From 900658976e42a1501b3d23d46b2ea3cab228bc4c Mon Sep 17 00:00:00 2001 From: Hanson Sun <182361798+bbMisk@users.noreply.github.com> Date: Tue, 13 Oct 2026 22:08:27 -0400 Subject: [PATCH] Prove the p25 complex decoder boundary and sharp cost-six completion --- .../CERTIFICATE.json | 7896 +++++++++++++++++ .../complex-general-normal-boundary/README.md | 250 + .../SOURCES.json | 54 + .../complex-general-normal-boundary/TASK.json | 192 + .../VALIDATION.json | 65 + .../VERIFY.json | 47 + .../check_gram.py | 257 + .../test_package.py | 99 + .../complex-general-normal-boundary/verify.py | 418 + 9 files changed, 9278 insertions(+) create mode 100644 research/complex-general-normal-boundary/CERTIFICATE.json create mode 100644 research/complex-general-normal-boundary/README.md create mode 100644 research/complex-general-normal-boundary/SOURCES.json create mode 100644 research/complex-general-normal-boundary/TASK.json create mode 100644 research/complex-general-normal-boundary/VALIDATION.json create mode 100644 research/complex-general-normal-boundary/VERIFY.json create mode 100644 research/complex-general-normal-boundary/check_gram.py create mode 100644 research/complex-general-normal-boundary/test_package.py create mode 100644 research/complex-general-normal-boundary/verify.py diff --git a/research/complex-general-normal-boundary/CERTIFICATE.json b/research/complex-general-normal-boundary/CERTIFICATE.json new file mode 100644 index 0000000000..c9c8310f64 --- /dev/null +++ b/research/complex-general-normal-boundary/CERTIFICATE.json @@ -0,0 +1,7896 @@ +{ + "all_boundary_cuts_sha256": "f1c3ecb1782f6142667c39489ce0db97350eee1492654b8a8ea83d58f8ab2fd9", + "assignment_seed": 31, + "base_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6", + "controls": [ + "universal term-cost-seven claim at p25", + "missing dual factor two", + "omitted unit label", + "omitted boundary cut", + "omitted dual functional", + "all 25 one-line restorations remove the cost-six exception", + 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No new +kappa or physical supplier.** + +The exact eight-term complex E8 decoder floor extends from `p<=24` to +`p<=25`. At 25 punctures the minimum, over puncture choices and decoders, is +**56 copies on 95 ports**. Parallel root normals can already give an individual +term of cost six, but every exact eight-term completion containing it costs +**at least 67**, and 67 is attained. The cheaper term does not lower the total. + +For **nonparallel** normals with nonzero ordinary bilinear product, including +arbitrary complex normals, the first cost-six puncture size is **26**. +PR1020 already supplies the attaining root-pair witness at 26. This result +extends its lower bound beyond root normals and classifies the parallel +exception; it does not contradict PR1020's distinct-root statement. + +## Reproduce + +From the repository root, Python 3.12, standard library only: + +```sh +python3.12 -B scripts/research_package.py --package research/complex-general-normal-boundary +``` + +The registered command runs the primary replay, separate Gram-minor checker +and scoped mutation tests. Expected final wrapper line: + +```text +PASS_REGISTERED_RESEARCH_PACKAGE research/complex-general-normal-boundary outcome=negative-result +``` + +Both mathematical checkers also support direct Python 3.10+ execution: + +```sh +python3 -I -O -B research/complex-general-normal-boundary/verify.py +python3 -I -O -B research/complex-general-normal-boundary/check_gram.py +``` + +Expected markers: `PASS_COMPLEX_GENERAL_NORMAL_BOUNDARY` and +`PASS_GENERAL_NORMAL_GRAM_CHECK`. Ordinary replay recomputes and compares +[CERTIFICATE.json](CERTIFICATE.json); `verify.py --write-certificate` deliberately +regenerates only that file. + +Coverage is **7,560** distinct boundary punctures, **63** for each of 120 +distinguished roots, with explicit symmetry transport. All **32,193** rational +matrices (`63*511`) in the representative slice are checked. Two full decoder +examples are checked on **18,050** ordered live pairs. The primary controls +include an incorrect dual scale, the omitted unit label, the parallel-root +counterexample and all 25 single-line restorations. + +## Model and theorem + +Use PR717/PR968's 120 real E8 root lines, with norm-squared-two representatives +`r_i`, and label pairing `N_ij=(r_i dot r_j+1) mod 2`. Delete a common set `P` +of `p` lines; `R` denotes the live-root matrix. Complex matrices `D,T`, each +with eight columns, must satisfy **every entry** of + +`D T^transpose = 2 R R^transpose`. + +Transpose is ordinary transpose. Each term costs the F2 rank of `N` on the +two column supports. Masked identities, arbitrary outside-code overcomplete +factorizations and different label pairings are excluded. + +**Theorem, with the inherited real root-flat classification F6.** + +1. At `p<=25`, normals with nonzero bilinear product can have term cost at most + six only at `p=25`, with both normals proportional to one real root `alpha` + and `P` one of its 63 boundary cuts. The cost is exactly six. Each of the + 7,560 cuts has a unique distinguished root. +2. For every such puncture, every exact eight-term decoder containing the + cost-six term costs at least 67. The certificate supplies attaining frames + for all representative cuts and verified transport to every root. +3. The exact minimum at `p=25`, over punctures and decoders, is 56. Combined + with PR968/PR1077, `cost>=max(72-p,56)` holds through `p=25`, and the best + copy/port ratio remains `7/13` at `p=16`. +4. Nonparallel normals with nonzero product first attain cost six at `p=26`, + using PR1020's witness. + +Statement 3 does not say every puncture admits 56. Statement 2 concerns +decoders containing a cost-six term, not all decoders on that puncture. + +## Proof + +### Reduce an arbitrary complex support to the first boundary + +The label space has nondegenerate dimension nine. For support spans `U,V`, + +`rank(N|U x V) >= dim(U)+dim(V)-9`. + +A term of cost at most six has one side of dimension at most seven. A generic +real combination of a complex normal's real and imaginary parts preserves +its exact root-zero pattern, as in PR1077. This is a support statement; it +does not independently replace two dual normals by real representatives. + +Extend a root basis of the real zero flat to seven independent roots. The +resulting root-spanned hyperplane has a normal `omega` with +`C(omega) subset C(a)`. Its surviving support also has dimension at most seven. + +The inherited **F6 classification of real root-spanned E8 hyperplanes** says +that every such hyperplane is Weyl-conjugate to one of eight fundamental +types. The primary checker recomputes their complete two-functional census: + +| Hyperplane root lines | Minimum deletions for label dimension at most seven | +|---:|---:| +| 63, root-normal E7 | 25 | +| 42 | 42 | +| 37 | 51 | +| 28 | 56 | +| 23 | 61 | +| 22 | 62 | +| 16 | 68 | +| 14 | 70 | + +Thus `p<=25` forces `p=25` and `omega` proportional to a root `alpha`. +Let `V_lambda` be the root set where a nonzero label functional evaluates to +one. Two independent functionals annihilating the surviving support require + +`C(alpha) intersect (V_lambda union V_mu) subset P`. + +All 130,305 independent functional pairs are enumerated. Minimum size 25 +occurs for 189 pairs, producing 63 distinct sets. Since `|P|=25`, containment +is equality. All 120 root censuses and their transport by eight explicit root +reflections are checked. Their 7,560 punctures are distinct; transitivity is +verified rather than assumed. + +### Recover every possible reduced normal, including complex ones + +For each cut and each of the 511 nonzero functionals, form `Z_lambda` with +the live roots in `V_lambda` as rows. A normal `b` has support label dimension +at most eight exactly when it belongs to the complex kernel of some +`Z_lambda`. + +For each of the 63 representative cuts, **483** matrices have rank eight and +**28** have rank seven. Every rank-seven kernel is a single real root +direction, extended to complex scalars. The 28 kernels represent **26** +distinct directions: only `alpha` has support dimension seven; the other +25 have dimension eight. Exactly one reduced direction, `beta`, lies in +`alpha^perp`. + +These are exhaustive linear constraints on arbitrary complex coefficients. +A rational matrix has the same rank over Q and C. The separate checker uses +exact integer Gram determinants: positive 8-by-8 determinants certify rank +eight; a positive 7-by-7 principal minor and an explicitly annihilated root +vector certify rank seven. These Gram matrices come from the **real rational +rows** of `Z_lambda`; no complex positive-definiteness assumption is made. + +The cheap term's smaller side must therefore be proportional to `alpha`. +If the other side has dimension nine, its cost is at least seven. Otherwise +it is one of the 26 directions. Exact binary ranks show that among directions +with nonzero product with `alpha`, only `alpha` has cost at most six, and its +cost is six. This proves statement 1 and excludes all nonparallel normals +through 25 punctures. + +### Charge the seven remaining terms + +At least 95 roots survive. F6 also gives a maximum of 63 lines in a real root +hyperplane, so the live-root matrix has rank eight. The all-pairs identity +implies `D=RA`, `T=RB` and + +`A^transpose B = A B^transpose = 2I`. + +The two normal lists are bases with paired products two. If one term uses +parallel `alpha` normals, the seven others on each side lie in `alpha^perp` +and are independent. At most one on either side can be reduced: it must be +proportional to the unique `beta`. Each side's support-dimension sum is at +least `8+6*9=62`. The remaining seven term costs sum to at least + +`62+62-7*9 = 61`. + +Including the cost-six term yields **67**. No blanket prohibition on complex +parallel cross-terms is used; duality and same-side independence suffice. + +Every representative cut has an explicit orthogonal root frame beginning with +`alpha,beta`. Using that frame on both sides gives +`(6,7,9,9,9,9,9,9)`. All frames and costs are checked, and the displayed +seed-selected example is checked entrywise on all 9,025 live pairs. Verified +symmetries transport these constructions, proving sharpness. + +If no term costs six, all eight cost at least seven. Otherwise 67 applies. +Hence the total is always at least 56. A second example extends PR968's +frame-pair construction by nine explicit punctures, with costs +`(7,7,7,7,7,7,7,7)` and all 9,025 entries checked. PR1020's nonparallel +26-puncture witness is independently reranked. Statements 2--4 follow. + +## Budget and limits + +The hypothesis was that the first seven-dimensional support could beat the +copy/port objective `7/13`. Even the favorable copy-only price fails: +`67/95 > 56/95 > 7/13`. Physical synthesis stopped at this budget gate. +This is the full **eight-term copy invoice**, not a recursive supplier +invoice. Source access, helper work, frame changes, restoration, recursive +children, hosting, precision/tape and assembly costs have not been priced or +removed. A different contract could change the comparison. + +F6 is cited, not proved here; the primary replay recomputes its eight-type +table. The written argument quantifies over arbitrary complex normals, while +the finite checks establish its stated premises and examples. Neither is a +formal proof of multiplication. The Gram checker imports no producer or +predecessor code, but both implementations were prepared in the same +AI-assisted session; no independent human review is claimed. + +No physical word, native admission, full selected-supplier cold replay, +all-size compiler proof, bit H1--H5 proof or analytic transfer is supplied. +The selected record remains unchanged at the base's conditional kappa +`836330098538459/500000000000000000`. + +## Provenance, overlap and credit + +Base: `ef5f39403610d231a0c27e8a6b4f08c10209bcd6`. OS assignment seed **31** +selected the complex-supplier track, candidate `31 mod 3 = 1`, root line 31 +and witness index `31 mod 63`. Randomization is not evidence of novelty. +The declared census is complete and has no randomized optimizer. +[TASK.json](TASK.json) records the three candidates, overlap and scope change +from the catalogue's masked-decoder card. No pinned sprint issue existed; +the donor's local-claim fallback was used. + +[SOURCES.json](SOURCES.json) binds base Git objects and exact byte hashes. +The selected record and catalogue are historical provenance. Proof sources +and the retained witness are explicit inputs in [VERIFY.json](VERIFY.json). + +- **DreamingOfClouds, PR717:** B14 label pairing and decoder cost model. +- **themilkman216, PR968:** eight-type census, F6 premise, floor through 24 + punctures and the frame-pair attaining construction. +- **woklerx, PR1077:** arbitrary-complex support bridge and isotropic caution. +- **woklerx, PR1020**, head `bf4ee95df4ee6e3101a88f5d015c8aa85b688b6d`: + distinct odd-root threshold 26 and its attaining witness. This package + extends its lower-bound scope and credits the existing witness. +- **PR1180:** overlapping factor-linear overcomplete work through `p<=24`; + its exact pending head is recorded for overlap, not used as a premise. +- **PR1187/PR1455:** thresholds and aggregate drops for a fixed P104 decoder, + with a different prescribed decoder and puncture range; no proof dependency. +- **Rohan Arun, Jeremy Kothe, Hao Yan and the PR1094/1087/k8/T16, + PR589/PR555 lineage:** reviewed supplier context. + +Prepared by **Hanson Sun (@bbMisk)** with substantial **OpenAI Codex assistance** +in task selection, research, proof, both implementations, verification and +writing. New material is **Apache-2.0**; predecessor sources remain unchanged. + +## Open next questions + +1. **What is the least complete eight-term cost at 26 punctures?** The known + nonparallel term costs six, but its retained completion costs 69. Start + with its annihilator-kernel inventory and an optimistic full copy bill. + Dual-basis compatibility beyond one cheap term is the blocker; this + package does not enumerate 26-line punctures. +2. **Can F6 be replaced by a self-contained root-flat certificate?** That + would remove the cited classification premise. Start with a canonical + exact enumeration of maximal root-spanned hyperplanes modulo the verified + reflections. Completeness and certificate size remain open here. diff --git a/research/complex-general-normal-boundary/SOURCES.json b/research/complex-general-normal-boundary/SOURCES.json new file mode 100644 index 0000000000..ebb466bff8 --- /dev/null +++ b/research/complex-general-normal-boundary/SOURCES.json @@ -0,0 +1,54 @@ +{ + "schema_version": 1, + "base_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6", + "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/complex-frame-pair-decoders/PROOF.md", + "usage": "input", + "git_blob": "889f3a653ed1bdbb7df61ebe3af4f2103602a1f4", + "sha256": "ba53c1d8a9bc2d935d546bde46ca7341bd65f8515013e07b073296b4db2a9a74" + }, + { + "path": "research/complex-frame-pair-decoders/verify.py", + "usage": "input", + "git_blob": "57dd8993e8ac016e8a92a9a885392b58014137bb", + "sha256": "bdf7ee497dc04dfc634ee5b7848f6a2b39b0169668fb1a60a9715111dbd15395" + }, + { + "path": "research/root-pair-first-drop-seed48/CERTIFICATE.json", + "usage": "input", + "git_blob": "73c0e0cac357270d7fb9510d7a0093a7d8e359d2", + "sha256": "a429b191d7f033864e9fcd676dc575f551a2a4093aaed0ba2a90cae1d91935ff" + }, + { + "path": "research/root-pair-first-drop-seed48/PROOF.md", + "usage": "input", + "git_blob": "71de1701822918906bbce29d781f3ac8a75a3bf3", + "sha256": "9ba97ca028405eb5e88dc9158df34de9ca1ef409de9ad72f7ae9faa34722e9c3" + }, + { + "path": "research/root-pair-first-drop-seed48/explore.py", + "usage": "input", + "git_blob": "df5a45663f66d9c801454bdf4ddb2227037d7aa4", + "sha256": "3f079b89931b5b1e55f318ba7c34a7dc8d0f0b20b499f1c23a22a41b0367132e" + } + ] +} diff --git a/research/complex-general-normal-boundary/TASK.json b/research/complex-general-normal-boundary/TASK.json new file mode 100644 index 0000000000..36ac9ed18d --- /dev/null +++ b/research/complex-general-normal-boundary/TASK.json @@ -0,0 +1,192 @@ +{ + "schema_version": 1, + "task_id": "complex-masked-decoder", + "priority_revision": 2, + "base_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6", + "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": "changed", + "actual_scope": "Classify all cost-at-most-six terms in exact eight-term complex E8 decoders through 25 common punctures, and price their complete eight-term copy invoice. Sharp cost-six completion minimum 67; unrestricted decoder minimum at p=25 is 56; arbitrary nonparallel normals first attain cost six at p=26.", + "scope_change_reason": "The donor assigned the complex supplier track. The catalogue masked-decoder card is used for metadata, but this is explicitly a deeper-puncture all-pairs side result beyond its inherited p<=24 theorem, not a masked or overcomplete construction.", + "bottleneck_link": "Copied-information cost enters the binding complex supplier budget; this resolves whether the first allowed dimension-seven support drop opens a cheaper term for non-root normals.", + "budget_minutes": 60, + "stop_rule": "One bounded p<=25 experiment, including all root support codimension-two minima and all compatible dual functionals; stop this route if the optimistic copy/port ratio cannot beat 7/13. Reserve the final 15 of 60 minutes for verification and handoff.", + "outcome": "negative-result", + "evidence": "CERTIFICATE.json, verify.py and the code-disjoint Gram-minor checker check all 7560 boundary cuts, 32193 rational matrices and two complete 95-port decoders. README.md gives the quantified proof under inherited classification F6. Eight semantic package tests cover corrupt source, incomplete coverage, false kernels/ranks, wrong scale and undercharged total.", + "author": "Hanson Sun (@bbMisk)", + "ai_assistance": "Substantial OpenAI Codex assistance in research, proof, implementation, verification and writing; no independent human review.", + "open_questions": [ + { + "question": "What is the least complete eight-term copy cost at 26 punctures?", + "why_it_matters": "The first nonparallel cost-six term exists there, but the retained completion costs 69.", + "first_step": "Classify the 26-puncture witness annihilator kernels and price a complete exact decoder before physical synthesis.", + "blocked_by": "Dual-basis compatibility and a complete 26-puncture domain are unresolved." + }, + { + "question": "Can the inherited F6 root-flat classification be replaced by a self-contained certificate?", + "why_it_matters": "It is the principal cited geometric premise of the global reduction.", + "first_step": "Enumerate maximal root-spanned hyperplanes canonically modulo verified root reflections.", + "blocked_by": "Completeness and manageable certificate size have not been established in this package." + } + ], + "assignment_seed": 31, + "randomness": "Python secrets.randbelow(100); operating-system entropy; recorded once before task selection", + "assigned_track": "Complex supplier", + "claim": { + "status": "DONE", + "posted": "local: no pinned coordination issue exists", + "base_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6", + "expected_finish_utc": "2026-10-14T02:34:50Z", + "method": "Exact E8 label-space census and rational annihilator lifting; no floating-point acceptance", + "deliverable": "Scoped proof or exact counterexample, executable certificate and negative controls", + "result": "Scoped negative result: p25 complete decoder minimum 56; any cost-six term forces sharp completion cost 67; arbitrary nonparallel normals first reach cost six at p26.", + "link": "https://github.com/bbMisk/integer-mult-bounds/tree/research/complex-general-normal-boundary-seed31/research/complex-general-normal-boundary", + "updated_utc": "2026-10-14T02:07:08.298104+00:00" + }, + "candidate_tasks": [ + { + "index": 0, + "scope": "First two copy drops on the fixed preserved-frame P104 decoder", + "decision": "not selected: PR1187 and PR1455 cover this boundary" + }, + { + "index": 1, + "scope": "General non-root normals at the 25-puncture copy-six boundary", + "decision": "selected by 31 modulo 3 = 1; PR968/1077 stop at p=24 and the retained seed48 package handles only odd root pairs" + }, + { + "index": 2, + "scope": "Three-coordinate temporal gauge on the selected k8 word", + "decision": "not selected: PR1380/1397 and later-word follow-ups already cover the natural objects" + } + ], + "overlap": { + "snapshot_utc": "2026-10-14T01:37:31.249720+00:00", + "open_prs": 744, + "discussion_records": 174, + "pagination_complete": true, + "limitations": "titles, bodies, discussion replies and top-level package notes; not every code diff", + "nearest": [ + { + "pr": 1180, + "head": "75b408646d558147a4eb250fc407d1fbfcb8df4e", + "distinction": "overcomplete factor-linear extension only through p<=24" + }, + { + "pr": 1187, + "head": "15bee4c8ea0f54fea575a48441c6acc8bad263cd", + "distinction": "fixed preserved-frame P104 decoder; 38-puncture boundary" + }, + { + "pr": 1455, + "head": "0ebd49c31300e67698fd812fc00f2a3b2b805a67", + "distinction": "aggregate two-copy drop for that fixed P104 decoder" + } + ], + "final_refresh": { + "searched_at": "2026-10-14T02:01:39.406952+00:00", + "coverage": { + "open_prs": { + "fetched_at": "2026-10-14T02:01:18.071560+00:00", + "complete_pagination": true, + "records": 758 + }, + "discussions": { + "fetched_at": "2026-10-14T02:01:39.357787+00:00", + "complete_pagination": true, + "records": 191 + } + }, + "warnings": [], + "terms": [ + "25 punctures", + "p=25", + "67 cop", + "general normal", + "nonparallel", + "cost-six" + ], + "distinct_external_matches": [ + { + "kind": "local-readme", + "number": null, + "title": "research/root-pair-first-drop-seed48/README.md", + "head": null, + "url": null + }, + { + "kind": "local-readme", + "number": null, + "title": "research/complex-decoder-field-extension-seed44/README.md", + "head": null, + "url": null + }, + { + "kind": "pr", + "number": 1180, + "title": "Structural proof: overcomplete factor-linear E8 decoders retain the puncture copy-cost floor", + "head": "75b408646d558147a4eb250fc407d1fbfcb8df4e", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1180" + } + ], + "conclusion": "No competing p25 general-normal classification or conditional 67-copy completion bound identified; this is a bounded metadata/notes search, not a novelty proof." + } + }, + "predecessors": [ + { + "pr": 717, + "credit": "DreamingOfClouds", + "scope": "B14 decoder model" + }, + { + "pr": 968, + "credit": "themilkman216", + "scope": "p<=24 floor, F6 and frame-pair attainment" + }, + { + "pr": 1077, + "credit": "woklerx", + "scope": "complex support bridge" + }, + { + "pr": 1020, + "credit": "woklerx", + "head": "bf4ee95df4ee6e3101a88f5d015c8aa85b688b6d", + "scope": "26-puncture nonparallel root-pair witness" + } + ] +} diff --git a/research/complex-general-normal-boundary/VALIDATION.json b/research/complex-general-normal-boundary/VALIDATION.json new file mode 100644 index 0000000000..bbb8f316e8 --- /dev/null +++ b/research/complex-general-normal-boundary/VALIDATION.json @@ -0,0 +1,65 @@ +{ + "schema_version": 1, + "status": "scoped_checks_passed", + "checked_at_utc": "2026-10-14T02:07:08.298424+00:00", + "base_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6", + "submission_main_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6", + "environment": { + "system": "Darwin", + "release": "27.0.0", + "architecture": "arm64", + "primary_python": "3.12.2", + "secondary_python": "3.14.6", + "third_party_dependencies": [] + }, + "checks": [ + { + "command": "python3.12 -B scripts/research_package.py --package research/complex-general-normal-boundary", + "exit_code": 0, + "evidence": "PASS_REGISTERED_RESEARCH_PACKAGE research/complex-general-normal-boundary outcome=negative-result; both math engines and all 8 tests passed" + }, + { + "command": "python3 -I -O -B research/complex-general-normal-boundary/verify.py", + "context": "read-only sparse relocation, cwd outside relocated repository; Python 3.14.6", + "exit_code": 0, + "evidence": "PASS_COMPLEX_GENERAL_NORMAL_BOUNDARY cuts=7560 representatives=63 rational_matrices=32193 p25_minimum=56 cost6_completion_minimum=67 nonparallel_threshold=26 controls=8" + }, + { + "command": "python3 -I -O -B research/complex-general-normal-boundary/check_gram.py", + "context": "same read-only sparse relocation; Python 3.14.6", + "exit_code": 0, + "evidence": "PASS_GENERAL_NORMAL_GRAM_CHECK matrices=32193 boundary_cuts=7560 witness_entries=18050" + }, + { + "command": "make selected-record-check", + "exit_code": 0, + "evidence": "PASS selected record, exact kappa, original source inventory and proof supplement: 836330098538459/500000000000000000" + }, + { + "command": "python3 -B scripts/research_package.py --package research/complex-general-normal-boundary --check-only", + "exit_code": 0, + "evidence": "PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)" + }, + { + "command": "ruff check research/complex-general-normal-boundary --output-format concise", + "exit_code": 0, + "evidence": "All checks passed!" + } + ], + "core_sha256": { + "verify.py": "8cba06b3a94b1baee6a469cc5ad1e77dda65072f909186bef2b126e0a588863c", + "check_gram.py": "29c27ee8e33e2674439ff34d6cbc7d3a685435ae46e78e41d4a4a55c698b7ade", + "test_package.py": "a1cc6a2aa71bbe7863638af8f417a13d79dd1d2f295c85e9ce5c53ee21439135", + "CERTIFICATE.json": "ce57b168d6b91a1fe570731c3fafbd779ec34405ca0f1577acd6a1f4dada86bd", + "README.md": "b3f00b757edc2fad911b99bb6088cd7ff2d3985d73d3b36051093e11d6987a85", + "SOURCES.json": "1810ece7594ab2b9eda30eab813db5a369feb2b817a2ca25d285f19c8871189a", + "VERIFY.json": "2e3b9312dd25a2b115eeff02da2154eaa4e388388785be78841797dd88d80e8c", + "TASK.json": "1143345000ebf94a0f12dfe46230a942ab98484816913d66d4ef841644d6c426" + }, + "scope_limits": [ + "F6 root-hyperplane classification is inherited, not proved here.", + "Both checkers were written in the same AI-assisted session; no independent human review.", + "No full selected cold replay, native/physical admission, all-size theorem, bit-interface proof or analytic transfer.", + "No new kappa and no selected-frontier edit." + ] +} diff --git a/research/complex-general-normal-boundary/VERIFY.json b/research/complex-general-normal-boundary/VERIFY.json new file mode 100644 index 0000000000..633dd494b2 --- /dev/null +++ b/research/complex-general-normal-boundary/VERIFY.json @@ -0,0 +1,47 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [ + "research/complex-decoder-field-extension-seed44/PROOF.md", + "research/complex-frame-pair-decoders/PROOF.md", + "research/complex-frame-pair-decoders/verify.py", + "research/root-pair-first-drop-seed48/CERTIFICATE.json", + "research/root-pair-first-drop-seed48/PROOF.md", + "research/root-pair-first-drop-seed48/explore.py" + ], + "commands": [ + { + "argv": [ + "{python}", + "-I", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 120, + "checks": "Complete p25 boundary, exact rational kernel inventory, complete decoder identities and eight controls; saved certificate comparison." + }, + { + "argv": [ + "{python}", + "-I", + "-B", + "{package}/check_gram.py" + ], + "timeout_seconds": 180, + "checks": "Independent dense F2 ranks and exact integer Gram minors for every new kernel premise, global reflection closure and all displayed decoder entries." + }, + { + "argv": [ + "{python}", + "-I", + "-O", + "-B", + "{package}/test_package.py" + ], + "timeout_seconds": 60, + "checks": "Eight proof-relevant corruption and arithmetic tests, with assertions disabled." + } + ] +} diff --git a/research/complex-general-normal-boundary/check_gram.py b/research/complex-general-normal-boundary/check_gram.py new file mode 100644 index 0000000000..58db2a6615 --- /dev/null +++ b/research/complex-general-normal-boundary/check_gram.py @@ -0,0 +1,257 @@ +#!/usr/bin/env python3 +"""Separate exact Gram-minor checker for the p=25 certificate. + +Imports no producer or predecessor code. Dense binary elimination, Gram +determinants and root-reflection closure replace the producer's rank engines. +Prepared by Hanson Sun with substantial OpenAI Codex assistance. Apache-2.0. +This is implementation separation, not independent human review. +""" +from collections import Counter, deque +import hashlib +from itertools import combinations, product +import json +from pathlib import Path + +HERE = Path(__file__).resolve().parent + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def scalar(u, v): + return sum(x * y for x, y in zip(u, v)) + + +def roots_from_coordinates(): + # Fix the first nonzero coordinate positive; no set-based +/- quotient. + result = [] + for a in range(8): + for b in range(a + 1, 8): + for sign in (-1, 1): + row = [0] * 8 + row[a], row[b] = 2, 2 * sign + result.append(tuple(row)) + for tail in product((-1, 1), repeat=7): + if tail.count(-1) % 2 == 0: + result.append((1,) + tail) + return sorted(result) + + +def dense_binary_rank(matrix): + rows = [list(x) for x in matrix] + pivot = 0 + for col in range(len(rows[0]) if rows else 0): + found = next((i for i in range(pivot, len(rows)) if rows[i][col]), None) + if found is None: + continue + rows[pivot], rows[found] = rows[found], rows[pivot] + for i in range(pivot + 1, len(rows)): + if rows[i][col]: + rows[i] = [x ^ y for x, y in zip(rows[i], rows[pivot])] + pivot += 1 + if pivot == len(rows): + break + return pivot + + +def bareiss(matrix): + """Exact determinant, including row pivots; division is checked.""" + a = [row[:] for row in matrix] + n, sign, previous = len(a), 1, 1 + if n == 0: + return 1 + for k in range(n - 1): + p = next((i for i in range(k, n) if a[i][k]), None) + if p is None: + return 0 + if p != k: + a[p], a[k] = a[k], a[p] + sign = -sign + pivot = a[k][k] + for i in range(k + 1, n): + for j in range(k + 1, n): + numerator = a[i][j] * pivot - a[i][k] * a[k][j] + need(numerator % previous == 0, "nonintegral Bareiss step") + a[i][j] = numerator // previous + for i in range(k + 1, n): + a[i][k] = 0 + previous = pivot + return sign * a[-1][-1] + + +def decode_mask(text): + value = int(text, 16) + need(0 <= value < 1 << 120, "mask outside root universe") + return frozenset(i for i in range(120) if value >> i & 1) + + +def encode_mask(indices): + return hex(sum(1 << i for i in indices)) + + +def compute_supports(roots): + return [frozenset(j for j, other in enumerate(roots) if scalar(r, other)) for r in roots] + + +def check_witness(witness, roots, pairing): + puncture = set(witness["puncture"]) + need(len(puncture) == 25 and all(0 <= i < 120 for i in puncture), "witness puncture") + live = [i for i in range(120) if i not in puncture] + left, right = witness["left_columns_doubled"], witness["right_columns_doubled"] + need(len(left) == len(right) == 8 and all(len(c) == 8 for c in left + right), "witness shape") + d = [[scalar(roots[i], a) for a in left] for i in live] + t = [[scalar(roots[j], b) for b in right] for j in live] + for i, row in enumerate(d): + for j, col in enumerate(t): + need(scalar(row, col) == 8 * scalar(roots[live[i]], roots[live[j]]), "full witness identity") + costs = [] + for k in range(8): + rows = [live[i] for i in range(95) if d[i][k]] + cols = [live[j] for j in range(95) if t[j][k]] + costs.append(dense_binary_rank([[pairing[i][j] for j in cols] for i in rows])) + need(costs == witness["term_costs"] and sum(costs) == witness["copy_cost"], "full witness copy bill") + return costs + + +def check(certificate): + roots = roots_from_coordinates() + need(len(roots) == len(set(roots)) == 120, "independent root count") + encoded = json.dumps(roots, sort_keys=True, separators=(",", ":")) + need(hashlib.sha256(encoded.encode()).hexdigest() == certificate["root_order_sha256"], "root order") + pairing = [[(scalar(a, b) // 4 + 1) % 2 for b in roots] for a in roots] + need(dense_binary_rank(pairing) == 9, "independent label rank") + indices = [] + for i in range(120): + if dense_binary_rank([pairing[j] for j in indices + [i]]) > len(indices): + indices.append(i) + need(len(indices) == 9, "independent functional basis") + functions = [] + for coeffs in product((0, 1), repeat=9): + if not any(coeffs): + continue + functions.append(frozenset(i for i in range(120) + if sum(coeffs[k] * pairing[indices[k]][i] for k in range(9)) % 2)) + need(len(set(functions)) == 511, "independent complete dual") + support = compute_supports(roots) + alpha = certificate["root_index"] + need(alpha == 31, "assignment root") + cuts = set() + minimum, multiplicity = 120, 0 + restricted = [s & support[alpha] for s in functions] + for a, b in combinations(restricted, 2): + union = a | b + n = len(union) + if n < minimum: + minimum, multiplicity, cuts = n, 1, {union} + elif n == minimum: + multiplicity += 1 + cuts.add(union) + need((minimum, multiplicity, len(cuts)) == (25, 189, 63), "independent boundary exhaustion") + records = certificate["representative_cuts"] + need(len(records) == 63 and {decode_mask(r["puncture_hex"]) for r in records} == cuts, + "certificate must contain every representative exactly once") + outer = [[[a * b for b in r] for a in r] for r in roots] + tested = 0 + for record in records: + puncture = decode_mask(record["puncture_hex"]) + saved_kernels = record["kernel_by_functional_hex"] + need(set(saved_kernels) <= {encode_mask(f) for f in functions}, "kernel key domain") + counts = Counter() + seen_kernels = {} + for function in functions: + live = function - puncture + gram = [[sum(outer[i][a][b] for i in live) for b in range(8)] for a in range(8)] + key = encode_mask(function) + if key in saved_kernels: + k = saved_kernels[key] + need(type(k) is int and 0 <= k < 120, "kernel root index") + normal = roots[k] + need(all(scalar(roots[i], normal) == 0 for i in live), "claimed kernel annihilates rows") + drop = next(i for i in range(8) if normal[i]) + kept = [i for i in range(8) if i != drop] + minor = [[gram[i][j] for j in kept] for i in kept] + need(bareiss(minor) > 0, "rank-seven Gram minor") + seen_kernels[key] = k + counts[7] += 1 + else: + need(bareiss(gram) > 0, "rank-eight Gram determinant") + counts[8] += 1 + tested += 1 + need(counts == Counter({8: 483, 7: 28}) and seen_kernels == saved_kernels, + "complete independently certified rank inventory") + need(record["rank_counts"] == {str(k): v for k, v in sorted(counts.items())}, "saved rank counts") + reduced = sorted(set(seen_kernels.values())) + dims, costs = {}, {} + for k in reduced: + rows = sorted(support[k] - puncture) + dims[str(k)] = dense_binary_rank([[pairing[i][j] for j in indices] for i in rows]) + cols = sorted(support[alpha] - puncture) + costs[str(k)] = dense_binary_rank([[pairing[i][j] for j in cols] for i in rows]) + need(dims == record["support_dimensions"] and Counter(dims.values()) == Counter({8: 25, 7: 1}), + "independent support dimensions") + need(dims[str(alpha)] == 7 and costs == record["distinguished_partner_costs"], "independent term ranks") + special = [k for k in reduced if scalar(roots[alpha], roots[k]) and costs[str(k)] <= 6] + need(special == [alpha] and costs[str(alpha)] == 6, "nonparallel lower bound") + orthogonal = [k for k in reduced if scalar(roots[alpha], roots[k]) == 0] + need(orthogonal == [record["orthogonal_reduced_root"]], "one reduced orthogonal direction") + frame = record["attaining_frame"] + need(len(set(frame)) == 8 and frame[:2] == [alpha, orthogonal[0]], "completion frame") + need(all(scalar(roots[i], roots[j]) == 0 for i, j in combinations(frame, 2)), "orthogonal completion") + costs = [dense_binary_rank([[pairing[i][j] for j in sorted(support[k] - puncture)] + for i in sorted(support[k] - puncture)]) for k in frame] + need(costs == record["attaining_term_costs"] == [6, 7, 9, 9, 9, 9, 9, 9], "sharp completion bill") + need(tested == certificate["rational_matrices_checked"] == 32193, "complete rational matrix count") + # Rebuild the entire boundary family by joint (root, cut) reflection closure. + simple = [(1, -1, -1, -1, -1, -1, -1, 1), (2, 2, 0, 0, 0, 0, 0, 0), + (-2, 2, 0, 0, 0, 0, 0, 0)] + simple += [tuple(2 * ((i == k + 1) - (i == k)) for i in range(8)) for k in range(1, 6)] + lookup = {r: i for i, r in enumerate(roots)} + permutations = [] + for s in simple: + permutation = [] + for r in roots: + image = tuple(x - scalar(r, s) // 4 * y for x, y in zip(r, s)) + if next(x for x in image if x) < 0: + image = tuple(-x for x in image) + permutation.append(lookup[image]) + need(len(set(permutation)) == 120, "independent reflection permutation") + need(all(pairing[i][j] == pairing[permutation[i]][permutation[j]] and + (scalar(roots[i], roots[j]) == 0) == + (scalar(roots[permutation[i]], roots[permutation[j]]) == 0) + for i in range(120) for j in range(120)), "independent equivariance") + permutations.append(permutation) + seen = {(alpha, p) for p in cuts} + todo = deque(seen) + while todo: + a, p = todo.popleft() + for permutation in permutations: + state = (permutation[a], frozenset(permutation[i] for i in p)) + if state not in seen: + seen.add(state) + todo.append(state) + all_cuts = {p for _, p in seen} + need(len(seen) == len(all_cuts) == certificate["distinct_boundary_cuts"] == 7560, "global boundary closure") + need(Counter(a for a, _ in seen) == Counter({i: 63 for i in range(120)}), "all-root boundary coverage") + encoded = json.dumps(sorted(encode_mask(p) for p in all_cuts), sort_keys=True, separators=(",", ":")) + need(hashlib.sha256(encoded.encode()).hexdigest() == certificate["all_boundary_cuts_sha256"], "global cut digest") + need(check_witness(certificate["witness67"], roots, pairing) == [6, 7, 9, 9, 9, 9, 9, 9], "67 witness") + need(check_witness(certificate["witness56"], roots, pairing) == [7] * 8, "56 witness") + previous = certificate["predecessor_nonparallel_witness"] + a, b = previous["pair"] + p = set(previous["puncture"]) + need(a != b and scalar(roots[a], roots[b]) != 0 and len(p) == 26, "26 witness scope") + need(dense_binary_rank([[pairing[i][j] for j in sorted(support[b] - p)] + for i in sorted(support[a] - p)]) == 6, "26 witness cost") + return tested + + +def main(): + certificate = json.loads((HERE / "CERTIFICATE.json").read_text()) + tested = check(certificate) + print(f"PASS_GENERAL_NORMAL_GRAM_CHECK matrices={tested} boundary_cuts=7560 witness_entries=18050") + + +if __name__ == "__main__": + main() diff --git a/research/complex-general-normal-boundary/test_package.py b/research/complex-general-normal-boundary/test_package.py new file mode 100644 index 0000000000..5505906072 --- /dev/null +++ b/research/complex-general-normal-boundary/test_package.py @@ -0,0 +1,99 @@ +#!/usr/bin/env python3 +"""Failure controls for proof-relevant mutations; no expensive full-suite replay. + +Hanson Sun, with substantial OpenAI Codex assistance. Apache-2.0. +""" +from copy import deepcopy +import importlib.util +import json +from pathlib import Path +import tempfile +import unittest +from unittest.mock import patch + +HERE = Path(__file__).resolve().parent + + +def load(name): + spec = importlib.util.spec_from_file_location(name, HERE / (name + ".py")) + module = importlib.util.module_from_spec(spec) + spec.loader.exec_module(module) + return module + + +gram = load("check_gram") +primary = load("verify") + + +class ProofMutations(unittest.TestCase): + @classmethod + def setUpClass(cls): + cls.certificate = json.loads((HERE / "CERTIFICATE.json").read_text()) + cls.roots = gram.roots_from_coordinates() + cls.pairing = [[(gram.scalar(a, b) // 4 + 1) % 2 for b in cls.roots] for a in cls.roots] + + def test_wrong_dual_scale(self): + witness = deepcopy(self.certificate["witness56"]) + witness["right_columns_doubled"][0] = [2 * x for x in witness["right_columns_doubled"][0]] + with self.assertRaisesRegex(ValueError, "full witness identity"): + gram.check_witness(witness, self.roots, self.pairing) + + def test_undercharged_complete_bill(self): + witness = deepcopy(self.certificate["witness67"]) + witness["term_costs"][1] = 6 + witness["copy_cost"] = 66 + with self.assertRaisesRegex(ValueError, "full witness copy bill"): + gram.check_witness(witness, self.roots, self.pairing) + + def test_missing_boundary_case(self): + cert = deepcopy(self.certificate) + cert["representative_cuts"].pop() + with self.assertRaisesRegex(ValueError, "every representative"): + gram.check(cert) + + def test_false_kernel_direction(self): + cert = deepcopy(self.certificate) + kernels = cert["representative_cuts"][0]["kernel_by_functional_hex"] + key = next(iter(kernels)) + kernels[key] = (kernels[key] + 1) % 120 + with self.assertRaisesRegex(ValueError, "claimed kernel annihilates rows"): + gram.check(cert) + + def test_missing_singular_functional(self): + cert = deepcopy(self.certificate) + kernels = cert["representative_cuts"][0]["kernel_by_functional_hex"] + del kernels[next(iter(kernels))] + with self.assertRaisesRegex(ValueError, "rank-eight Gram determinant"): + gram.check(cert) + + def test_false_support_dimension(self): + cert = deepcopy(self.certificate) + cert["representative_cuts"][0]["support_dimensions"]["31"] = 8 + with self.assertRaisesRegex(ValueError, "independent support dimensions"): + gram.check(cert) + + def test_determinant_singularity_and_pivot_controls(self): + fixtures = [([[0, 2], [3, 4]], -6), ([[1, 2], [2, 4]], 0), + ([[4, 1, 2], [1, 3, 0], [2, 0, 5]], 43)] + for matrix, determinant in fixtures: + self.assertEqual(gram.bareiss(matrix), determinant) + + def test_changed_proof_source_rejected_without_touching_repository(self): + manifest = json.loads((HERE / "SOURCES.json").read_text()) + with tempfile.TemporaryDirectory(prefix="e8-boundary-source-control-") as directory: + root = Path(directory) + for entry in manifest["files"]: + if entry["usage"] == "input": + destination = root / entry["path"] + destination.parent.mkdir(parents=True, exist_ok=True) + destination.write_bytes((primary.ROOT / entry["path"]).read_bytes()) + with patch.object(primary, "ROOT", root): + primary.source_pins() + source = root / "research/complex-frame-pair-decoders/PROOF.md" + source.write_bytes(source.read_bytes() + b"\nmutation control\n") + with self.assertRaisesRegex(ValueError, "source pin mismatch"): + primary.source_pins() + + +if __name__ == "__main__": + unittest.main(verbosity=2) diff --git a/research/complex-general-normal-boundary/verify.py b/research/complex-general-normal-boundary/verify.py new file mode 100644 index 0000000000..d5e3707490 --- /dev/null +++ b/research/complex-general-normal-boundary/verify.py @@ -0,0 +1,418 @@ +#!/usr/bin/env python3 +"""Exact p=25 E8 decoder boundary; explicit checks also run under -O. + +Hanson Sun (@bbMisk), with substantial OpenAI Codex assistance. Apache-2.0. +Model and predecessor credit: DreamingOfClouds/PR717, themilkman216/PR968, +woklerx/PR1077 and the retained root-pair-first-drop-seed48 package. +The root-flat classification F6 is a cited premise, not proved by this replay. +""" +import argparse +from collections import Counter, deque +from fractions import Fraction +import hashlib +from itertools import combinations, product +import json +from math import gcd, lcm +from pathlib import Path + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +SEED_ROOT = 31 +SIMPLE = [(1, -1, -1, -1, -1, -1, -1, 1), + (2, 2, 0, 0, 0, 0, 0, 0), (-2, 2, 0, 0, 0, 0, 0, 0)] + [ + tuple(-2 if i == k else 2 if i == k + 1 else 0 for i in range(8)) + for k in range(1, 6)] +PAIRS56 = [(119, 84), (106, 54), (29, 17), (11, 4), + (118, 20), (39, 6), (75, 1), (49, 0)] + + +def require(condition, reason): + if not condition: + raise ValueError(reason) + + +def canonical(value): + # JSON object keys are strings; normalize integer-indexed rank maps before + # sorting so a serialized/reloaded receipt has the same canonical bytes. + normalized = json.loads(json.dumps(value, allow_nan=False)) + return json.dumps(normalized, sort_keys=True, separators=(",", ":"), allow_nan=False) + + +def digest(value): + return hashlib.sha256(canonical(value).encode()).hexdigest() + + +def dot(x, y): + return sum(a * b for a, b in zip(x, y)) + + +def direction(vector): + den = lcm(*(Fraction(x).denominator for x in vector)) + integers = [int(Fraction(x) * den) for x in vector] + common = gcd(*integers) + require(common != 0, "zero direction") + integers = tuple(x // common for x in integers) + return max(integers, tuple(-x for x in integers)) + + +def root_lines(): + roots = set() + for i, j in combinations(range(8), 2): + for x, y in product((-2, 2), repeat=2): + v = [0] * 8 + v[i], v[j] = x, y + roots.add(max(tuple(v), tuple(-a for a in v))) + for v in product((-1, 1), repeat=8): + if v.count(-1) % 2 == 0: + roots.add(max(v, tuple(-a for a in v))) + return sorted(roots) + + +def binary_basis(rows): + basis = {} + for row in rows: + while row: + p = row.bit_length() - 1 + if p not in basis: + basis[p] = row + break + row ^= basis[p] + return list(basis.values()) + + +def rank2(rows): + return len(binary_basis(rows)) + + +def bits(mask): + while mask: + low = mask & -mask + yield low.bit_length() - 1 + mask ^= low + + +def echelon(rows): + """Integer fraction-free row operations over Q; no numerical decisions.""" + pivots = {} + for values in rows: + row = list(values) + for col in sorted(pivots): + if row[col]: + pivot = pivots[col] + a, b = row[col], pivot[col] + row = [b * x - a * y for x, y in zip(row, pivot)] + divisor = gcd(*row) + if divisor > 1: + row = [x // divisor for x in row] + col = next((i for i, x in enumerate(row) if x), None) + if col is not None: + pivots[col] = row + return pivots + + +def kernel_direction(pivots): + require(len(pivots) == 7, "one-dimensional rational kernel required") + free = next(i for i in range(8) if i not in pivots) + vector = [Fraction(i == free) for i in range(8)] + for col, row in sorted(pivots.items(), reverse=True): + vector[col] = -sum((Fraction(row[j]) * vector[j] + for j in range(col + 1, 8)), Fraction(0)) / row[col] + return direction(vector) + + +def inverse(matrix): + n = len(matrix) + rows = [[Fraction(x) for x in row] + [Fraction(i == j) for j in range(n)] + for i, row in enumerate(matrix)] + for col in range(n): + p = next((i for i in range(col, n) if rows[i][col]), None) + require(p is not None, "singular matrix") + rows[col], rows[p] = rows[p], rows[col] + scale = rows[col][col] + rows[col] = [x / scale for x in rows[col]] + for i in range(n): + if i != col: + scale = rows[i][col] + rows[i] = [x - scale * y for x, y in zip(rows[i], rows[col])] + return [row[n:] for row in rows] + + +def model(): + roots = root_lines() + require(len(roots) == 120 and all(dot(r, r) == 8 for r in roots), "E8 roots") + require(all(dot(a, b) % 4 == 0 for a in roots for b in roots), "integral pairing") + pairing = [sum(((dot(a, b) // 4 + 1) & 1) << j for j, b in enumerate(roots)) + for a in roots] + require(rank2(pairing) == 9, "unit-coordinate pairing must have rank nine") + basis = binary_basis(pairing) + # Symmetry identifies the row space with every functional evaluation mask. + words = [0] + for row in basis: + words += [x ^ row for x in words] + words = sorted(words[1:]) + require(len(words) == len(set(words)) == 511, "complete nonzero dual space") + supports = [sum((dot(a, b) != 0) << j for j, b in enumerate(roots)) for a in roots] + return roots, pairing, words, supports + + +def two_annihilator_census(support, words): + restricted = [support & w for w in words] + minimum, count, masks = 121, 0, set() + for i, x in enumerate(restricted): + for y in restricted[i + 1:]: + puncture = x | y + size = puncture.bit_count() + if size < minimum: + minimum, count, masks = size, 1, {puncture} + elif size == minimum: + count += 1 + masks.add(puncture) + return minimum, count, masks + + +def flat_census(roots, words): + inv = inverse(SIMPLE) + normals = [direction([row[i] for row in inv]) for i in range(8)] + table = [] + for normal in normals: + require(sum(dot(normal, s) == 0 for s in SIMPLE) == 7, "fundamental flat") + support = sum((dot(normal, r) != 0) << i for i, r in enumerate(roots)) + minimum, _, _ = two_annihilator_census(support, words) + table.append({"normal": list(normal), "flat_lines": 120 - support.bit_count(), + "minimum_punctures_for_span_at_most_seven": minimum}) + table.sort(key=lambda x: x["flat_lines"], reverse=True) + require([(x["flat_lines"], x["minimum_punctures_for_span_at_most_seven"]) + for x in table] == [(63, 25), (42, 42), (37, 51), (28, 56), + (23, 61), (22, 62), (16, 68), (14, 70)], + "inherited flat census") + return table + + +def symmetry(roots, pairing): + index = {r: i for i, r in enumerate(roots)} + generators = [] + for normal in SIMPLE: + require(dot(normal, normal) == 8, "reflection normal") + permutation = [] + for r in roots: + require(dot(r, normal) % 4 == 0, "integral reflection") + s = tuple(x - dot(r, normal) // 4 * y for x, y in zip(r, normal)) + permutation.append(index[max(s, tuple(-x for x in s))]) + require(len(set(permutation)) == 120, "root permutation") + for i, j in product(range(120), repeat=2): + require(((pairing[i] >> j) & 1) == + ((pairing[permutation[i]] >> permutation[j]) & 1), "pairing equivariance") + require((dot(roots[i], roots[j]) == 0) == + (dot(roots[permutation[i]], roots[permutation[j]]) == 0), + "support equivariance") + generators.append(permutation) + transports = {SEED_ROOT: list(range(120))} + queue = deque([SEED_ROOT]) + while queue: + a = queue.popleft() + for g in generators: + b = g[a] + if b not in transports: + transports[b] = [g[i] for i in transports[a]] + queue.append(b) + require(set(transports) == set(range(120)), "all-root orbit coverage") + return generators, transports + + +def term_cost(pairing, left, right, puncture): + return rank2(pairing[i] & right & ~puncture for i in bits(left & ~puncture)) + + +def decoder(roots, pairing, puncture, left_columns, right_columns): + """Columns are doubled normal coordinates. Check every surviving pair.""" + require(len(left_columns) == len(right_columns) == 8, "eight terms required") + for i, j in product(range(8), repeat=2): + require(sum(a[i] * b[j] for a, b in zip(left_columns, right_columns)) == 8 * (i == j), + "exact dual identity") + live = [i for i in range(120) if not puncture >> i & 1] + left = [[dot(roots[i], a) for a in left_columns] for i in live] + right = [[dot(roots[i], b) for b in right_columns] for i in live] + for ii, i in enumerate(live): + for jj, j in enumerate(live): + require(dot(left[ii], right[jj]) == 8 * dot(roots[i], roots[j]), + "all-pairs decoder identity") + costs = [] + for k in range(8): + rows = sum((left[ii][k] != 0) << i for ii, i in enumerate(live)) + cols = sum((right[jj][k] != 0) << j for jj, j in enumerate(live)) + costs.append(term_cost(pairing, rows, cols, 0)) + return {"puncture": list(bits(puncture)), "ports": len(live), + "left_columns_doubled": [list(a) for a in left_columns], + "right_columns_doubled": [list(b) for b in right_columns], + "term_costs": costs, "copy_cost": sum(costs), + "copy_per_port": str(Fraction(sum(costs), len(live))), + "identity_entries": len(live) ** 2} + + +def reduced_inventory(roots, pairing, words, supports, puncture): + index = {direction(r): i for i, r in enumerate(roots)} + ranks = Counter() + kernels = {} + for word in words: + rows = [roots[i] for i in bits(word & ~puncture)] + pivots = echelon(rows) + rank = len(pivots) + require(rank in (7, 8), "boundary rational ranks must be seven or eight") + ranks[rank] += 1 + if rank == 7: + normal = kernel_direction(pivots) + require(normal in index and all(dot(r, normal) == 0 for r in rows), + "reduced normal must be a root direction") + kernels[hex(word)] = index[normal] + reduced = sorted(set(kernels.values())) + dimensions = {i: rank2(pairing[j] for j in bits(supports[i] & ~puncture)) for i in reduced} + require(ranks == Counter({8: 483, 7: 28}), "complete rational rank inventory") + require(Counter(dimensions.values()) == Counter({8: 25, 7: 1}), "support inventory") + require(dimensions[SEED_ROOT] == 7, "distinguished rank-seven support") + orthogonal = [i for i in reduced if dot(roots[i], roots[SEED_ROOT]) == 0] + require(len(orthogonal) == 1, "unique reduced direction perpendicular to distinguished root") + costs = {i: term_cost(pairing, supports[SEED_ROOT], supports[i], puncture) for i in reduced} + admissible_six = [i for i in reduced if dot(roots[i], roots[SEED_ROOT]) and costs[i] <= 6] + require(admissible_six == [SEED_ROOT] and costs[SEED_ROOT] == 6, + "only parallel root normals give an admissible cost-six term") + frame = [SEED_ROOT, orthogonal[0]] + for i in range(120): + if all(dot(roots[i], roots[j]) == 0 for j in frame): + frame.append(i) + if len(frame) == 8: + break + require(len(frame) == 8, "orthogonal completion exists") + frame_costs = [term_cost(pairing, supports[i], supports[i], puncture) for i in frame] + require(frame_costs == [6, 7, 9, 9, 9, 9, 9, 9], "sharp 67-copy completion") + return {"puncture_hex": hex(puncture), "rank_counts": dict(sorted(ranks.items())), + "kernel_by_functional_hex": kernels, + "support_dimensions": dimensions, "distinguished_partner_costs": costs, + "orthogonal_reduced_root": orthogonal[0], "attaining_frame": frame, + "attaining_term_costs": frame_costs} + + +def source_pins(): + sources = json.loads((HERE / "SOURCES.json").read_text()) + require(sources["base_commit"] == "ef5f39403610d231a0c27e8a6b4f08c10209bcd6", "base pin") + for entry in sources["files"]: + if entry["usage"] == "input": + require(hashlib.sha256((ROOT / entry["path"]).read_bytes()).hexdigest() == entry["sha256"], + "source pin mismatch: " + entry["path"]) + + +def run(): + source_pins() + roots, pairing, words, supports = model() + table = flat_census(roots, words) + generators, transports = symmetry(roots, pairing) + minimum, count, cuts = two_annihilator_census(supports[SEED_ROOT], words) + require((minimum, count, len(cuts)) == (25, 189, 63), "first boundary is complete") + all_cuts = set() + root_cut_counts = [] + for a in range(120): + m, n, actual = two_annihilator_census(supports[a], words) + transported = {sum(1 << transports[a][i] for i in bits(p)) for p in cuts} + require((m, n, len(actual)) == (25, 189, 63) and actual == transported, + "transported cuts equal complete root census") + require(not (all_cuts & actual), "distinguished root is unique for every cut") + all_cuts.update(actual) + root_cut_counts.append(len(actual)) + records = [reduced_inventory(roots, pairing, words, supports, p) for p in sorted(cuts)] + record = records[SEED_ROOT % len(records)] + puncture67 = int(record["puncture_hex"], 16) + frame = [roots[i] for i in record["attaining_frame"]] + witness67 = decoder(roots, pairing, puncture67, frame, frame) + require(witness67["copy_cost"] == 67, "completed exceptional bill") + deleted = set(i for pair in PAIRS56 for i in pair) + require(len(deleted) == 16, "P104 parent puncture") + deleted.update([i for i in range(120) if i not in deleted][:9]) + puncture56 = sum(1 << i for i in deleted) + a_columns = [roots[a] for a, _ in PAIRS56] + b_columns = [tuple(2 * (dot(roots[a], roots[b]) // 4) * x for x in roots[b]) + for a, b in PAIRS56] + witness56 = decoder(roots, pairing, puncture56, a_columns, b_columns) + require(witness56["term_costs"] == [7] * 8, "attaining 56-copy decoder at p=25") + predecessor = json.loads((ROOT / "research/root-pair-first-drop-seed48/CERTIFICATE.json").read_text()) + p26 = sum(1 << i for i in predecessor["witness_puncture"]) + a26, b26 = predecessor["canonical_pair"] + require(p26.bit_count() == 26 and a26 != b26 and dot(roots[a26], roots[b26]) != 0, + "predecessor nonparallel witness scope") + require(term_cost(pairing, supports[a26], supports[b26], p26) == 6, "sharp nonparallel threshold") + require(Fraction(67, 95) > Fraction(56, 95) > Fraction(7, 13), "copy/port budget gate") + controls = controls_for(roots, pairing, words, supports, cuts, witness56, puncture67) + return {"schema_version": 1, "status": "PASS_COMPLEX_GENERAL_NORMAL_BOUNDARY", + "assignment_seed": 31, "base_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6", + "root_order_sha256": digest(roots), "root_index": SEED_ROOT, + "flat_census": table, "functional_pairs_per_support": 130305, + "weyl_generators": len(generators), "root_orbit": len(transports), + "equivariance_pairs": len(generators) * 120 * 120, + "root_cut_counts": root_cut_counts, "distinct_boundary_cuts": len(all_cuts), + "all_boundary_cuts_sha256": digest(sorted(hex(p) for p in all_cuts)), + "representative_functional_pairs_attaining_25": count, + "representative_cuts": records, "rational_matrices_checked": len(records) * len(words), + "witness67": witness67, "witness56": witness56, + "predecessor_nonparallel_witness": {"pair": [a26, b26], "puncture": list(bits(p26)), "cost": 6}, + "controls": controls, + "scope": "Written proof plus exact finite premises; inherited real root-flat classification F6. No physical supplier, new kappa, cold replay, or formal multiplication theorem."} + + +def controls_for(roots, pairing, words, supports, cuts, witness56, puncture67): + passed = [] + + def reject(name, action, reason): + try: + action() + except ValueError as error: + require(reason in str(error), "wrong control failure: " + name) + passed.append(name) + return + raise ValueError("accepted negative control: " + name) + + reject("universal term-cost-seven claim at p25", + lambda: require(term_cost(pairing, supports[SEED_ROOT], supports[SEED_ROOT], puncture67) >= 7, + "parallel-root counterexample"), "parallel-root counterexample") + bad = [[x for x in col] for col in witness56["right_columns_doubled"]] + bad[0] = [x // 2 for x in bad[0]] + reject("missing dual factor two", lambda: decoder(roots, pairing, + sum(1 << i for i in witness56["puncture"]), witness56["left_columns_doubled"], bad), + "exact dual identity") + wrong_pairing = [sum(((dot(a, b) // 4) & 1) << j for j, b in enumerate(roots)) for a in roots] + reject("omitted unit label", lambda: require(rank2(wrong_pairing) == 9, "pairing rank"), "pairing rank") + reject("omitted boundary cut", lambda: require(len(cuts - {min(cuts)}) == 63, "cut coverage"), "cut coverage") + reject("omitted dual functional", lambda: require(len(words[:-1]) == 511, "dual coverage"), "dual coverage") + for i in bits(puncture67): + restored = puncture67 ^ (1 << i) + require(term_cost(pairing, supports[SEED_ROOT], supports[SEED_ROOT], restored) >= 7, + "single-line restoration must remove the cost-six term") + passed.append("all 25 one-line restorations remove the cost-six exception") + def gaussian_dot(xs, ys): + return (sum(a * c - b * d for (a, b), (c, d) in zip(xs, ys)), + sum(a * d + b * c for (a, b), (c, d) in zip(xs, ys))) + + aa, bb = [(1, 0), (0, 1)], [(1, 0), (0, -1)] + require(gaussian_dot(aa, bb) == (2, 0) and + gaussian_dot(aa, [(a, -b) for a, b in bb]) == (0, 0), + "ordinary-transpose complex convention") + passed.append("conjugate transpose changes the exact identity") + require(gaussian_dot(aa, aa) == (0, 0), + "complex parallel cross-terms need not be forbidden") + passed.append("isotropic parallel cross-term is not assumed impossible") + return passed + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--write-certificate", action="store_true") + args = parser.parse_args() + result = run() + path = HERE / "CERTIFICATE.json" + if args.write_certificate: + path.write_text(json.dumps(result, sort_keys=True, indent=2) + "\n") + else: + require(canonical(json.loads(path.read_text())) == canonical(result), "saved certificate mismatch") + print("PASS_COMPLEX_GENERAL_NORMAL_BOUNDARY cuts=7560 representatives=63 rational_matrices=32193 " + "p25_minimum=56 cost6_completion_minimum=67 nonparallel_threshold=26 controls=8") + + +if __name__ == "__main__": + main()