From a34723742801f877c9636ed72f176a7ffa7bf005 Mon Sep 17 00:00:00 2001 From: Hanson Sun <182361798+bbMisk@users.noreply.github.com> Date: Tue, 13 Oct 2026 23:57:53 -0400 Subject: [PATCH] Prove the three-mode singleton P112 COPY floor with independent checks --- .../CERTIFICATE.json | 6739 +++++++++++++++++ .../p112-three-mode-singleton/OVERLAP.json | 126 + research/p112-three-mode-singleton/README.md | 296 + .../p112-three-mode-singleton/SOURCES.json | 48 + research/p112-three-mode-singleton/TASK.json | 76 + .../p112-three-mode-singleton/VALIDATION.json | 94 + .../p112-three-mode-singleton/VERIFY.json | 44 + .../independent_check.py | 205 + .../p112-three-mode-singleton/test_package.py | 84 + research/p112-three-mode-singleton/verify.py | 305 + 10 files changed, 8017 insertions(+) create mode 100644 research/p112-three-mode-singleton/CERTIFICATE.json create mode 100644 research/p112-three-mode-singleton/OVERLAP.json create mode 100644 research/p112-three-mode-singleton/README.md create mode 100644 research/p112-three-mode-singleton/SOURCES.json create mode 100644 research/p112-three-mode-singleton/TASK.json create mode 100644 research/p112-three-mode-singleton/VALIDATION.json create mode 100644 research/p112-three-mode-singleton/VERIFY.json create mode 100644 research/p112-three-mode-singleton/independent_check.py create mode 100644 research/p112-three-mode-singleton/test_package.py create mode 100644 research/p112-three-mode-singleton/verify.py diff --git a/research/p112-three-mode-singleton/CERTIFICATE.json b/research/p112-three-mode-singleton/CERTIFICATE.json new file mode 100644 index 0000000000..425bfeef4c --- /dev/null +++ b/research/p112-three-mode-singleton/CERTIFICATE.json @@ -0,0 +1,6739 @@ +{ + "controls": { + "hermitian_substitution_rejected": true, + "omitted_compensation_entry": [ + 9, + 9 + ], + "rank_eight_per_active_term_refuted": true, + "singular_mixing_rejected": true, + "wrong_inverse_rejected": true + }, + "fixtures": [ + { + "checked_entries": 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b/research/p112-three-mode-singleton/OVERLAP.json new file mode 100644 index 0000000000..700e493518 --- /dev/null +++ b/research/p112-three-mode-singleton/OVERLAP.json @@ -0,0 +1,126 @@ +{ + "status": "SHARED_SCOPE_WITH_EARLIER_PARALLEL_CLAIM_NOT_A_NOVELTY_CERTIFICATE", + "initial": { + "searched_at": "2026-10-14T03:33:49.077562+00:00", + "coverage": { + "open_prs": { + "fetched_at": "2026-10-14T03:33:23.608178+00:00", + "complete_pagination": true, + "records": 726 + }, + "discussions": { + "fetched_at": "2026-10-14T03:33:49.033812+00:00", + "complete_pagination": true, + "records": 233 + } + }, + "terms": [ + "complex-masked-decoder", + "masked decoder", + "overcomplete", + "eight-term", + "complex-decoder-field-extension-seed44", + "complex-frame-pair-decoders" + ], + "warnings": [], + "limits": "Remote PR scope is open PRs including drafts. Closed work is searched through local READMEs, TASK metadata and ledger only. No code diffs or arbitrary package files are searched. Recheck relevant heads before submission." + }, + "final": { + "searched_at": "2026-10-14T03:52:09.694842+00:00", + "coverage": { + "open_prs": { + "fetched_at": "2026-10-14T03:51:44.051691+00:00", + "complete_pagination": true, + "records": 733 + }, + "discussions": { + "fetched_at": "2026-10-14T03:52:09.642498+00:00", + "complete_pagination": true, + "records": 244 + } + }, + "terms": [ + "singleton", + "three-mode", + "one-column", + "complex-masked-decoder", + "masked decoder", + "overcomplete", + "eight-term", + "complex-decoder-field-extension-seed44", + "complex-frame-pair-decoders" + ], + "warnings": [], + "limits": "Remote PR scope is open PRs including drafts. Closed work is searched through local READMEs, TASK metadata and ledger only. No code diffs or arbitrary package files are searched. Recheck relevant heads before submission." + }, + "related": [ + { + "kind": "pr", + "number": 1512, + "title": "Scoped negative result: three-mode P112 COPY64 despite four span-loss exceptions", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1512", + "head": "403158808f075dfafcd80da09f630b9a3fde6baa", + "updated_at": "2026-10-14T02:06:52Z" + }, + { + "kind": "pr", + "number": 1473, + "title": "Scoped negative result: two-anchor span preservation for P112 decoders", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1473", + "head": "ea2f6bf0940580dfaeb393ec0786abc8f7bbb7b4", + "updated_at": "2026-10-14T01:20:30Z" + }, + { + "kind": "pr", + "number": 1468, + "title": "Research: independent COPY floor for arbitrary one-column P112 corrections", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1468", + "head": "5e2d8ec8ebff17c85e813ef1bd6f06312c1fe16a", + "updated_at": "2026-10-14T01:12:52Z" + }, + { + "kind": "pr", + "number": 1432, + "title": "Scoped negative result: fixed P112 24-row boundary retains COPY64", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1432", + "head": "4bcc49e3f4041018b0607da46a87d8928a6e0fd0", + "updated_at": "2026-10-14T00:47:58Z" + }, + { + "kind": "pr", + "number": 1375, + "title": "Research: COPY floor 64 for two-coordinate P112 mixing with small free rectangles (no new kappa)", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1375", + "head": "4a533aff2a7a87c9b36e077a31774b332a7f64bd", + "updated_at": "2026-10-13T23:46:30Z" + }, + { + "kind": "discussion", + "number": 1360, + "title": "CLAIM: sharp copy floor for all rank-nine factorizations of a single odd-entry P112 completion", + "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1360", + "updated_at": "2026-10-14T03:39:44Z" + }, + { + "kind": "discussion", + "number": 1581, + "title": "Unrestricted two-factor P112 masked decoders: exact tangent-rank audit", + "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1581", + "updated_at": "2026-10-14T03:34:59Z" + }, + { + "kind": "discussion", + "number": 1550, + "title": "CLAIM: three-mode P112 singleton-column correction with at least two rows", + "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1550", + "updated_at": "2026-10-14T02:48:20Z" + } + ], + "discussion1550": { + "scope": "Same designated triple and arbitrary GL(4,C) with a one-column free correction and support size at least two. Its earlier claim was recognized at the final targeted check. No proof or artifact link was present in that snapshot.", + "body_sha256": "4c631db113b200775dcea39acffa078943390e67158ee37c382d8bb91e476282", + "author_credit": "Andrey Mas (@an664), with GPT-6 Pro research assistance and Codex coordination.", + "disposition": "Submit as an explicitly parallel independent proof and finite check, not as an unclaimed question. All 56 triples and zero/single-entry boundaries are included; no claim that these extensions supersede the parallel investigation." + }, + "new_artifact": "Written compensation proof, complete 170352 one-anchor pairing table, code-disjoint recomputation, all 56 explicit transports, attaining and failure fixtures. No pending theorem is imported." +} diff --git a/research/p112-three-mode-singleton/README.md b/research/p112-three-mode-singleton/README.md new file mode 100644 index 0000000000..96fcadf236 --- /dev/null +++ b/research/p112-three-mode-singleton/README.md @@ -0,0 +1,296 @@ +# Three-mode P112 mixing with a one-column correction retains COPY64 + +**Scoped negative result: a written quantified proof with exact finite premises.** +Mix any three P112 coordinate terms and one arbitrary free one-column correction +by any `GL(4,C)` change of factors. The complete separate COPY sum has sharp floor +**64**, and the complete COPY power bill has sharp floor `8*8^q` for `0=2`. +It was recognized at the final targeted overlap check; its snapshot contains a +claim, not a proof or artifact to compare. This submission credits that shared +scope and does not claim to have originated the follow-up. Its independently +reviewable contribution is the explicit proof, complete finite certificate, +second implementation, all-triple transport and zero/single-entry boundaries. + +No new supplier or improved kappa is claimed. The selected record and frozen +inputs remain unchanged. + +## Reproduce + +From the repository root, Python 3.12 and the standard library: + +```sh +python3.12 -I -B scripts/research_package.py --package research/p112-three-mode-singleton +``` + +The registered commands check source pins, recompute the primary receipt, run a +second implementation with assertions disabled, and run eight external tests. +Expected final marker: + +```text +PASS_REGISTERED_RESEARCH_PACKAGE research/p112-three-mode-singleton outcome=negative-result +``` + +[verify.py](verify.py) uses PR845's pinned simple-root labels and enumerates +normal intersections. [independent_check.py](independent_check.py) imports no +primary or predecessor code: it constructs Euclidean E8 roots, closes proper +rational row spaces by RREF, and eliminates full 112-bit pairing rows from the +opposite pivot direction. They compare every finite support and pairing, rather +than only their minimum or histogram. Both implementations have the same +AI-assisted author; their agreement is not independent human review. + +[CERTIFICATE.json](CERTIFICATE.json) contains all support representatives, +character spectra, pairing tables, coordinate transports and fixture results. +Default replay only reads files. To regenerate a new receipt: + +```sh +python3.12 -I -B research/p112-three-mode-singleton/verify.py --write /tmp/new-singleton-receipt.json +``` + +The output path must not already exist. [VALIDATION.json](VALIDATION.json) +records the actual checks and their limits. + +## Model and exact statement + +Use the PR845/PR717 P112 geometry: choose the lexicographically larger sign of +each doubled E8 root line, remove the eight lines +`f_(2k)=2e_(2k)+2e_(2k+1)` and `f_(2k+1)=2e_(2k)-2e_(2k+1)`, then sort the +112 remaining lines. Define + +```text +G_ij = (r_i dot r_j)/4, +X_ik = (r_i dot f_k)/4, +B_ij = (1+G_ij) mod 2. +``` + +Then `XX^T=2G`. The entries of X are `0,+1,-1`. B is the nondegenerate +pairing on the nine-dimensional binary label space; the 112 labels are distinct +and nonzero. For supports A,C, let `rho(A,C)=rank_F2 B[A,C]`. A separately +charged scalar term `d t^T` costs `rho(supp(d),supp(t))`. Its complex values +determine its supports and are never reduced modulo two. All transposes below +are ordinary transposes. + +Choose any three coordinate indices J and put `Y=X[:,J]`. Fix a live site j, +write `O_j={i:B_ij=0}`, and take arbitrary complex u supported on O_j. Set +`v=lambda e_j`, where lambda is any nonzero complex number. For any `M in GL(4,C)`, +put + +```text +N=M^(-T), D=[Y,u]M, T=[Y,v]N. +``` + +The four terms have product `YY^T+uv^T`. Together with the other five coordinate +terms they agree with `2G` on every constrained entry `B_ij=1`. Let r_h be the +COPY ranks of these four terms. + +**Theorem.** For every such choice and every real `0= 3*8^q. +``` + +Adding the five unchanged rank-eight terms gives the stated floor `8*8^q`, +and at q=1 gives 64. Identity mixing attains both for every allowed u and v: +its four ranks are `(8,8,8,0)`. + +## Exhaustive finite geometry + +It suffices to study `J=(2,4,5)`. The certificate provides 56 explicit +coordinate permutations carrying this triple to every triple. Both checkers +verify their signed bijections on all 112 root lines and all 12,544 entries of +B. Row signs and coordinate order can be absorbed into u, v and M without +changing supports. This transport uses no assumed group-classification result. + +The 112 projected rows have 13 nonzero projective directions, precisely the +directions of `{0,+1,-1}^3`. For any nonzero `a in C^3`, the rational span of +the projected rows annihilating a is a proper rational flat of rank at most +two. Rational and complex ranks of rational matrices agree. The zero rows +are exactly those lying in this flat. Conversely, a generic normal to any +proper flat avoids all directions outside it; the primary checker supplies +an explicit representative for each resulting support. Therefore enumerating +these flats exhausts complex supports, not just a finite coefficient grid. + +There are one rank-zero, 13 rank-one and 25 rank-two flats, giving **39 supports**. +This is the three-mode arrangement from PR1512, recomputed in both implementations. +For these supports S,T, the exact new one-anchor pairing table and retained +baseline facts are: + +| Finite premise | Complete evidence | +|---|---| +| Original `rho(S,T)>=8` | 1,521 pairs: 225 of rank 8, 1,296 of rank 9 | +| Minimum positive character weight on S is at least 12 | 19,968 evaluations, including zero characters/weights | +| `rho(S intersect O_j-complement,T)>=7` | 170,352 cases: 19,464 of rank 7, 150,888 of rank 8 | +| Singleton deletion preserves each support's label span | 4,368 direct independent deletion checks | + +The minimum positive character weights have distribution +`{12:6,24:10,30:12,32:6,38:4,40:1}` across the 39 supports. A separating +binary functional proves that deleting at most eleven rows from any support +preserves its full label span: otherwise a nonzero character restriction of +weight at most eleven would exist. This also proves singleton preservation. +The primary enumerates all 512 binary functionals; the independent checker +constructs the full dual code from the pairing matrix. + +The one-anchor survivor dimensions are seven in 344 cases and eight in 4,024. +A dimension-only estimate would give pairing rank six in some cases. The exact +**pairing** minimum seven is what lets the three-mode compensation bill close. + +## Proof of the complete four-term bill + +Write each column of M as `(a_h,c_h)` and each column of N as `(b_h,d_h)`, +with `a_h,b_h in C^3`. From `MN^T=I_4`, + +```text +sum_h a_h b_h^T = I_3. (1) +``` + +Call a product active when both a_h and b_h are nonzero. There are at least +three active products by rank subadditivity in (1). Each projection matrix +has rank three, hence has at most one zero column. + +An active left factor contains every row of `supp(Ya_h)` outside U=supp(u). +In particular it contains `supp(Ya_h) intersect O_j-complement`. An active +right factor contains `supp(Yb_h)` outside `{j}`, whose label span is unchanged +by singleton deletion. The one-anchor pairing table and rank monotonicity +therefore give **r_h>=7 for every active product**. If `|U|<=11`, the full +left span also survives, so every active product instead has r_h>=8. This +already proves the theorem for small U, including u=0. + +Assume now `|U|>=12`. The following cases exhaust the projection patterns. + +1. **Some right projection b_h is zero.** Its column of N is a nonzero + multiple of the fourth coordinate vector. `M^T N=I_4` forces `c_k=0` + for every k different from h. The other three left factors are thus + uncorrected, nonzero combinations of Y. Their right projections are nonzero + and retain their full spans after singleton deletion. All three have + pairing rank at least eight by the original pairing table. The fourth + term has nonnegative cost. +2. **Every right and left projection is nonzero.** All four terms cost at + least seven, so their rank sum is at least 28. +3. **Every right projection is nonzero and one left projection is zero.** + This pure left factor is a nonzero scalar multiple of u. Its support U + has at least twelve distinct nonzero labels, so its span has dimension + at least four: dimension three has only seven nonzero vectors. Its right + partner has span at least eight. In a nondegenerate nine-dimensional + pairing space, subspaces A,C satisfy + `rank B[A,C] >= dim(A)+dim(C)-9`. Consequently this pure factor costs + at least three. The other three products are active and each costs at + least seven. The complete rank sum is at least `3+3*7=24`. + +Up to permutation, the four actual ranks therefore dominate one of + +```text +(8,8,8,0), (7,7,7,7), (7,7,7,3). (2) +``` + +Every entry t in these lower envelopes is at most eight. For `0= t/8`, including t=0. Each envelope sums to at least 24, so its +q-power sum is at least `3*8^q`. Monotonicity transfers this to the actual +ranks, including those exceeding eight. This proves the theorem. Symmetry +of B and transposition give the one-row version. + +For fixed width m, normalization and multiplicity, set `q=1-a`, `0<=a<1`. +The affected optimistic moment contribution cannot beat `3*(8/m)^q`, even +when additional preparation, frames, helpers, source access, restoration and +hosting are granted zero cost. This screens the stated COPY mechanism; it +does not bound other work or a changed billing contract. + +## Why the compensation must remain in the bill + +In the specified ordering, take j=0 and +`u_i=X_i,2` on O_0, zero elsewhere, with v=e_0. Its actual support has size +24. Set `M=I_4-E_(3,0)`. The four factors are + +```text +D = [x_2-u, x_4, x_5, u], +T = [x_2, x_4, x_5, x_2+e_0]. +``` + +Their ranks are exactly `(7,8,8,7)`, so the whole COPY bill is 70. Only two +children retain rank eight; the stronger per-child assertion from the +two-sided case would be false here. Dropping the compensating fourth term +would give the apparent bill 63, but violates the constrained diagonal entry +`(9,9)`. Both implementations check the full 12,544-entry identity and all four +ranks. The cancellation mechanism is credited to PR1432/PR1468; the additional +coordinate term and three-mode quantified bound are the scope here. + +Seven primary fixtures check 87,808 scalar entries: identity, compensation, +dense complex mixing, a single-entry correction, u=0, a pure right projection, +and transposition. Controls reject inverse/transpose confusion, Hermitian +substitution, singular mixing and omitted compensation. The external tests +also reject a missing stratum, wrong support representative, changed pairing +rank, forged floor 63, invalid transport, missing source pin and changed +geometry bytes. Their rejection checks remain active under `-O`. + +## Provenance, overlap and limits + +Initial main and mathematical base: `1f327c6f3e3a35271c5c141d381b540bab8ad005`. +The selector nonce was `097f8f53ac691a8f49688424ca8eba6b`, priority revision 2, +card `complex-masked-decoder`. The scope is refined to the open singleton case; +it changes the eight-term all-pairs premise to a nine-term masked decoder with +five original coordinate terms retained. [TASK.json](TASK.json) records the +bounded question and stop rule. The card's complex-field starting example was +replayed successfully. + +The isolated branch was fast-forwarded to submission main `81e81af6` after +revision 3 of the priorities arrived. The selected record and both live pinned +inputs were unchanged. The original assignment remains historical provenance; +the present COPY power obstruction makes no new supplier or kappa claim. + +[SOURCES.json](SOURCES.json) pins main inputs and the contextual predecessor +heads: PR1512 `403158808f075dfafcd80da09f630b9a3fde6baa`, PR1468 +`5e2d8ec8ebff17c85e813ef1bd6f06312c1fe16a`. [OVERLAP.json](OVERLAP.json) +records live coverage and limitations. Discussion1360 varies all nine factors +for a **single-entry** completion; this result varies four factors for an +**arbitrary one-column** correction. Neither scope contains the other. +Discussion1581 concerns the rank-eight tangent problem. PR1512's both-sides +size-at-least-two result remains complementary and separately reviewable. + +Four or more mixed coordinate modes, multiple correction terms, other punctures +and shared COPY billing are outside this proof. The universal argument is +written, not kernel checked. There is no independent human review, physical +word, native admission, selected-construction cold replay, all-size compiler, +common-label/mutable-source realization, owner/hosting proof, precision/tape +analysis or outer analytic transfer here. No multiplication theorem or frontier +change follows from this finite certificate. + +Credit to Hanson Sun / PR1512 for the three-mode arrangement and open question; +Andrey Mas (@an664) / Discussion1550 for the earlier parallel investigation; +Gareth Davidson (@bitplane) / PR1468 for the singleton compensation argument +with two modes; commdata2338 / PR1375 and PR1432/PR1460/PR1473 for preceding +support-erasure and compensation work; Mihir Srivastava / PR845 for the pinned +geometry; DreamingOfClouds / PR717 and PR446/595 for the decoder/COPY model; +themilkman216 / PR968 for the pairing-cost floor; Joel Pulikkan / GamingPuzzled / +PR620 for the coordinate COPY64 construction; and woklerx / PR1077 for the +complex-field baseline. Existing source attribution is retained. + +Prepared for **Hanson Sun (@bbMisk)** with substantial **OpenAI Codex** assistance +in mathematics, both implementations, checking and writing. No subagents were +used. New work is Apache-2.0. + +## Open next questions + +- **Four coordinate modes with one singleton-sided correction.** The + one-anchor pairing table might retain a useful bound, but even if the same + rank-seven premise held, `(7,7,7,7,3)` sums to 31, below the four-mode baseline 32. + First enumerate proper flats of the four-coordinate projected rows and + classify the low-pairing cases together with the pure-factor support. + A stronger coupled bound or an exact improving factorization is needed; + simply repeating the present dimension estimate cannot close it. Any + favorable COPY result would still need the full physical supplier budget. +- **More than one free correction.** Multiple correction directions can + erase larger support spans and invalidate the single pure-factor split. + A feasible first test is a specified two-column support and exact `GL(5,C)` + mixing with three retained modes. The blocker is a new complete support + and compensation classification, followed by full charging of any candidate. diff --git a/research/p112-three-mode-singleton/SOURCES.json b/research/p112-three-mode-singleton/SOURCES.json new file mode 100644 index 0000000000..5e0ec48a85 --- /dev/null +++ b/research/p112-three-mode-singleton/SOURCES.json @@ -0,0 +1,48 @@ +{ + "schema_version": 1, + "base_commit": "1f327c6f3e3a35271c5c141d381b540bab8ad005", + "files": [ + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11", + "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32" + }, + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "c05fafaa25ba95b2d5af86b1800427198e3d20eb", + "sha256": "89b5928bc92c81e2647b6d702a7ad85a6795b8f44e2eb267b2ff3145d8f2b2ce" + }, + { + "path": "research/complex-decoder-field-extension-seed44/PROOF.md", + "usage": "input", + "git_blob": "4a3063de8fe606a88bd5bdb9b7d2e5dccb70f73f", + "sha256": "a89530576d8a5e1ae8778d601b37140d954212cd46e06b87a3f565294ee251a2" + }, + { + "path": "research/p112-single-lift/geometry.py", + "usage": "input", + "git_blob": "34c1abf059d7be5fe5e5044cf01a09b843071c68", + "sha256": "3857c11ec731a602b8059d85d58fbb3b641bab687bb33016a0e54f7567752c8a" + } + ], + "context_predecessors": [ + { + "pr": 1512, + "head": "403158808f075dfafcd80da09f630b9a3fde6baa", + "path": "research/p112-three-mode-seed40/README.md", + "git_blob": "c80b963aced8a36c4f6ea3d2b941977bfd866dbb", + "sha256": "ca6b23d64f53086b0095c0104c2ea559198bdab8f943ce4c2722bb45207c548a", + "usage": "read for scope, proof comparison and attribution; no runtime import or theorem assumption" + }, + { + "pr": 1468, + "head": "5e2d8ec8ebff17c85e813ef1bd6f06312c1fe16a", + "path": "research/p112-column-correction-floor/PROOF.md", + "git_blob": "baf7ebcdae4b5d33426014d27eead333c387ca13", + "sha256": "ed210c3f03b56dd973fb507d2ff99a04512e37043399f43b3e9d7f78f871c499", + "usage": "read for scope, proof comparison and attribution; no runtime import or theorem assumption" + } + ] +} diff --git a/research/p112-three-mode-singleton/TASK.json b/research/p112-three-mode-singleton/TASK.json new file mode 100644 index 0000000000..befa0ec1da --- /dev/null +++ b/research/p112-three-mode-singleton/TASK.json @@ -0,0 +1,76 @@ +{ + "schema_version": 1, + "task_id": "complex-masked-decoder", + "priority_revision": 2, + "base_commit": "1f327c6f3e3a35271c5c141d381b540bab8ad005", + "original_card": { + "id": "complex-masked-decoder", + "question": "Can one specified masked or overcomplete decoder improve the complete moment budget?", + "starting_points": [ + "research/complex-decoder-field-extension-seed44/README.md", + "research/complex-frame-pair-decoders/README.md" + ], + "methods": [ + "Synthesize a decoder with exact coefficients and fully priced helper endpoints.", + "Bound the best possible budget for a fixed support alphabet before synthesis." + ], + "acceptance_gate": "Change an assumption of the eight-term all-pairs bound and count all copied information, frames and restoration.", + "first_step": { + "command": [ + "python3", + "-B", + "research/complex-decoder-field-extension-seed44/verify.py" + ], + "expected": "PASS_COMPLEX_DECODER_FIELD_EXTENSION_S44", + "inspect": "Locate the excluded classes: masked identities, at least nine terms, and deeper punctures.", + "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay." + }, + "first_experiment": "Choose one finite support alphabet and test exact decoding constraints before building a whole network.", + "bottleneck_link": "A cheaper fully realized decoder could increase deficit or lower recursive cost on the binding complex side.", + "allowed_variations": [ + "Select a specific masked or overcomplete family with exact labels and supports." + ], + "scope_change_triggers": [ + "Repeating the exact eight-term all-pairs search inside its proved scope." + ], + "overlap_terms": [ + "masked decoder", + "overcomplete", + "eight-term" + ] + }, + "scope_status": "refined", + "actual_scope": "P112 three-coordinate terms plus one arbitrary free one-column or one-row correction, mixed by GL(4,C), under separate COPY billing.", + "scope_change_reason": "PR1512 explicitly leaves singleton-sided corrections open; PR1468 treats only two modes, and discussion1360 fixes a one-entry correction while varying all nine factors. An earlier matching claim in Discussion1550 was recognized at the final targeted refresh; this contribution is explicitly parallel and independently checkable, without a claim of exclusive novelty.", + "bottleneck_link": "Test the complete separate COPY power bill against three rank-eight children before spending effort on physical realization; all added overhead is optimistically zero.", + "budget_minutes": 60, + "stop_rule": "One 60-minute session. Enumerate all support strata and one-anchor survivors first. Stop or report a precise gap if coupled compensation cannot be bounded; reserve final quarter for verification.", + "outcome": "negative-result", + "evidence": "CERTIFICATE.json: 39 exhaustive complex support strata, 170352 one-anchor pairing ranks, all 56 coordinate transports, seven exact fixtures; independent_check.py recomputes the geometric premises without primary or predecessor imports. README supplies the quantified compensation proof.", + "author": "Hanson Sun (@bbMisk), with substantial OpenAI Codex assistance", + "ai_assistance": "Substantial OpenAI Codex assistance in mathematics, code, checking and writing; no subagents or independent human review.", + "open_questions": [ + { + "question": "Can four coordinate modes and one singleton-sided correction lower the separate COPY power bill?", + "why_it_matters": "This changes the mode-count hypothesis; even a preserved rank-seven bound plus rank-three compensation gives only 31 against baseline 32.", + "first_step": "Enumerate proper rational flats of the four-coordinate root projections and classify low-pairing survivor supports jointly with compensation.", + "blocked_by": "A stronger coupled lower bound or an exact improving factorization, followed by complete physical pricing." + }, + { + "question": "Can two free correction directions improve the three-mode COPY bill?", + "why_it_matters": "The one-correction proof has only one possible pure projection and a single odd-neighborhood survivor.", + "first_step": "Choose two explicit correction columns and enumerate the projected support losses before testing GL(5,C) mixing.", + "blocked_by": "A new support and compensation classification, plus full physical costs for any positive candidate." + } + ], + "selection_nonce": "097f8f53ac691a8f49688424ca8eba6b", + "scope_key": "7acfe4d9cac04f7ba4cb:p112-three-mode-singleton-GL4", + "coordination": "Local scope card; read-only live Discussion/PR overlap check, 726 open PRs and 233 Discussion records with complete pagination at 2026-10-14T03:33Z. No public claim posted. Final check: 733 open PRs and 244 Discussion records, complete pagination and no warnings. Earlier Discussion1550 has the same core scope and is explicitly credited.", + "predecessors": { + "PR1512": "403158808f075dfafcd80da09f630b9a3fde6baa", + "PR1468": "5e2d8ec8ebff17c85e813ef1bd6f06312c1fe16a" + }, + "submission_main": "81e81af67714bdc8d1693b0bf5c29468b92f62ff", + "submission_priority_revision": 3, + "priority_update": "Main advanced to revision 3 during the session. The original revision-2 task/source provenance is retained; the exact COPY power bound remains a scoped negative screen, with no new kappa or supplier claim." +} diff --git a/research/p112-three-mode-singleton/VALIDATION.json b/research/p112-three-mode-singleton/VALIDATION.json new file mode 100644 index 0000000000..79fdc889eb --- /dev/null +++ b/research/p112-three-mode-singleton/VALIDATION.json @@ -0,0 +1,94 @@ +{ + "status": "LOCAL_SCOPED_CHECKS_PASSED", + "timestamp_utc": "2026-10-14T03:57:02.605256+00:00", + "initial_main": "1f327c6f3e3a35271c5c141d381b540bab8ad005", + "submission_main": "81e81af67714bdc8d1693b0bf5c29468b92f62ff", + "environment": { + "platform": "Darwin", + "machine": "arm64", + "registered_python": "Python 3.12.13", + "optimized_python": "3.14.6", + "dependencies": "Python standard library only" + }, + "commands": [ + { + "argv": [ + "python3.12", + "-I", + "-B", + "scripts/research_package.py", + "--package", + "research/p112-three-mode-singleton" + ], + "exit_code": 0, + "coverage": "Primary deterministic receipt and source pins; independent geometry/flat/pairing checks with -O; eight tests (one positive, seven external corruptions).", + "log_sha256": "ea1178bbdc3a885d006b2fe7db890fd771519d4bb81c7fe3564c025ea3e33a95" + }, + { + "argv": [ + "python3", + "-I", + "-O", + "-B", + "research/p112-three-mode-singleton/verify.py" + ], + "exit_code": 0, + "coverage": "Optimized primary replay under Python 3.14.6; identical committed certificate.", + "log_sha256": "ef3fcacd20f4e3bab047c335d0902696d61bd66af4f719ec747173e6e331bcc4" + }, + { + "argv": [ + "python3.12", + "-B", + "scripts/research_baseline.py" + ], + "exit_code": 0, + "coverage": "Selected source bindings and arithmetic only, not a cold construction replay." + }, + { + "argv": [ + "python3.12", + "-B", + "scripts/verify_selected_result.py" + ], + "exit_code": 0, + "coverage": "Selected record, exact kappa, original source inventory and proof supplements." + }, + { + "argv": [ + "python3", + "-B", + "research/complex-decoder-field-extension-seed44/verify.py" + ], + "exit_code": 0, + "coverage": "Original task card baseline: 25 puncture examples, 14400 pairings and seven controls." + } + ], + "package_file_sha256": { + "verify.py": "a163071dc3d2a34be1ea88f6367a6c963adf9074d16f57181ec09eef8a01e036", + "independent_check.py": "ddda267f66f6cade87f97781f60e195538a3ce4eaff8226638fbf81e4d71fde8", + "test_package.py": "126b144b13bb485696af1e0a0ca199619ee24e192f714dcef8c00819507a6655", + "CERTIFICATE.json": "a43cc5291e82fb6af6c13539ff30cc7a896c4df6c3fae148a76bf5603fdc769b", + "SOURCES.json": "2cebbbbbe6b5427ed9b99956ebee5070736467bfe1aff90ad8ca9958ab7f8e01", + "VERIFY.json": "bb6c33f22bb943af05c264bc9c6b0e38357d0381ecd0f96ae4821bdab6e0f21e" + }, + "scope": { + "finite_pairings": 170352, + "singleton_erasures_independent": 4368, + "coordinate_transports": 56, + "primary_fixture_entries": 87808, + "independent_compensation_entries": 12544, + "sharp_separate_COPY_sum": 64, + "power_exponents": "0 < q <= 1", + "theorem_kind": "Written quantified proof with completely recomputed finite premises, not kernel formalization." + }, + "not_claimed": [ + "Independent human review", + "Hosted CI success at the time of this local receipt", + "Physical supplier or native admission", + "Selected-construction cold replay", + "All-size/compiler, hosting, precision/tape or outer analytic obligations", + "New kappa or selected-frontier change" + ], + "overlap": "Earlier Discussion1550 has the same core scope; this is explicitly parallel independent work with retained attribution." +} diff --git a/research/p112-three-mode-singleton/VERIFY.json b/research/p112-three-mode-singleton/VERIFY.json new file mode 100644 index 0000000000..d4ae98d421 --- /dev/null +++ b/research/p112-three-mode-singleton/VERIFY.json @@ -0,0 +1,44 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [ + "research/complex-decoder-field-extension-seed44/PROOF.md", + "research/p112-single-lift/geometry.py" + ], + "commands": [ + { + "argv": [ + "{python}", + "-I", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 120, + "checks": "Exact source pins, complete finite premises, ordinary-transpose fixtures, compensation and orientation controls, deterministic receipt comparison." + }, + { + "argv": [ + "{python}", + "-I", + "-O", + "-B", + "{package}/independent_check.py" + ], + "timeout_seconds": 120, + "checks": "Code-disjoint Euclidean roots and rational-flat enumeration, all pairing tables, singleton erasures, transports and integer compensation." + }, + { + "argv": [ + "{python}", + "-I", + "-O", + "-B", + "{package}/test_package.py" + ], + "timeout_seconds": 120, + "checks": "Positive independent replay and seven external source, coverage, rank, support, transport and claimed-floor corruption controls." + } + ] +} diff --git a/research/p112-three-mode-singleton/independent_check.py b/research/p112-three-mode-singleton/independent_check.py new file mode 100644 index 0000000000..bfb9d9e0eb --- /dev/null +++ b/research/p112-three-mode-singleton/independent_check.py @@ -0,0 +1,205 @@ +#!/usr/bin/env python3 +"""Rebuild the finite premises without importing primary or predecessor code. + +Direct Euclidean roots, rational flat closure, and full pairing-row elimination. +Hanson Sun, substantial OpenAI Codex assistance. Apache-2.0. +This and verify.py have the same AI-assisted author, not independent human review. +""" +import argparse +from collections import Counter +from fractions import Fraction +import hashlib +from itertools import combinations, product +import json +from pathlib import Path + +HERE = Path(__file__).resolve().parent +COORDINATES = (2,4,5) + + +def check(ok, message): + if not ok: + raise ValueError(message) + + +def fingerprint(value): + return hashlib.sha256(json.dumps(value,sort_keys=True,separators=(',', ':')).encode()).hexdigest() + + +def reduce_binary(vectors): + pivots={} + for v in vectors: + while v: + bit=v & -v + if bit in pivots: + v ^= pivots[bit] + else: + pivots[bit]=v + break + return tuple(pivots[b] for b in sorted(pivots)) + + +def dot(a,b): + return sum(u*v for u,v in zip(a,b)) + + +def rref(vectors): + rows=[list(map(Fraction,v)) for v in vectors] + used=0 + for col in range(3): + pivot=next((i for i in range(used,len(rows)) if rows[i][col]),None) + if pivot is None: + continue + rows[used],rows[pivot]=rows[pivot],rows[used] + divisor=rows[used][col] + rows[used]=[v/divisor for v in rows[used]] + for i in range(len(rows)): + if i!=used: + scalar=rows[i][col] + rows[i]=[v-scalar*w for v,w in zip(rows[i],rows[used])] + used+=1 + return tuple(tuple(r) for r in rows[:used]) + + +def geometry(): + all_roots=set() + for i,j in combinations(range(8),2): + for a,b in product((-2,2),repeat=2): + r=[0]*8;r[i]=a;r[j]=b + all_roots.add(tuple(r)) + all_roots.update(r for r in product((-1,1),repeat=8) if r.count(-1)%2==0) + check(len(all_roots)==240,'240 doubled E8 roots') + frame=[] + for i in range(0,8,2): + for s in (2,-2): + r=[0]*8;r[i]=2;r[i+1]=s;frame.append(tuple(r)) + roots=sorted({max(r,tuple(-v for v in r)) for r in all_roots}-set(frame)) + check(len(roots)==112,'112 live lines') + check(all(dot(r,r)==8 for r in roots),'root norms') + check(all(dot(r,f)%4==0 for r in roots for f in frame),'integral coordinate entries') + x=[[dot(r,f)//4 for f in frame] for r in roots] + check(all(dot(r,s)%4==0 for r in roots for s in roots),'integral Gram entries') + matrix=[[(1+dot(r,s)//4)%2 for s in roots] for r in roots] + rows=[sum(value<len(flat)] + supports.add(sum(1<>i&1] + full=reduce_binary(rows[i] for i in members) + weights=Counter((support&c).bit_count() for c in code) + distance=min(w for w in weights if w);distances[distance]+=1 + check(distance>=12 and record['minimum_erasure']==distance,'erasure minimum') + check(record['erasure_spectrum']=={str(k):v for k,v in weights.items()},'erasure spectrum') + check(record['span']==len(full),'support span') + original=[len(reduce_binary(r&t for r in full)) for t in supports] + check(min(original)>=8,'original pairing minimum') + check(record['original_pair_ranks']==''.join(map(str,original)),'original pairing table') + orig_counts.update(original) + dimensions=[];table=[] + for j in range(112): + # Direct deletion check: equal dimensions of nested spans prove equality. + check(len(reduce_binary(rows[i] for i in members if i!=j))==len(full),'singleton erasure') + singleton_cases+=1 + surviving=reduce_binary(rows[i] for i in members if matrix[i][j]) + dimensions.append(len(surviving));slice_counts[len(surviving)]+=1 + values=[len(reduce_binary(r&t for r in surviving)) for t in supports] + check(min(values)>=7,'one-anchor pairing minimum') + pair_counts.update(values);table.append(''.join(map(str,values))) + check(record['survivor_dimensions']==''.join(map(str,dimensions)),'survivor dimensions') + check(record['survivor_pair_ranks']==table,'full survivor pairing table') + summary=cert['summary'] + for key,counts in [('erasure_distance_counts',distances),('original_pair_counts',orig_counts), + ('survivor_dimension_counts',slice_counts),('survivor_pair_counts',pair_counts)]: + check(summary[key]=={str(k):v for k,v in counts.items()},'aggregate '+key) + check(summary['survivor_pair_cases']==sum(pair_counts.values())==170352,'pairing case count') + check(summary['minimum_erasure']==min(distances)==12,'distance summary') + check(summary['minimum_survivor_pair_rank']==min(pair_counts)==7,'pair minimum summary') + check(summary['sharp_whole_copy_sum']==64,'claimed floor') + targets=set();lookup={max(tuple(r),tuple(-v for v in r)):i for i,r in enumerate(x)} + for transport in cert['transports']: + p=transport['permutation'] + check(sorted(p)==list(range(8)),'coordinate permutation') + target=tuple(sorted(p[k] for k in COORDINATES)) + check(list(target)==transport['target'] and target not in targets,'triple transport uniqueness') + targets.add(target);indices=[];signs=[] + for r in x: + image=[r[p.index(k)] for k in range(8)] + normalized=max(tuple(image),tuple(-v for v in image)) + check(normalized in lookup,'transport root image') + i=lookup[normalized];indices.append(i);signs.append(1 if image==x[i] else -1) + check(len(set(indices))==112,'transport line bijection') + check(all(matrix[i][j]==matrix[indices[i]][indices[j]] for i in range(112) for j in range(112)), + 'transport preserves pairing') + check(transport['signed_map_sha256']==fingerprint([indices,signs]),'transport digest') + check(targets==set(combinations(range(8),3)),'all 56 coordinate triples') + # Separate integer derivation of the decisive cancellation/compensation example. + y=[[r[k] for k in COORDINATES] for r in x] + u=[r[0] if matrix[i][0]==0 else 0 for i,r in enumerate(y)] + d=[[r[0]-u[i],r[1],r[2],u[i]] for i,r in enumerate(y)] + t=[[r[0],r[1],r[2],r[0]+(i==0)] for i,r in enumerate(y)] + check(all(dot(d[i],t[j])==dot(y[i],y[j])+u[i]*(j==0) for i in range(112) for j in range(112)), + 'independent compensation identity') + costs=[] + for k in range(4): + target_mask=sum(1<=8)==2,'stronger per-child assertion is false') + # The bilinear rank oracle must distinguish a rank-one full block from two rows. + check(len(reduce_binary([3,3]))==1 and len(reduce_binary([1,2]))==2,'binary rank controls') + print('PASS_INDEPENDENT_SINGLETON_THREE_MODE pairings=170352 singleton_erasures='+str(singleton_cases) + +' transports=56 compensation_entries=12544') + + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--certificate',type=Path,default=HERE/'CERTIFICATE.json') + args=parser.parse_args() + check_receipt(json.loads(args.certificate.read_text())) + + +if __name__=='__main__': main() diff --git a/research/p112-three-mode-singleton/test_package.py b/research/p112-three-mode-singleton/test_package.py new file mode 100644 index 0000000000..3295cdc10f --- /dev/null +++ b/research/p112-three-mode-singleton/test_package.py @@ -0,0 +1,84 @@ +#!/usr/bin/env python3 +"""Positive replay and external corruption controls, active under Python -O. + +Hanson Sun, with substantial OpenAI Codex assistance. Apache-2.0. +""" +import json +from pathlib import Path +import shutil +import subprocess +import sys +import tempfile +import unittest + +PACKAGE=Path(__file__).resolve().parent +ROOT=PACKAGE.parents[1] + + +class CorruptionControls(unittest.TestCase): + def execute(self, script, *args): + return subprocess.run([sys.executable,'-I','-O','-B',str(script),*map(str,args)], + text=True,capture_output=True,timeout=120) + + def test_positive_independent_replay(self): + result=self.execute(PACKAGE/'independent_check.py') + self.assertEqual(result.returncode,0,result.stderr) + self.assertIn('PASS_INDEPENDENT_SINGLETON_THREE_MODE',result.stdout) + + def corrupted_certificate(self, mutate, reason): + receipt=json.loads((PACKAGE/'CERTIFICATE.json').read_text()) + mutate(receipt) + with tempfile.TemporaryDirectory(prefix='singleton-control-') as tmp: + path=Path(tmp)/'bad.json';path.write_text(json.dumps(receipt)) + result=self.execute(PACKAGE/'independent_check.py','--certificate',path) + self.assertNotEqual(result.returncode,0,'corrupt receipt was accepted') + self.assertIn(reason,result.stderr) + + def test_missing_stratum(self): + self.corrupted_certificate(lambda r:r['strata'].pop(),'support coverage') + + def test_wrong_support_representative(self): + self.corrupted_certificate(lambda r:r['strata'][0].update(coefficient=[0,0,0]), + 'coefficient-to-support binding') + + def test_changed_pair_rank(self): + def mutate(r): + row=r['strata'][0]['survivor_pair_ranks'][0] + r['strata'][0]['survivor_pair_ranks'][0]='6'+row[1:] + self.corrupted_certificate(mutate,'full survivor pairing table') + + def test_false_copy_floor(self): + self.corrupted_certificate(lambda r:r['summary'].update(sharp_whole_copy_sum=63),'claimed floor') + + def test_invalid_transport(self): + self.corrupted_certificate(lambda r:r['transports'][0].update(permutation=[0]*8), + 'coordinate permutation') + + def corrupted_source(self, missing_pin): + with tempfile.TemporaryDirectory(prefix='singleton-source-') as tmp: + root=Path(tmp);package=root/'research'/'p112-three-mode-singleton' + package.mkdir(parents=True) + shutil.copy2(PACKAGE/'verify.py',package/'verify.py') + pins=json.loads((PACKAGE/'SOURCES.json').read_text()) + for row in pins['files']: + if row['usage']=='input': + dest=root/row['path'];dest.parent.mkdir(parents=True,exist_ok=True) + shutil.copy2(ROOT/row['path'],dest) + geometry='research/p112-single-lift/geometry.py' + if missing_pin: + pins['files']=[r for r in pins['files'] if r['path']!=geometry] + else: + with (root/geometry).open('a') as f:f.write('\n# deliberate corruption\n') + (package/'SOURCES.json').write_text(json.dumps(pins)) + result=self.execute(package/'verify.py') + self.assertNotEqual(result.returncode,0,'corrupt source accepted') + self.assertIn('incomplete live source inventory' if missing_pin else 'source SHA256',result.stderr) + + def test_missing_geometry_pin(self): + self.corrupted_source(True) + + def test_changed_geometry_bytes(self): + self.corrupted_source(False) + + +if __name__=='__main__': unittest.main() diff --git a/research/p112-three-mode-singleton/verify.py b/research/p112-three-mode-singleton/verify.py new file mode 100644 index 0000000000..d2a56716a4 --- /dev/null +++ b/research/p112-three-mode-singleton/verify.py @@ -0,0 +1,305 @@ +#!/usr/bin/env python3 +"""Finite premises for the three-mode, singleton-sided P112 COPY theorem. + +Hanson Sun, with substantial OpenAI Codex assistance. Apache-2.0. +Geometry: Mihir Srivastava / PR845, following DreamingOfClouds / PR717. +Support arrangements and compensation: PR1512 and Gareth Davidson / PR1468. +The quantified proof is in README.md; examples alone are not that proof. +""" +import argparse +from collections import Counter +from fractions import Fraction +import hashlib +import importlib.util +from itertools import combinations, permutations, product +import json +from math import gcd +from pathlib import Path + +PACKAGE = Path(__file__).resolve().parent +ROOT = PACKAGE.parents[1] +GEOMETRY = 'research/p112-single-lift/geometry.py' +INPUTS = {GEOMETRY, 'research/complex-decoder-field-extension-seed44/PROOF.md'} +TRIPLE = (2, 4, 5) + + +def require(ok, message): + if not ok: + raise ValueError(message) + + +def digest(obj): + return hashlib.sha256(json.dumps(obj, sort_keys=True, separators=(',', ':')).encode()).hexdigest() + + +def bits(n): + while n: + b = n & -n + yield b.bit_length() - 1 + n ^= b + + +def mask(indices): + return sum(1 << i for i in indices) + + +def dot(a, b): + return sum(x*y for x, y in zip(a, b)) + + +def basis(rows): + pivots = {} + for r in rows: + while r: + k = r.bit_length() - 1 + if k not in pivots: + pivots[k] = r + break + r ^= pivots[k] + return [pivots[k] for k in sorted(pivots, reverse=True)] + + +def pairing(left_duals, right_labels): + return len(basis(sum(((a & b).bit_count() % 2) << k + for k, a in enumerate(left_duals)) for b in right_labels)) + + +def canonical(v): + g = gcd(*v) + if not g: + return None + v = tuple(x//g for x in v) + return v if next(x for x in v if x) > 0 else tuple(-x for x in v) + + +def cross(a, b): + return (a[1]*b[2]-a[2]*b[1], a[2]*b[0]-a[0]*b[2], a[0]*b[1]-a[1]*b[0]) + + +def load_geometry(): + rows = json.loads((PACKAGE/'SOURCES.json').read_text())['files'] + live = [r for r in rows if r['usage'] == 'input'] + require(len(live) == len(INPUTS) and {r['path'] for r in live} == INPUTS, + 'incomplete live source inventory') + for r in live: + p = ROOT/r['path'] + require(p.resolve().is_relative_to(ROOT.resolve()), 'external source path') + data = p.read_bytes() + require(hashlib.sha256(data).hexdigest() == r['sha256'], 'source SHA256: '+r['path']) + require(hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest() == r['git_blob'], + 'source Git blob: '+r['path']) + spec = importlib.util.spec_from_file_location('pinned_p112_geometry', ROOT/GEOMETRY) + module = importlib.util.module_from_spec(spec) + spec.loader.exec_module(module) + raw = module.build() + module.check_geometry(raw) + order = sorted(range(112), key=lambda i: max(tuple(raw['roots'][i]), tuple(-z for z in raw['roots'][i]))) + roots, x, labels, duals = [], [], [], [] + for i in order: + sign = 1 if tuple(raw['roots'][i]) > tuple(-z for z in raw['roots'][i]) else -1 + roots.append([sign*z for z in raw['roots'][i]]) + x.append([sign*z for z in raw['X'][i]]) + labels.append(raw['labels'][i]); duals.append(raw['duals'][i]) + b = [[(duals[i]&labels[j]).bit_count() % 2 for j in range(112)] for i in range(112)] + require(len(set(labels)) == 112 and all(labels), 'distinct nonzero labels') + require(len(basis(labels)) == len(basis(duals)) == 9, 'nondegenerate nine-dimensional labels') + require(all(dot(x[i], x[j]) == dot(roots[i], roots[j])//2 + and b[i][j] == (1+dot(roots[i], roots[j])//4) % 2 + for i in range(112) for j in range(112)), 'geometry identities') + return roots, x, labels, duals, b + + +def strata(x): + rows = [tuple(r[k] for k in TRIPLE) for r in x] + normals = sorted({canonical(r) for r in rows}-{None}) + rank_two = {canonical(cross(a,b)) for a,b in combinations(normals,2)}-{None} + require(len(normals) == 13 and len(rank_two) == 25, 'proper flat census') + coefficients = set(rank_two) + for n in normals: + a = next((a for a in product(range(-4,5), repeat=3) + if any(a) and dot(a,n) == 0 and all(dot(a,m) for m in normals if m != n)), None) + require(a is not None, 'rank-one flat representative') + coefficients.add(canonical(a)) + require(all(dot((1,3,9),n) for n in normals), 'generic flat representative') + coefficients.add((1,3,9)) + out = sorted((a,mask(i for i,r in enumerate(rows) if dot(r,a))) for a in coefficients) + require(len(out) == len({s for _,s in out}) == 39, '39 distinct supports') + return out + + +def coordinate_transports(x, b): + tetrads = {mask(k for k,v in enumerate(r) if v) for r in x} + lookup = {max(tuple(r),tuple(-v for v in r)):i for i,r in enumerate(x)} + out = {} + for perm in permutations(range(8)): + target = tuple(sorted(perm[k] for k in TRIPLE)) + if target in out or any(mask(perm[k] for k in bits(t)) not in tetrads for t in tetrads): + continue + line_map, signs = [], [] + for r in x: + new = [0]*8 + for k in range(8): new[perm[k]] = r[k] + key = max(tuple(new),tuple(-v for v in new)) + require(key in lookup, 'invalid coordinate transport') + i = lookup[key]; line_map.append(i); signs.append(1 if new == x[i] else -1) + require(len(set(line_map)) == 112, 'transport bijection') + require(all(b[i][j] == b[line_map[i]][line_map[j]] for i in range(112) for j in range(112)), + 'transport pairing') + out[target] = {'target':list(target), 'permutation':list(perm), 'signed_map_sha256':digest([line_map,signs])} + if len(out) == 56: break + require(set(out) == set(combinations(range(8),3)), 'all triples covered') + return [out[t] for t in sorted(out)] + + +# Exact Gaussian-rational examples test the factor contract, not the quantifier. +ZERO, ONE, IMAG = (0,0), (1,0), (0,1) +def add(a,b): return (a[0]+b[0],a[1]+b[1]) +def neg(a): return (-a[0],-a[1]) +def mul(a,b): return (a[0]*b[0]-a[1]*b[1],a[0]*b[1]+a[1]*b[0]) +def conj(a): return (a[0],-a[1]) +def divide(a,b): + n = b[0]*b[0]+b[1]*b[1] + require(bool(n), 'singular fixture matrix') + z = mul(a,conj(b)) + return (Fraction(z[0],n),Fraction(z[1],n)) +def identity(n): return [[ONE if i==j else ZERO for j in range(n)] for i in range(n)] +def transpose(a): return list(map(list,zip(*a))) +def mm(a,b): + out=[] + for row in a: + values=[] + for col in zip(*b): + v=ZERO + for s,t in zip(row,col): + if s != ZERO and t != ZERO: v=add(v,mul(s,t)) + values.append(v) + out.append(values) + return out +def inverse(a): + n=len(a); rows=[list(r)+e for r,e in zip(a,identity(n))] + for k in range(n): + pivot=next((i for i in range(k,n) if rows[i][k] != ZERO),None) + require(pivot is not None,'singular fixture matrix') + rows[k],rows[pivot]=rows[pivot],rows[k] + z=rows[k][k]; rows[k]=[divide(v,z) for v in rows[k]] + for i in range(n): + if i != k: + z=rows[i][k]; rows[i]=[add(v,neg(mul(z,w))) for v,w in zip(rows[i],rows[k])] + return [r[n:] for r in rows] + + +def fixtures(x,labels,duals,b): + y=[[(r[k],0) for k in TRIPLE] for r in x] + site=0 + boundary=[(r[TRIPLE[0]],0) if not b[i][site] else ZERO for i,r in enumerate(x)] + dense=[(1,i%3-1) if not b[i][site] else ZERO for i in range(112)] + small=[ZERO]*112; small[next(i for i in range(112) if not b[i][site])]=(1,1) + v=[ONE if i==site else ZERO for i in range(112)] + shear=identity(4); shear[3][0]=(-1,0) + complex_m=identity(4) + for i,j,z in [(3,0,IMAG),(0,1,ONE),(2,3,(1,1)),(1,2,ONE)]: + e=identity(4);e[i][j]=z; complex_m=mm(complex_m,e) + right_pure=identity(4);right_pure[0][3]=ONE;right_pure[1][3]=IMAG + cases=[('identity',boundary,identity(4),False),('compensation',boundary,shear,False), + ('dense_complex',dense,complex_m,False),('singleton_entry',small,complex_m,False), + ('zero_correction',[ZERO]*112,complex_m,False),('right_pure',dense,right_pure,False), + ('transposed_compensation',boundary,shear,True)] + results=[]; saved={} + for name,u,m,transposed in cases: + require(all(u[i]==ZERO or not b[i][site] for i in range(112)), 'nonfree correction') + n=transpose(inverse(m)) + require(mm(m,transpose(n))==identity(4),'coupled inverse identity') + left=[r+[z] for r,z in zip(y,u)];right=[r+[z] for r,z in zip(y,v)] + d,t=mm(left,m),mm(right,n) + expected=[[add((dot([x[i][k] for k in TRIPLE],[x[j][k] for k in TRIPLE]),0),mul(u[i],v[j])) + for j in range(112)] for i in range(112)] + if transposed: d,t=t,d;expected=transpose(expected) + require(mm(d,transpose(t))==expected,'full ordinary-transpose product') + require(all(expected[i][j]==(dot([x[i][k] for k in TRIPLE],[x[j][k] for k in TRIPLE]),0) + for i in range(112) for j in range(112) if b[i][j]),'masked identity') + ranks=[pairing(basis(duals[i] for i in range(112) if d[i][k]!=ZERO), + basis(labels[i] for i in range(112) if t[i][k]!=ZERO)) for k in range(4)] + require(sum(ranks)>=24,'fixture COPY bound') + a0=sum(all(m[k][h]==ZERO for k in range(3)) for h in range(4)) + b0=sum(all(n[k][h]==ZERO for k in range(3)) for h in range(4)) + sizes=[sum(z!=ZERO for z in u),1] + if transposed: a0,b0=b0,a0;sizes.reverse() + results.append({'name':name,'site':site,'correction_support_sizes':sizes, + 'replacement_ranks':ranks,'whole_copy_sum':40+sum(ranks), + 'zero_left_projections':a0,'zero_right_projections':b0,'checked_entries':112*112}) + saved[name]=(left,right,m,d,t,expected) + require(results[0]['replacement_ranks']==[8,8,8,0],'attaining positive fixture') + require(results[1]['replacement_ranks']==[7,8,8,7],'per-child rank-eight assertion must fail') + require(results[2]['zero_left_projections']==results[2]['zero_right_projections']==0,'all-active fixture') + left,right,m,d,t,target=saved['compensation'] + omitted=mm([r[:3] for r in d],transpose([r[:3] for r in t])) + bad=next(((i,j) for i in range(112) for j in range(112) if b[i][j] and omitted[i][j]!=target[i][j]),None) + require(bad is not None,'omitted compensation must violate constrained entry') + require(mm(mm(left,m),transpose(mm(right,inverse(m))))!=target,'wrong inverse orientation must fail') + left,right,m,d,t,target=saved['dense_complex'] + hermitian=transpose([[conj(z) for z in row] for row in inverse(m)]) + require(mm(mm(left,m),transpose(mm(right,hermitian)))!=target,'Hermitian substitution must fail') + singular=identity(4);singular[3]=singular[2][:];rejected=False + try: inverse(singular) + except ValueError: rejected=True + require(rejected,'singular mixing must fail') + return results, {'omitted_compensation_entry':list(bad),'wrong_inverse_rejected':True, + 'hermitian_substitution_rejected':True,'singular_mixing_rejected':True, + 'rank_eight_per_active_term_refuted':True} + + +def build_certificate(): + roots,x,labels,duals,b=load_geometry();ss=strata(x) + right=[basis(labels[i] for i in bits(s)) for _,s in ss] + chars=[mask(i for i,l in enumerate(labels) if (l&q).bit_count()%2) for q in range(512)] + require(len(set(chars))==512,'all label characters') + spectra=Counter();slice_counts=Counter();pair_counts=Counter();original=Counter();rows=[] + for a,s in ss: + full=basis(duals[i] for i in bits(s));weights=Counter((s&c).bit_count() for c in chars) + distance=min(w for w in weights if w);spectra[distance]+=1 + require(distance>=12,'singleton and small-erasure premise') + orig=[pairing(full,c) for c in right] + require(min(orig)>=8,'unmodified support pairing');original.update(orig) + dimensions=[];table=[] + for site in range(112): + surviving=basis(duals[i] for i in bits(s) if b[i][site]) + dimensions.append(len(surviving));slice_counts[len(surviving)]+=1 + ranks=[pairing(surviving,c) for c in right] + require(min(ranks)>=7,'one-anchor pairing bound') + table.append(''.join(map(str,ranks)));pair_counts.update(ranks) + rows.append({'coefficient':list(a),'support_hex':format(s,'028x'),'span':len(full), + 'erasure_spectrum':{str(k):v for k,v in sorted(weights.items())},'minimum_erasure':distance, + 'original_pair_ranks':''.join(map(str,orig)),'survivor_dimensions':''.join(map(str,dimensions)), + 'survivor_pair_ranks':table}) + transports=coordinate_transports(x,b);examples,controls=fixtures(x,labels,duals,b) + envelopes=[[8,8,8,0],[7,7,7,7],[7,7,7,3]] + require(all(sum(v)>=24 and all(0<=t<=8 for t in v) for v in envelopes),'power-bill envelopes') + require(2**3-1<12,'large-support pure-factor dimension') + return {'schema_version':1,'status':'FINITE_PREMISES_FOR_WRITTEN_SINGLETON_THREE_MODE_THEOREM', + 'triple':list(TRIPLE),'geometry':{'roots_sha256':digest(roots),'coordinates_sha256':digest(x), + 'pairing_sha256':digest(b),'ports':112,'label_rank':9}, + 'strata':rows,'transports':transports,'fixtures':examples,'controls':controls, + 'summary':{'strata':39,'proper_flats_by_rank':[1,13,25],'sites':112,'coordinate_triples':56, + 'character_support_evaluations':39*512,'minimum_erasure':min(spectra), + 'erasure_distance_counts':dict(sorted(spectra.items())), + 'original_pair_counts':dict(sorted(original.items())), + 'survivor_dimension_counts':dict(sorted(slice_counts.items())), + 'survivor_pair_counts':dict(sorted(pair_counts.items())), + 'survivor_pair_cases':39*112*39,'minimum_survivor_pair_rank':min(pair_counts), + 'capped_lower_envelopes':envelopes,'sharp_whole_copy_sum':64, + 'fixture_entries':len(examples)*112*112}} + + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + group=parser.add_mutually_exclusive_group() + group.add_argument('--write',type=Path,help='write a new deterministic receipt; refuses overwrite') + group.add_argument('--check',type=Path,default=PACKAGE/'CERTIFICATE.json') + args=parser.parse_args();result=json.loads(json.dumps(build_certificate())) + if args.write: + with args.write.open('x') as f: json.dump(result,f,indent=2,sort_keys=True);f.write('\n') + else: require(result==json.loads(args.check.read_text()),'certificate mismatch') + print('PASS_P112_THREE_MODE_SINGLETON strata=39 sites=112 pairings=170352 copy_floor=64 controls=5') + + +if __name__=='__main__': main()