Research: quadratic five-cube copy floor and sharp rank-cap tradeoff (no new kappa)

research/five-cube-quadratic-code-seed44/CLAIM.json

@@ -0,0 +1,25 @@
+{
+  "seed": 44,
+  "random_source": "Python secrets.randbelow(100), operating-system randomness, first draw",
+  "track": "structural decoder research",
+  "base_commit": "1b59b624d21924f8e3537cdc1ac52d3be6eda2fe",
+  "scope_key": "five-cube/quadratic-augmented-parity-code/all-finite-binary-factorizations/rational-pairing-rank",
+  "candidates": [
+    "Rank-one off-mask perturbations of the P112 complex decoder; adjacent to PR1191/1197 and historical masked searches",
+    "Deeper punctures retaining a P104 frame pair; substantially covered by PR1187",
+    "All finite factorizations in span(1, parity, h) for every Boolean h of degree at most two on the five-cube; chosen"
+  ],
+  "seed_diversification": "44 modulo 3 is 2: choose the third candidate, with a five-cube object rather than the eight-term E8 frontier",
+  "changed_restriction": "PR1046 restricts masks to affine forms. Allow a quadratic Boolean mask h, with both factor spaces in the same three-dimensional augmented parity code. No term-count bound.",
+  "method": "Enumerate all 16383 degree-at-most-two masks modulo span(1,parity), quotient only by verified cube isometries, compute exact rational pairing ranks, and certify the weighted shortest XOR path on all 512 coefficient matrices.",
+  "hypothesis": "The algebraic pairing-rank sum still cannot fall below 32 in this class.",
+  "optimistic_budget": "Inherited five-cube ledger D <= (s-1)*32 - s*C; grant every other nonnegative helper, source-access, frame, restoration and recursive expense zero. A copy sum C>=32 kills this mechanism in that ledger.",
+  "protected_invariant": "Every admitted factorization is covered and every charged rank is exact; a finite search is not a physical supplier or a full multiplication theorem.",
+  "failure_modes": ["omitted quadratic codes", "invalid orbit symmetry", "rational rank confused with binary rank", "wrong all-pairs identity", "overcomplete decompositions excluded by a term cap", "distance certificate accepted without every edge inequality"],
+  "overlap": "811 open PRs inspected through PR1199; PR1177 covers at most three arbitrary masks, while PR1046 covers arbitrary finite affine-code factorizations. No matching quadratic-code investigation found. Discussion1163/1166/1167 read; no pinned coordination issue exists.",
+  "coordination": "Local claim, following the user's no-pinned-issue fallback; no public reservation issue created.",
+  "expected_deliverable": "One scoped structural result with exact certificate, independently implemented checker, meaningful fault controls, source pins and at most one PR.",
+  "stop_rule": "Complete this finite domain or report exact incomplete coverage; one evidence-driven pivot at most. Freeze research by 20:29 UTC for verification.",
+  "expected_finish_utc": "2026-10-13T20:41:00Z",
+  "attribution": "Hanson Sun (@bbMisk), substantial OpenAI Codex assistance; credit martinxxx9 PR1046, Ross Richardson PR958/1177, Joel Pulikkan/GamingPuzzled PR231, and the retained accounting lineage. Apache-2.0."
+}

research/five-cube-quadratic-code-seed44/NOTICE.md

@@ -0,0 +1,41 @@
+# Attribution, source identity and scope
+
+New finite census, graph certificates, proof, implementations and packaging:
+**Hanson Sun (@bbMisk), 2026**, with substantial **OpenAI Codex assistance**
+in research, mathematics, implementation, checking and writing. Both
+implementations are AI-assisted. Code and arithmetic independence does
+not establish independent human review. Apache-2.0.
+
+Predecessor credit:
+
+- **Martin Rv (@martinxxx9), PR1046**, head
+  `7cadcb0a2fc71fdedc0066704dec3fec58c2ca2c`: the affine five-cube
+  factorization model, cost-32 parity witness, four rank-eight affine
+  terms and conditional stage-ledger obstruction. Its proof and complete
+  attribution notice are retained unchanged.
+- **Ross Richardson (@RossRichardson), PR958**, head
+  `28d68c88f0184835d8c8a7266ab8d0dad3f18a8a`: the five-port quadratic
+  Gram and explicit geometry/accounting premises. Its proof and notice,
+  already preserved by PR1046, are retained byte for byte.
+- **Ross Richardson, pending PR1177**, inspected at
+  `e878ef36109bc6d7e21e62d03c22d8ec36f6f936`: arbitrary masks with at
+  most three totals, parity-basis analysis and the canonical three-total
+  family containing our displayed rank-34 witness. No code or theorem
+  from that pending PR is imported. Our certificate treats every finite
+  term count in common quadratic-augmented parity codes.
+- **Joel Pulikkan (@GamingPuzzled), PR231**, and **icekylinx/eumemic**:
+  the five-port parity identity and paired-cube lineage.
+- **DreamingOfClouds, PR446/PR595**, **Jacob Sussman** and
+  **Zhihao Chen (@jacklightChen)**: the retained common-label,
+  separate-total copying and explicit stage/deficit interfaces.
+- **Rohan Arun, Jeremy Kothe, Hao Yan**, and the selected PR1094/PR589
+  construction lineage: benchmark context only, not newly replayed here.
+
+All retained files are pinned to their exact repository bytes in
+`SOURCES.json`; their original licenses and assistance disclosures remain.
+The selected-result snapshot is context only. No upstream or predecessor
+file is modified, and no predecessor executable implementation is used.
+
+This is a scoped finite structural result, not a new supplier, kappa,
+physical implementation, formal multiplication proof or literature-wide
+priority claim. Physical and analytic obligations are listed in PROOF.md.

research/five-cube-quadratic-code-seed44/OVERLAP.json

@@ -0,0 +1,80 @@
+{
+  "observed_utc": "2026-10-13T20:01:13Z",
+  "open_pr_count": 811,
+  "highest_pr": 1199,
+  "pinned_coordination_issues": 0,
+  "discussion_numbers": [
+    1163,
+    1166,
+    1167
+  ],
+  "relevant_prs": [
+    {
+      "number": 1197,
+      "title": "Scoped negative result: two P112 totals need overlapping independent lifts (no new kappa)",
+      "head_sha": "a0935afff6cc5540a582c5881bc8d5f98b587357",
+      "updated_at": "2026-10-13T19:57:10Z",
+      "html_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1197"
+    },
+    {
+      "number": 1191,
+      "title": "Research: one-sided E8 mask rigidity on 1,373 punctured units (no new kappa)",
+      "head_sha": "640b43bb13d3d476a2d83b74ef0d33f14d1b5711",
+      "updated_at": "2026-10-13T19:54:49Z",
+      "html_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1191"
+    },
+    {
+      "number": 1187,
+      "title": "Research: sharp preserved-frame P104 COPY-drop threshold at 38 punctures (no new kappa)",
+      "head_sha": "15bee4c8ea0f54fea575a48441c6acc8bad263cd",
+      "updated_at": "2026-10-13T19:54:00Z",
+      "html_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1187"
+    },
+    {
+      "number": 1180,
+      "title": "Structural proof: overcomplete factor-linear E8 decoders retain the puncture copy-cost floor",
+      "head_sha": "75b408646d558147a4eb250fc407d1fbfcb8df4e",
+      "updated_at": "2026-10-13T19:46:21Z",
+      "html_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1180"
+    },
+    {
+      "number": 1177,
+      "title": "Research: exact arbitrary-mask three-total five-cube boundary (no new kappa)",
+      "head_sha": "e878ef36109bc6d7e21e62d03c22d8ec36f6f936",
+      "updated_at": "2026-10-13T19:43:35Z",
+      "html_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1177"
+    },
+    {
+      "number": 912,
+      "title": "Research/seed42 decoder",
+      "head_sha": "945b3da1f1dd87cebd456b3a0459601b15f7ea8b",
+      "updated_at": "2026-10-13T12:00:26Z",
+      "html_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/912"
+    },
+    {
+      "number": 765,
+      "title": "Scoped negative result: tensor E8 basis mixing cannot lower 72^d copy rank",
+      "head_sha": "d8eaa3db32f76111b39db9b930f347ef30118a28",
+      "updated_at": "2026-10-13T05:18:29Z",
+      "html_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/765"
+    }
+  ],
+  "selected_task": "quadratic-augmented parity code on the five-cube, both factors in one common three-dimensional code, arbitrary finite term count",
+  "code_or_theorem_imports_from_pending_prs": false,
+  "final_refresh": {
+    "observed_utc": "2026-10-13T20:19:57Z",
+    "open_pr_count": 823,
+    "highest_pr": 1214,
+    "main_head": "1b59b624d21924f8e3537cdc1ac52d3be6eda2fe",
+    "discussion_numbers": [
+      1200,
+      1202,
+      1210,
+      1163,
+      1166,
+      1167
+    ],
+    "pinned_coordination_issues": 0,
+    "overlap_result": "No matching quadratic five-cube code investigation. PR1177 unchanged at e878ef36109bc6d7e21e62d03c22d8ec36f6f936; its three-total proof inspected. New decoder PR1212/1213 and discussion1210 concern E8, not the five-cube."
+  }
+}

research/five-cube-quadratic-code-seed44/PROOF.md

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+# Quadratic augmentation of the five-cube parity code
+
+This is a finite, computer-assisted structural result. It is not a formal
+proof of the multiplication theorem or an implemented supplier.
+
+## Statement
+
+Use PR958/PR1046's 32 ports `x in F2^5`, with
+
+```text
+j(x,y) = 5 - HammingDistance(x,y)
+G[x,y] = 2(j(x,y)-1)(j(x,y)-3)                 over Q
+B[x,y] = 1 + parity(x) + parity(y)            over F2.
+```
+
+Let `E=1+parity` and `O=parity`. For a Boolean polynomial `h` of degree
+at most two, outside `U=span_F2{E,O}`, define `C_h=span_F2{E,O,h}`.
+Allow any finite list of nonzero functions `f_i,g_i in C_h` satisfying
+
+```text
+sum_i f_i(x) g_i(y) = B[x,y]                  over F2, for all x,y.
+cost = sum_i rank_Q G[supp(f_i),supp(g_i)].
+```
+
+Both factors must lie in the **same** code `C_h`. The factors may differ,
+repeat, cancel or have zero pairing rank. There is no term-count cap.
+The nonlinear functions here select ports; messages remain linear XOR
+sums of input data.
+
+**Result.** The minimum cost is exactly **32** for each of the **16,383**
+distinct codes. Of these, **16,368** have a nonzero quadratic part. In
+every such code, requiring each term's rank to be at most **15** forces
+cost at least **34**, or makes the factorization impossible. The bound
+34 is attained in exactly **60** of these codes. In particular, any
+cost-32 factorization in a genuinely quadratic code has a rank-16 term.
+
+The two parity totals already attain 32; that value and witness belong
+to PR1046 and its predecessors. PR1177 separately treats arbitrary
+masks with at most three terms. The new coverage here is every term
+count in every common quadratic-augmented parity code.
+
+## Complete domain and symmetry reduction
+
+Boolean polynomials of degree at most two in five variables have the
+16 independent squarefree monomials `1`, five `x_i`, and ten `x_i*x_j`.
+Quotienting by `U=span{1,parity}` leaves dimension 14. Each nonzero
+quotient element defines one three-dimensional code, since the scalar
+field is F2. Thus there are `2^14-1=16383` codes. Exactly `2^4-1=15`
+are affine; the remaining 16368 are genuinely quadratic.
+
+The generator represents each coset by constant coefficient zero and
+`x_4` coefficient zero. Its fourteen bits encode `x_0,...,x_3`, followed
+by the ten quadratic monomials in lexicographic pair order.
+
+Coordinate flips and coordinate permutations preserve Hamming distance,
+G and B. They preserve degree and U and consequently carry a code's
+factorizations to those of the image code, with identical termwise
+costs. The nine generators are the five flips and four adjacent swaps.
+The certificate contains a transport tree for **every** nonzero quotient
+element, rooted at **64** representatives. Closure under all nine
+generators proves these connected classes are precisely the orbits.
+
+The separate checker evaluates all 65536 polynomials and all four
+members of each quotient class. It checks the transport edges using
+polynomial substitution, whereas the generator uses truth-table
+permutations. It also checks each proposed isometry on all port pairs.
+No unverified symmetry classification is a premise.
+
+## Why 512 states cover arbitrarily many terms
+
+Fix a representative. Let F be the 32-by-3 binary evaluation matrix
+with columns E,O,h. It has column rank three. Consequently the map
+`M -> F M F^t` on 3-by-3 binary matrices is injective: a binary left
+inverse L of F recovers `M=L(FMF^t)L^t`.
+
+A nonzero factor is specified by one of seven coefficient vectors.
+There are therefore **49 ordered** allowed dyads `a b^t`, with exact
+weights `w(a,b)=rank_Q G[supp(Fa),supp(Fb)]`. Factorization of B is
+equivalent to their XOR sum being
+
+```text
+M_B = diag(1,1,0).
+```
+
+The 512 binary 3-by-3 matrices form a graph: from state M, a term
+`(a,b)` moves to `M + a b^t` and costs `w(a,b)`. Every finite
+factorization is a path from zero to M_B, and conversely. This includes
+overcomplete decompositions and arbitrary repetitions. Zero-weight
+edges are retained. No positive per-term bound or bounded word length
+is used to exclude a cheaper factorization.
+
+For every representative, the package computes all 49 rational ranks
+and emits a potential d on the reachable states, together with an
+attaining word. The independent checker verifies
+
+```text
+d(0)=0,
+|d(M+a b^t)-d(M)| <= w(a,b)                   for every allowed edge.
+```
+
+Summing along any path proves its cost is at least `d(M_B)`.
+The displayed word attains this value and is checked on all 1024 port
+pairs. This proves the minimum without trusting Dijkstra's algorithm
+or its stopping rule. The unrestricted certificates check **1,605,632**
+directed edge inequalities. A second set retains only edges of rank
+at most 15. Unreachable targets are certified by the exact binary span
+of the allowed dyads; the checker recomputes that span.
+
+Integer elimination with gcd normalization generates the weights.
+The separate checker uses rational `Fraction` elimination on all
+**3,136 ordered** restricted Gram matrices. Neither uses floating-point
+ranks, a modular-rank guess, or predecessor executable code.
+
+## Sharp rank-15 tradeoff
+
+The capped minima, weighted by the number of codes in each orbit, are:
+
+| Minimum total with each rank at most 15 | Codes |
+|---|---:|
+| 32 (affine codes only) | 15 |
+| 34 | 60 |
+| 35 | 180 |
+| 38 | 240 |
+| 39 | 690 |
+| 41 | 900 |
+| 44 | 80 |
+| 45 | 3660 |
+| 52 | 3808 |
+| No such factorization | 6750 |
+
+An attaining representative for 34 is code 7920:
+
+```text
+h = x0*x1 + x0*x2 + x0*x3 + x0*x4
+  + x1*x3 + x1*x4 + x2*x3 + x2*x4              over F2.
+
+B = E h^t + O (E+O+h)^t + (E+O)(E+h)^t.
+term ranks = 13, 13, 8; total = 34.
+```
+
+This witness lies in the three-total parameterization described by
+Ross Richardson's pending PR1177. The new certificate proves that
+adding more terms within these codes cannot beat 34 under the cap.
+It does not claim a new three-total identity or priority for that family.
+
+## Accounting implication and limits
+
+Retain PR1046's explicit additive per-total copy pricing and per-replica
+stage ledger, with `v=32` and `s>=3`:
+
+```text
+D <= (s-1)*v - s*cost <= (s-1)*32 - s*32 = -32.
+```
+
+This grants source access, gathers, arithmetic, helpers, frame changes,
+restoration, idle work and all other recursive/precision/hosting expenses
+zero additional cost. Nonnegative extra expenses cannot create a positive
+deficit. With a genuinely quadratic code and the rank-15 cap the bound
+is stronger: `D <= -32-2s`. This is an optimistic exclusion within the
+stated ledger, not a proposed complete supplier invoice.
+
+The result excludes different source/reader codes, more than one
+additional independent mask, higher-degree masks, other cubes or Grams,
+masked rather than all-pairs identities, interacting data messages, and
+changed copying/stage accounting. Source-access realization, native
+frames, physical precision, restoration, whole-call hosting, bit H1-H5,
+all-size compilation and analytic transfer are not discharged. The
+selected frontier is unchanged. No new kappa, runtime gain, full cold
+supplier replay, proof-assistant verification or human review is claimed.

research/five-cube-quadratic-code-seed44/README.md

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+# Quadratic five-cube recoding: exact copy floor and rank-cap tradeoff
+
+**Status: scoped structural negative result. No new kappa.**
+
+Adding one quadratic Boolean mask to the five-cube parity code does not
+reduce its minimum copy-rank sum below **32**. The certificate covers all
+**16,383** augmented codes and every finite binary factorization within
+each code, including redundant and zero-cost terms. For the **16,368**
+genuinely quadratic codes, limiting each term's rank to 15 raises the
+minimum total to at least **34**; a **13,13,8** witness is exact. Separate
+integer and rational implementations verify the coverage and certificates.
+Under the inherited additive stage ledger, the optimistic deficit remains
+negative. This is a scoped structural result, with no new kappa or physical
+supplier.
+
+Both factor spaces must lie in the same `span_F2{1,parity,h}`, with
+`degree(h)<=2`. This changes PR1046's affine restriction and differs from
+PR1177's at-most-three-term arbitrary-mask result. See [PROOF.md](PROOF.md)
+for the complete domain, graph certificate argument, exact capped-cost
+table, accounting premises and exclusions.
+
+## Reproduce
+
+Python 3.10+; standard library only; from the repository root:
+
+```sh
+python3 -I -O -B research/five-cube-quadratic-code-seed44/test_package.py
+```
+
+Expected: **six test groups pass**, ending in `Ran 6 tests` and `OK`.
+This regenerates the certificate, runs the separate mathematical checker,
+tests corrupted inputs, and repeats both CLIs from a standalone copy.
+The scripts themselves can also be run separately:
+
+```sh
+python3 -I -O -B research/five-cube-quadratic-code-seed44/verify.py
+python3 -I -O -B research/five-cube-quadratic-code-seed44/independent.py
+```
+
+Expected markers: `PASS_FIVE_CUBE_QUADRATIC_CODES` and
+`PASS_INDEPENDENT_QUADRATIC_CODES`.
+
+`verify.py --write /new/path.json` writes a fresh certificate and refuses
+an occupied output. `CERTIFICATE.json` contains 64 exact rank tables,
+all 16,383 symmetry transports, two potential certificates per orbit,
+and attaining words. The separate checker imports no generator code and
+does not run its shortest-path algorithm.
+
+## Pins and attribution
+
+- Base: `1b59b624d21924f8e3537cdc1ac52d3be6eda2fe`.
+- Track: structural decoder research; first OS-random seed **44**.
+- [SOURCES.json](SOURCES.json): exact source hashes, Git blobs and heads.
+- [CLAIM.json](CLAIM.json): candidates, overlap, domain and stopping rule.
+- [VALIDATION.json](VALIDATION.json): executed checks and their limits.
+- [NOTICE.md](NOTICE.md): predecessor credit and AI-assistance disclosure.
+
+Prepared by **Hanson Sun (@bbMisk)** with substantial **OpenAI Codex**
+assistance. Apache-2.0. PR1046's attaining value, PR958's geometry and
+accounting, and PR1177's adjacent three-total family retain their credit.
+The selected construction and `upstream/` are unchanged.