and proves, for every possible input, that the original circuit has a successful input exactly when the translated comparison passes its threshold.</p><p>Lean also records that this translation runs within an explicit polynomial limit, so the construction does not hide an impractical exhaustive search. This is an important complexity-theory bridge, but it does not yet connect ordinary CNF-SAT to this exact source format, prove that the target comparison can be solved efficiently, discharge the remaining assumptions, or prove P = NP. The 55% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="55">Editorial progress estimate at publication: 55%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-30-concrete-locked-nand-polynomial-reduction">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-30-concrete-locked-nand-target-emitter 2026-07-29T18:18:47Z 2026-07-29T18:18:47Z

A circuit description is a list of simple yes-or-no operations. The previous milestone added a machine that checks this description before it is used. This milestone adds a second fixed, inspectable machine that turns an accepted description into the larger comparison circuit required by the mathematical argument. Lean proves that every output bit matches the construction already defined in the formal development. Malformed or invalid descriptions are rejected without leaving a result. Lean also proves that the running time and output size are bounded by explicit formulas based only on the input length, and that the checker and builder can be joined into one verified path. The remaining work includes packaging that path as the formal reduction needed by complexity theory and closing the still-open threshold, hardness, and final P = NP steps. The 54% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 54 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit description is a list of simple yes-or-no operations. The previous milestone added a machine that checks this description before it is used. This milestone adds a second fixed, inspectable machine that turns an accepted description into the larger comparison circuit required by the mathematical argument. Lean proves that every output bit matches the construction already defined in the formal development.</p><p>Malformed or invalid descriptions are rejected without leaving a result. Lean also proves that the running time and output size are bounded by explicit formulas based only on the input length, and that the checker and builder can be joined into one verified path. The remaining work includes packaging that path as the formal reduction needed by complexity theory and closing the still-open threshold, hardness, and final P = NP steps. The 54% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="54">Editorial progress estimate at publication: 54%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-30-concrete-locked-nand-target-emitter">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-29-concrete-locked-nand-source-parser 2026-07-28T20:38:41Z 2026-07-28T20:38:41Z

A circuit description is a string of zeros and ones that tells a computer which simple yes-or-no steps to perform. Lean now proves that a fixed, inspectable machine checks every possible description from beginning to end. It verifies the format version, the stated counts, each operation, every reference to earlier data, all closing markers, and the exact end of the input. Valid descriptions are accepted and returned unchanged. Anything malformed, including a reference to something that has not been defined yet, is rejected and produces no output. Lean also proves that this check always finishes within a stated size-based limit. This completes the input-checking half of the planned conversion; it does not yet build the comparison object, complete the full conversion, or prove P = NP. The 53% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 53 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit description is a string of zeros and ones that tells a computer which simple yes-or-no steps to perform. Lean now proves that a fixed, inspectable machine checks every possible description from beginning to end. It verifies the format version, the stated counts, each operation, every reference to earlier data, all closing markers, and the exact end of the input.</p><p>Valid descriptions are accepted and returned unchanged. Anything malformed, including a reference to something that has not been defined yet, is rejected and produces no output. Lean also proves that this check always finishes within a stated size-based limit. This completes the input-checking half of the planned conversion; it does not yet build the comparison object, complete the full conversion, or prove P = NP. The 53% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="53">Editorial progress estimate at publication: 53%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-29-concrete-locked-nand-source-parser">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-28-concrete-locked-nand-encoded-semantic-boundary 2026-07-28T05:12:17Z 2026-07-28T05:12:17Z

A circuit is a list of very simple yes-or-no operations. To turn a mathematical construction into a real algorithm, its input and output need one exact bit format. Lean now fixes a strict version-zero format for the source circuit and the complete comparison object, proves that valid data can be encoded and decoded without changing its meaning, and normalizes outputs that were only an input or a constant. The resulting pure transformation preserves the yes-or-no comparison: for valid encoded circuits, the constructed bytes cross the target size threshold exactly when the original circuit has a solution, while malformed inputs are rejected. This is a strategic bridge from the mathematical circuit theorem toward an algorithm, but it is not yet a bounded parser or emitter machine, a polynomial-time reduction, CNF-SAT in P, or P = NP. The 52% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 52 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a list of very simple yes-or-no operations. To turn a mathematical construction into a real algorithm, its input and output need one exact bit format. Lean now fixes a strict version-zero format for the source circuit and the complete comparison object, proves that valid data can be encoded and decoded without changing its meaning, and normalizes outputs that were only an input or a constant.</p><p>The resulting pure transformation preserves the yes-or-no comparison: for valid encoded circuits, the constructed bytes cross the target size threshold exactly when the original circuit has a solution, while malformed inputs are rejected. This is a strategic bridge from the mathematical circuit theorem toward an algorithm, but it is not yet a bounded parser or emitter machine, a polynomial-time reduction, CNF-SAT in P, or P = NP. The 52% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="52">Editorial progress estimate at publication: 52%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-28-concrete-locked-nand-encoded-semantic-boundary">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-28-locked-nand-global-semantic-threshold 2026-07-27T20:42:01Z 2026-07-27T20:42:01Z

A circuit is a sequence of simple yes-or-no steps. Earlier work proved what happens when no successful input exists. Lean now also proves the other direction: when a successful input does exist, the enlarged circuit cannot shrink back to the established baseline size. Its smallest equivalent form must use at least one extra step, while the construction adds no more than four. Together, these results give an exact yes-or-no comparison at the mathematical circuit level: the minimum size rises above the baseline exactly when the original circuit has a solution. The efficient encoded procedure needed to turn arbitrary inputs into these circuits is still missing, as are the remaining reduction, CNF-SAT in P, and P = NP. The 51% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 51 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a sequence of simple yes-or-no steps. Earlier work proved what happens when no successful input exists. Lean now also proves the other direction: when a successful input does exist, the enlarged circuit cannot shrink back to the established baseline size. Its smallest equivalent form must use at least one extra step, while the construction adds no more than four.</p><p>Together, these results give an exact yes-or-no comparison at the mathematical circuit level: the minimum size rises above the baseline exactly when the original circuit has a solution. The efficient encoded procedure needed to turn arbitrary inputs into these circuits is still missing, as are the remaining reduction, CNF-SAT in P, and P = NP. The 51% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="51">Editorial progress estimate at publication: 51%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-28-locked-nand-global-semantic-threshold">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-27-locked-nand-global-unsatisfiable-final-zero 2026-07-27T16:01:13Z 2026-07-27T16:01:13Z

A circuit is a sequence of simple yes-or-no steps. This project adds a final check that can turn on only when the original circuit has a successful input and all of the recorded intermediate work agrees. Lean now proves that if no successful input exists, that final check stays off for every possible filling of the workspace, including deliberately inconsistent ones. Lean also proves that, in this no-solution case, the smallest equivalent implementation has exactly the established baseline size. This closes only the no-solution half of the comparison. The successful-solution separation argument, complete size threshold, uniform polynomial-time builder, CNF-SAT in P, and P = NP remain unproved. The 50% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 50 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A circuit is a sequence of simple yes-or-no steps. This project adds a final check that can turn on only when the original circuit has a successful input and all of the recorded intermediate work agrees. Lean now proves that if no successful input exists, that final check stays off for every possible filling of the workspace, including deliberately inconsistent ones.</p><p>Lean also proves that, in this no-solution case, the smallest equivalent implementation has exactly the established baseline size. This closes only the no-solution half of the comparison. The successful-solution separation argument, complete size threshold, uniform polynomial-time builder, CNF-SAT in P, and P = NP remain unproved. The 50% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="50">Editorial progress estimate at publication: 50%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-27-locked-nand-global-unsatisfiable-final-zero">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-27-locked-nand-global-baseline-distinct 2026-07-27T04:27:24Z 2026-07-27T04:27:24Z

The previous milestone assembled a baseline circuit with one exposed output for every gate. That head count alone did not show every gate was necessary: an output might have been fixed, might merely copy an input, or might duplicate another output. This milestone proves none of those shortcuts occurs, for every finite circuit in the family. That makes the baseline's exact minimum size match its stated size and completes another required part of the later threshold argument. Two final-output laws and the uniform polynomial-time construction are still missing, so the locked-circuit threshold, CNF-SAT in P, and P = NP remain unproved. The 49% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 49 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>The previous milestone assembled a baseline circuit with one exposed output for every gate. That head count alone did not show every gate was necessary: an output might have been fixed, might merely copy an input, or might duplicate another output. This milestone proves none of those shortcuts occurs, for every finite circuit in the family.</p><p>That makes the baseline&#39;s exact minimum size match its stated size and completes another required part of the later threshold argument. Two final-output laws and the uniform polynomial-time construction are still missing, so the locked-circuit threshold, CNF-SAT in P, and P = NP remain unproved. The 49% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="49">Editorial progress estimate at publication: 49%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-27-locked-nand-global-baseline-distinct">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-26-locked-nand-global-candidates 2026-07-26T00:51:21Z 2026-07-26T00:51:21Z

The previous milestone established a reliable way to represent and check the step-by-step behaviour of any finite circuit built from simple logic gates. This milestone uses that representation to assemble the two complete circuit families required by the next stage: a baseline whose number of inputs matches its number of gates, and a slightly larger version with four additional gates. It proves the exact size of each construction and that every original output is preserved. The construction also avoids hidden fixed values inside the circuits and shows that changing the new final control input cannot alter any original output. This turns an abstract outline into a concrete family that works for circuits of any finite size. Important comparison and threshold arguments are still missing, as is the uniform polynomial-time procedure that would generate the construction from encoded input. The 48% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved. Editorial progress estimate at publication: 48 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>The previous milestone established a reliable way to represent and check the step-by-step behaviour of any finite circuit built from simple logic gates. This milestone uses that representation to assemble the two complete circuit families required by the next stage: a baseline whose number of inputs matches its number of gates, and a slightly larger version with four additional gates. It proves the exact size of each construction and that every original output is preserved.</p><p>The construction also avoids hidden fixed values inside the circuits and shows that changing the new final control input cannot alter any original output. This turns an abstract outline into a concrete family that works for circuits of any finite size. Important comparison and threshold arguments are still missing, as is the uniform polynomial-time procedure that would generate the construction from encoded input. The 48% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved.</p><p data-progress-estimate-percent="48">Editorial progress estimate at publication: 48%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-26-locked-nand-global-candidates">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-25-locked-nand-carrier-trace 2026-07-25T00:00:50Z 2026-07-25T00:00:50Z

A computer circuit can be pictured as a row of small logic gates. Each gate reads two earlier values and produces a new value. This milestone gives every finite circuit arranged in that order a fixed set of labelled spaces for its inputs, connections, intermediate results, and final check. It proves that filling those spaces consistently is exactly the same as evaluating the circuit gate by gate. This is a rule for circuits of any finite size, not just another fixed-size example. It still does not build the complete locked circuit family, prove the required size threshold, supply the full polynomial-time construction, or prove P = NP. The 47% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct. Editorial progress estimate at publication: 47 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>A computer circuit can be pictured as a row of small logic gates. Each gate reads two earlier values and produces a new value. This milestone gives every finite circuit arranged in that order a fixed set of labelled spaces for its inputs, connections, intermediate results, and final check. It proves that filling those spaces consistently is exactly the same as evaluating the circuit gate by gate.</p><p>This is a rule for circuits of any finite size, not just another fixed-size example. It still does not build the complete locked circuit family, prove the required size threshold, supply the full polynomial-time construction, or prove P = NP. The 47% figure is a revisable estimate of known reconstruction work, not a probability that the claim is correct.</p><p data-progress-estimate-percent="47">Editorial progress estimate at publication: 47%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-25-locked-nand-carrier-trace">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-25-second-constraint-seventh-padding-or-unary-opportunity 2026-07-24T16:11:18Z 2026-07-24T16:11:18Z

The machine is building a long checklist one position at a time. In wider layouts it has already written the opening mark for the following item. At the next position, the smallest supported layout still has intentional empty spacing, so it moves past without writing. In wider layouts, it writes the first mark of that item’s number. In both cases, it stops at the exact following position. Only this one position is handled. The machine does not write the remaining number marks or closing mark, finish the second block, or finish the full checklist. The 46% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved. Editorial progress estimate at publication: 46 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>The machine is building a long checklist one position at a time. In wider layouts it has already written the opening mark for the following item. At the next position, the smallest supported layout still has intentional empty spacing, so it moves past without writing. In wider layouts, it writes the first mark of that item’s number. In both cases, it stops at the exact following position.</p><p>Only this one position is handled. The machine does not write the remaining number marks or closing mark, finish the second block, or finish the full checklist. The 46% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved.</p><p data-progress-estimate-percent="46">Editorial progress estimate at publication: 46%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-25-second-constraint-seventh-padding-or-unary-opportunity">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-24-second-constraint-sixth-padding-or-opening-unary-opportunity 2026-07-24T12:05:57Z 2026-07-24T12:05:57Z

The machine is building a long checklist one position at a time. It has reached the position where another item would begin. In the smallest supported layout there is no item here, so it leaves the position blank. In wider layouts it writes the opening mark for the following item. In both cases it stops at the exact next position. Only this one position is handled. The machine does not write the following item's number or closing mark, finish the second block, or finish the full checklist. The 45% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved. Editorial progress estimate at publication: 45 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>The machine is building a long checklist one position at a time. It has reached the position where another item would begin. In the smallest supported layout there is no item here, so it leaves the position blank. In wider layouts it writes the opening mark for the following item. In both cases it stops at the exact next position.</p><p>Only this one position is handled. The machine does not write the following item&#39;s number or closing mark, finish the second block, or finish the full checklist. The 45% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved.</p><p data-progress-estimate-percent="45">Editorial progress estimate at publication: 45%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-24-second-constraint-sixth-padding-or-opening-unary-opportunity">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-24-second-constraint-fifth-padding-or-terminator-opportunity 2026-07-24T08:51:40Z 2026-07-24T08:51:40Z

The machine is building a long checklist one position at a time. It has now reached the next reserved position. In the smallest supported layout, there is no extra item here, so the position stays blank. In wider layouts, the machine writes the closing mark that finishes the next item. In both cases, it stops at the exact following position. Only this one position is handled. The machine does not write the first mark of whatever comes next, finish the second block, or finish the full checklist. The 44% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved. Editorial progress estimate at publication: 44 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>The machine is building a long checklist one position at a time. It has now reached the next reserved position. In the smallest supported layout, there is no extra item here, so the position stays blank. In wider layouts, the machine writes the closing mark that finishes the next item. In both cases, it stops at the exact following position.</p><p>Only this one position is handled. The machine does not write the first mark of whatever comes next, finish the second block, or finish the full checklist. The 44% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved.</p><p data-progress-estimate-percent="44">Editorial progress estimate at publication: 44%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-24-second-constraint-fifth-padding-or-terminator-opportunity">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-24-second-constraint-fourth-padding-or-unary-opportunity 2026-07-24T04:47:50Z 2026-07-24T04:47:50Z

The machine is building a long checklist one position at a time. It has now reached a fourth reserved position after a completed item. In the smallest supported layout, that position is still intentional empty spacing, so the machine moves past it without writing. In wider layouts, it writes the fourth and final number mark of the next item. In both cases, it stops at the exact following position. Only this one additional position is handled. The machine does not write the following closing mark, complete the next item, finish the second block, or finish the full checklist. The 43% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved. Editorial progress estimate at publication: 43 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>The machine is building a long checklist one position at a time. It has now reached a fourth reserved position after a completed item. In the smallest supported layout, that position is still intentional empty spacing, so the machine moves past it without writing. In wider layouts, it writes the fourth and final number mark of the next item. In both cases, it stops at the exact following position.</p><p>Only this one additional position is handled. The machine does not write the following closing mark, complete the next item, finish the second block, or finish the full checklist. The 43% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved.</p><p data-progress-estimate-percent="43">Editorial progress estimate at publication: 43%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-24-second-constraint-fourth-padding-or-unary-opportunity">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-24-second-constraint-third-padding-or-unary-opportunity 2026-07-24T00:38:35Z 2026-07-24T00:38:35Z

The machine is building a long checklist one position at a time. It has now reached a third reserved position after a completed item. In the smallest supported layout, that position is still intentional empty spacing, so the machine moves past it without writing. In wider layouts, it writes the third mark of the next numbered item. In both cases, it stops at the exact following position. Only this one additional position is handled. The machine does not write the following mark, complete the next item, finish the second block, or finish the full checklist. The 42% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved. Editorial progress estimate at publication: 42 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>The machine is building a long checklist one position at a time. It has now reached a third reserved position after a completed item. In the smallest supported layout, that position is still intentional empty spacing, so the machine moves past it without writing. In wider layouts, it writes the third mark of the next numbered item. In both cases, it stops at the exact following position.</p><p>Only this one additional position is handled. The machine does not write the following mark, complete the next item, finish the second block, or finish the full checklist. The 42% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved.</p><p data-progress-estimate-percent="42">Editorial progress estimate at publication: 42%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-24-second-constraint-third-padding-or-unary-opportunity">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-second-padding-or-unary-opportunity 2026-07-23T13:55:51Z 2026-07-23T13:55:51Z

The machine is building a long checklist one position at a time. It had already handled the first reserved position after a completed item. At the next position, the smallest supported layout still needs intentional empty spacing, so the machine moves past it without writing. In wider layouts, it writes the second mark of the next numbered item. In both cases, it stops at the exact following position. Only this one additional position is handled. The machine does not write the following mark, complete the next item, finish the second block, or finish the full checklist. The 41% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved. Editorial progress estimate at publication: 41 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>The machine is building a long checklist one position at a time. It had already handled the first reserved position after a completed item. At the next position, the smallest supported layout still needs intentional empty spacing, so the machine moves past it without writing. In wider layouts, it writes the second mark of the next numbered item. In both cases, it stops at the exact following position.</p><p>Only this one additional position is handled. The machine does not write the following mark, complete the next item, finish the second block, or finish the full checklist. The 41% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved.</p><p data-progress-estimate-percent="41">Editorial progress estimate at publication: 41%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-second-padding-or-unary-opportunity">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-padding-or-unary-opportunity 2026-07-23T11:20:59Z 2026-07-23T11:20:59Z

The machine has finished one numbered item in a long checklist and has already chosen the mark that follows it. This milestone verifies what happens at the very next reserved position. In the smallest supported layout, that position is intentional empty spacing, so the machine moves past it without writing. In wider layouts, it writes the first mark of the next numbered item. In both cases, it finishes at the exact following position. Only this one position is handled. The machine does not write the next mark, complete the next item, finish the second block, or finish the full checklist. The 40% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved. Editorial progress estimate at publication: 40 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>The machine has finished one numbered item in a long checklist and has already chosen the mark that follows it. This milestone verifies what happens at the very next reserved position. In the smallest supported layout, that position is intentional empty spacing, so the machine moves past it without writing. In wider layouts, it writes the first mark of the next numbered item. In both cases, it finishes at the exact following position.</p><p>Only this one position is handled. The machine does not write the next mark, complete the next item, finish the second block, or finish the full checklist. The 40% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved.</p><p data-progress-estimate-percent="40">Editorial progress estimate at publication: 40%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-padding-or-unary-opportunity">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-first-literal-successor-token 2026-07-23T07:39:29Z 2026-07-23T07:39:29Z

Imagine the machine has just finished one numbered item in its long checklist. The next mark depends on how wide the checklist is. In the smallest supported layout, it writes an end-of-line mark. In wider layouts, it writes the opening mark for the next item. This milestone verifies that it makes that choice, writes exactly one mark, and moves to the following position. It does not write the next item's number, finish the second block, or finish the full checklist. The 39% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved. Editorial progress estimate at publication: 39 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>Imagine the machine has just finished one numbered item in its long checklist. The next mark depends on how wide the checklist is. In the smallest supported layout, it writes an end-of-line mark. In wider layouts, it writes the opening mark for the next item. This milestone verifies that it makes that choice, writes exactly one mark, and moves to the following position.</p><p>It does not write the next item&#39;s number, finish the second block, or finish the full checklist. The 39% figure is a revisable planning estimate of known reconstruction work, not a probability that the claim is correct. P = NP remains unproved.</p><p data-progress-estimate-percent="39">Editorial progress estimate at publication: 39%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-first-literal-successor-token">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-first-literal-terminator 2026-07-23T03:34:55Z 2026-07-23T03:34:55Z

Imagine a machine writing a numbered item in a long checklist, one mark at a time. The marks saying how to use the item and which item it is were already in place. This milestone verifies that the machine adds the single closing mark that completes the first item in the second major block, then moves to the next required position. Only that closing mark is added. In the smallest supported layout, the next position closes the line; in wider layouts, it begins another item. The machine identifies which case applies but writes neither next mark. It has not finished the second block or the full checklist, and this update does not prove that P equals NP. Editorial progress estimate at publication: 38 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>Imagine a machine writing a numbered item in a long checklist, one mark at a time. The marks saying how to use the item and which item it is were already in place. This milestone verifies that the machine adds the single closing mark that completes the first item in the second major block, then moves to the next required position.</p><p>Only that closing mark is added. In the smallest supported layout, the next position closes the line; in wider layouts, it begins another item. The machine identifies which case applies but writes neither next mark. It has not finished the second block or the full checklist, and this update does not prove that P equals NP.</p><p data-progress-estimate-percent="38">Editorial progress estimate at publication: 38%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-first-literal-terminator">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-first-literal-third-unary-unit 2026-07-23T01:06:25Z 2026-07-23T01:06:25Z

Imagine a machine writing a numbered item in a long checklist, one mark at a time. Two of the three marks needed for the first item's number in the second major block were already in place. This milestone verifies that it writes the third and final number mark, then moves to the closing mark for that item. Only that final number mark is added. The machine has not written the closing mark, completed the item, finished the second block, or finished the full checklist. This update does not prove that P equals NP. Editorial progress estimate at publication: 37 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>Imagine a machine writing a numbered item in a long checklist, one mark at a time. Two of the three marks needed for the first item&#39;s number in the second major block were already in place. This milestone verifies that it writes the third and final number mark, then moves to the closing mark for that item.</p><p>Only that final number mark is added. The machine has not written the closing mark, completed the item, finished the second block, or finished the full checklist. This update does not prove that P equals NP.</p><p data-progress-estimate-percent="37">Editorial progress estimate at publication: 37%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-first-literal-third-unary-unit">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-first-literal-second-unary-unit 2026-07-22T22:24:57Z 2026-07-22T22:24:57Z

Imagine a machine writing a long checklist, one mark at a time. It had already written the first mark of the first item's number in the second major block. This milestone verifies that it now writes the second mark and moves to the exact place for the third mark. Only that one additional number mark is added. The machine has not written the third mark, completed the item, finished the second block, or finished the full checklist. This update does not prove that P equals NP. Editorial progress estimate at publication: 36 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>Imagine a machine writing a long checklist, one mark at a time. It had already written the first mark of the first item&#39;s number in the second major block. This milestone verifies that it now writes the second mark and moves to the exact place for the third mark.</p><p>Only that one additional number mark is added. The machine has not written the third mark, completed the item, finished the second block, or finished the full checklist. This update does not prove that P equals NP.</p><p data-progress-estimate-percent="36">Editorial progress estimate at publication: 36%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-first-literal-second-unary-unit">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-first-literal-first-unary-unit 2026-07-22T15:55:00Z 2026-07-22T15:55:00Z

Imagine a machine writing a long checklist, one mark at a time. It had already placed the divider for the second major block and marked that its first item should be used as written. This milestone verifies that it now writes the first mark of that item's number and moves to the next mark. Only that one number mark is added. The machine has not written the next mark, completed the item, finished the second block, or finished the full checklist. This update does not prove that P equals NP. Editorial progress estimate at publication: 35 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>Imagine a machine writing a long checklist, one mark at a time. It had already placed the divider for the second major block and marked that its first item should be used as written. This milestone verifies that it now writes the first mark of that item&#39;s number and moves to the next mark.</p><p>Only that one number mark is added. The machine has not written the next mark, completed the item, finished the second block, or finished the full checklist. This update does not prove that P equals NP.</p><p data-progress-estimate-percent="35">Editorial progress estimate at publication: 35%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-23-second-constraint-first-literal-first-unary-unit">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-22-second-constraint-first-literal-sign 2026-07-22T13:14:00Z 2026-07-22T13:14:00Z

Imagine a machine writing a long checklist, one mark at a time. It had already placed the divider that opens the second major block. This milestone verifies that it now writes the single mark saying the first item in that block should be used as written, rather than as its opposite, then moves to the exact place where the item's number begins. Only that one mark is added. The machine has not written the item's number, completed the item, finished the second block, or finished the full checklist. This update does not prove that P equals NP. Editorial progress estimate at publication: 34 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>Imagine a machine writing a long checklist, one mark at a time. It had already placed the divider that opens the second major block. This milestone verifies that it now writes the single mark saying the first item in that block should be used as written, rather than as its opposite, then moves to the exact place where the item&#39;s number begins.</p><p>Only that one mark is added. The machine has not written the item&#39;s number, completed the item, finished the second block, or finished the full checklist. This update does not prove that P equals NP.</p><p data-progress-estimate-percent="34">Editorial progress estimate at publication: 34%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-22-second-constraint-first-literal-sign">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-22-second-constraint-separator 2026-07-22T08:06:00Z 2026-07-22T08:06:00Z

Imagine this project as a machine that writes a very long checklist of simple logical conditions. The first major checklist block was already complete. This milestone verifies that the machine writes one divider marking the start of the second block, then moves to the exact position where the next item belongs. This step adds only that divider. It does not write the next checklist item, finish the second block, complete the full checklist-building machine, or prove that P equals NP. Editorial progress estimate at publication: 33 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>Imagine this project as a machine that writes a very long checklist of simple logical conditions. The first major checklist block was already complete. This milestone verifies that the machine writes one divider marking the start of the second block, then moves to the exact position where the next item belongs.</p><p>This step adds only that divider. It does not write the next checklist item, finish the second block, complete the full checklist-building machine, or prove that P equals NP.</p><p data-progress-estimate-percent="33">Editorial progress estimate at publication: 33%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-22-second-constraint-separator">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-22-first-constraint-padding 2026-07-22T02:20:57Z 2026-07-22T02:20:57Z

This project is checking a step-by-step method that turns a computer task into a long checklist of simple yes-or-no conditions. Each major block has a fixed amount of reserved space. This update confirms that the builder can cross all the unused space left in the first major block without changing the checklist, then stop at the exact marker where the second block begins. Think of completing the first section of a very large fixed-size form and moving across every unused box until you reach the next section divider. The divider is seen but not printed by this step, and the next item is not started. The full builder is still incomplete, and this update does not prove that P equals NP. Editorial progress estimate at publication: 32 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>This project is checking a step-by-step method that turns a computer task into a long checklist of simple yes-or-no conditions. Each major block has a fixed amount of reserved space. This update confirms that the builder can cross all the unused space left in the first major block without changing the checklist, then stop at the exact marker where the second block begins.</p><p>Think of completing the first section of a very large fixed-size form and moving across every unused box until you reach the next section divider. The divider is seen but not printed by this step, and the next item is not started. The full builder is still incomplete, and this update does not prove that P equals NP.</p><p data-progress-estimate-percent="32">Editorial progress estimate at publication: 32%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-22-first-constraint-padding">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-22-fifth-clause-padding 2026-07-21T15:49:29Z 2026-07-21T15:49:29Z

This project is building, and checking, a step-by-step method that turns a computer task into a long list of simple yes-or-no checks. Some places in that list are deliberately left empty so every section has a predictable size. This update confirms that the builder can cross one whole empty section without changing the list and stop at the next exact boundary. Think of a form with fixed-size boxes: the machine can now move across one unused box and arrive at the next unused box without writing anything. This is a small construction milestone. It does not reach the next meaningful check, finish the builder, or prove that P equals NP. Editorial progress estimate at publication: 30 percent; this is not a probability, confidence score, or theorem-correctness claim.

<p>This project is building, and checking, a step-by-step method that turns a computer task into a long list of simple yes-or-no checks. Some places in that list are deliberately left empty so every section has a predictable size. This update confirms that the builder can cross one whole empty section without changing the list and stop at the next exact boundary.</p><p>Think of a form with fixed-size boxes: the machine can now move across one unused box and arrive at the next unused box without writing anything. This is a small construction milestone. It does not reach the next meaningful check, finish the builder, or prove that P equals NP.</p><p data-progress-estimate-percent="30">Editorial progress estimate at publication: 30%. This estimate is revisable and is not a probability, confidence score, or statement of theorem correctness.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-22-fifth-clause-padding">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-21-fourth-clause-padding 2026-07-21T12:42:16Z 2026-07-21T12:42:16Z

To check a computer program with mathematics, this project turns the program and its input into a long checklist of yes-or-no conditions. The fourth checklist section was already complete. This milestone verifies that the builder can move across every blank position reserved after it without writing anything, stop at the exact boundary of the next reserved block, and fail safely if its workspace is damaged. Think of a document template with fixed-size boxes: the builder has crossed the unused part of the fourth box and landed at the next box, which is also intentionally blank. It has not started the next meaningful checklist section, finished the full builder, or established the project’s P-versus-NP claim.

<p>To check a computer program with mathematics, this project turns the program and its input into a long checklist of yes-or-no conditions. The fourth checklist section was already complete. This milestone verifies that the builder can move across every blank position reserved after it without writing anything, stop at the exact boundary of the next reserved block, and fail safely if its workspace is damaged.</p><p>Think of a document template with fixed-size boxes: the builder has crossed the unused part of the fourth box and landed at the next box, which is also intentionally blank. It has not started the next meaningful checklist section, finished the full builder, or established the project’s P-versus-NP claim.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-21-fourth-clause-padding">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-21-fourth-clause-complete 2026-07-21T09:45:32Z 2026-07-21T09:45:32Z

To check a computer program with mathematics, this project turns the program and its input into a long checklist of yes-or-no conditions. This milestone verifies that the checklist builder can add the closing marker to the fourth section, move to the first unused space after it, and stop safely if its workspace is damaged. Think of a document generator: the fourth section now has its heading, both required lines, and its closing mark. The larger document is still far from finished. This update does not complete the full checklist builder or establish the project’s P-versus-NP claim.

<p>To check a computer program with mathematics, this project turns the program and its input into a long checklist of yes-or-no conditions. This milestone verifies that the checklist builder can add the closing marker to the fourth section, move to the first unused space after it, and stop safely if its workspace is damaged.</p><p>Think of a document generator: the fourth section now has its heading, both required lines, and its closing mark. The larger document is still far from finished. This update does not complete the full checklist builder or establish the project’s P-versus-NP claim.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-21-fourth-clause-complete">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-21-fourth-clause-second-literal 2026-07-21T02:28:32Z 2026-07-21T02:28:32Z

To check a computer program with mathematics, this project turns the program and its input into a long checklist of yes-or-no conditions. This milestone verifies that the checklist builder can add the second complete item in the fourth section, move to the exact place where the section-ending marker belongs, and fail safely if its workspace is damaged. Think of a document generator: the fourth section now has two checked lines, but its closing marker has not yet been written. This is one small verified construction step. The fourth section and the full checklist are still unfinished, and the project’s much larger claim about P versus NP is not established.

<p>To check a computer program with mathematics, this project turns the program and its input into a long checklist of yes-or-no conditions. This milestone verifies that the checklist builder can add the second complete item in the fourth section, move to the exact place where the section-ending marker belongs, and fail safely if its workspace is damaged.</p><p>Think of a document generator: the fourth section now has two checked lines, but its closing marker has not yet been written. This is one small verified construction step. The fourth section and the full checklist are still unfinished, and the project’s much larger claim about P versus NP is not established.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-21-fourth-clause-second-literal">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-21-fourth-clause-first-literal 2026-07-20T15:50:45Z 2026-07-20T15:50:45Z

To check a computer program with mathematics, this project turns the program and its input into a long checklist of yes-or-no conditions. This milestone verifies that the checklist builder can place the first complete item in the fourth section, stop exactly where the next item should begin, and fail safely if its workspace is damaged. Think of a document generator: the fourth section heading was already in place, and this update checks the first full line beneath it. This is one small verified construction step. The fourth section and the full checklist are still unfinished, and the project’s much larger claim about P versus NP is not established.

<p>To check a computer program with mathematics, this project turns the program and its input into a long checklist of yes-or-no conditions. This milestone verifies that the checklist builder can place the first complete item in the fourth section, stop exactly where the next item should begin, and fail safely if its workspace is damaged.</p><p>Think of a document generator: the fourth section heading was already in place, and this update checks the first full line beneath it. This is one small verified construction step. The fourth section and the full checklist are still unfinished, and the project’s much larger claim about P versus NP is not established.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-21-fourth-clause-first-literal">Read the technical details on PNPLabs.</a></p> https://pnplabs.com.au/updates.html#2026-07-20-fourth-clause-separator 2026-07-20T13:11:02Z 2026-07-20T13:11:02Z

This project turns a computer program and its input into a long checklist of logical conditions. The work is being verified one small construction step at a time. This milestone confirms that the builder can add the marker that starts the fourth section of that checklist and then move to the correct place for the next symbol. In everyday terms, it is like checking that a document generator placed the next section break in exactly the right spot. It is useful progress, but it does not complete the fourth section, finish the full generator, or establish the project’s overall mathematical claim.

<p>This project turns a computer program and its input into a long checklist of logical conditions. The work is being verified one small construction step at a time. This milestone confirms that the builder can add the marker that starts the fourth section of that checklist and then move to the correct place for the next symbol.</p><p>In everyday terms, it is like checking that a document generator placed the next section break in exactly the right spot. It is useful progress, but it does not complete the fourth section, finish the full generator, or establish the project’s overall mathematical claim.</p><p><a href="https://pnplabs.com.au/updates.html#2026-07-20-fourth-clause-separator">Read the technical details on PNPLabs.</a></p>