diff --git a/research/five-cube-quadratic-code-seed44/CERTIFICATE.json b/research/five-cube-quadratic-code-seed44/CERTIFICATE.json
new file mode 100644
index 0000000000..ab66b8582f
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/CERTIFICATE.json
@@ -0,0 +1 @@
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08,3920,6,1008],[6032,3920,8,1008],[10064,5936,7,1008],[9456,5360,7,1008],[10480,8944,6,1008],[7952,6384,5,1008],[6527,1919,6,1008],[1976,1919,8,1008],[3903,2559,5,1008],[2815,2559,7,1008],[4600,2559,8,1008],[1780,1528,7,1008],[6969,6522,5,1008],[10870,6522,7,1008],[6589,6522,8,1008],[2007,1979,7,1008],[3925,3897,7,1008],[5950,3897,8,1008],[5366,2806,8,1008],[8957,4605,7,1008],[6391,1783,6,1008],[6962,6513,5,1008],[10865,6513,7,1008],[6577,6513,8,1008],[2001,1969,7,1008],[3928,3892,7,1008],[5940,3892,8,1008],[5375,2808,8,1008],[8952,4596,7,1008],[6388,1778,6,1008],[11096,6964,7,1008],[7476,6964,8,1008],[13503,10872,8,1008],[10968,6580,7,1008],[6628,2002,6,1008],[7012,3922,6,1008],[6034,3922,8,1008],[10066,5938,7,1008],[9457,5361,7,1008],[10481,8945,6,1008],[7954,6385,5,1008],[6971,6523,5,1008],[10871,6523,7,1008],[6588,6523,8,1008],[2006,1978,7,1008],[3933,3901,7,1008],[5946,3901,8,1008],[5369,2814,8,1008],[8949,4601,7,1008],[6387,1781,6,1008],[11101,6973,7,1008],[7482,6973,8,1008],[13497,10878,8,1008],[10965,6585,7,1008],[6627,2005,6,1008],[7015,3927,6,1008],[6039,3927,8,1008],[10076,5948,7,1008],[9467,5367,7,1008],[10490,8956,6,1008],[7957,6390,5,1008],[11098,6966,7,1008],[7478,6966,8,1008],[13502,10873,8,1008],[10969,6581,7,1008],[6629,2003,6,1008],[7020,3930,6,1008],[6045,3930,8,1008],[10074,5942,7,1008],[9470,5374,7,1008],[10489,8953,6,1008],[7958,6389,5,1008],[11104,11088,6,1008],[13712,11088,8,1008],[11856,7472,7,1008],[13520,13488,7,1008],[10976,10960,6,1008],[7968,6624,5,1008],[7584,7008,8,1008],[10128,6032,7,1008],[14688,10064,6,1008],[12528,9456,6,1008],[12048,10480,5,1008],[6975,6527,5,1008],[10879,6527,7,1008],[6584,6527,8,1008],[2004,1976,7,1008],[3935,3903,7,1008],[5944,3903,8,1008],[5368,2815,8,1008],[8948,4600,7,1008],[6386,1780,6,1008],[11093,6969,7,1008],[7486,6969,8,1008],[13494,10870,8,1008],[10973,6589,7,1008],[6631,2007,6,1008],[7011,3925,6,1008],[6037,3925,8,1008],[10078,5950,7,1008],[9466,5366,7,1008],[10491,8957,6,1008],[7959,6391,5,1008],[11090,6962,7,1008],[7474,6962,8,1008],[13489,10865,8,1008],[10961,6577,7,1008],[6625,2001,6,1008],[7016,3928,6,1008],[6047,3928,8,1008],[10072,5940,7,1008],[9471,5375,7,1008],[10488,8952,6,1008],[7956,6388,5,1008],[11112,11096,6,1008],[13727,11096,8,1008],[11864,7476,7,1008],[13535,13503,7,1008],[10984,10968,6,1008],[7972,6628,5,1008],[7588,7012,8,1008],[10130,6034,7,1008],[14692,10066,6,1008],[12529,9457,6,1008],[12050,10481,5,1008],[11095,6971,7,1008],[7484,6971,8,1008],[13495,10871,8,1008],[10972,6588,7,1008],[6630,2006,6,1008],[7019,3933,6,1008],[6042,3933,8,1008],[10070,5946,7,1008],[9461,5369,7,1008],[10483,8949,6,1008],[7955,6387,5,1008],[11115,11101,6,1008],[13722,11101,8,1008],[11862,7482,7,1008],[13525,13497,7,1008],[10979,10965,6,1008],[7971,6627,5,1008],[7591,7015,8,1008],[10139,6039,7,1008],[14698,10076,6,1008],[12541,9467,6,1008],[12057,10490,5,1008],[11116,11098,6,1008],[13725,11098,8,1008],[11866,7478,7,1008],[13534,13502,7,1008],[10985,10969,6,1008],[7974,6629,5,1008],[7595,7020,8,1008],[10141,6045,7,1008],[14700,10074,6,1008],[12542,9470,6,1008],[12058,10489,5,1008],[13728,11104,8,1008],[13968,13712,7,1008],[14944,11856,6,1008],[13536,13520,6,1008],[12096,10976,5,1008],[11968,7584,7,1008],[14752,10128,6,1008],[15136,14688,5,1008],[14096,12528,5,1008],[11103,6975,7,1008],[7480,6975,8,1008],[13496,10879,8,1008],[10964,6584,7,1008],[6626,2004,6,1008],[7023,3935,6,1008],[6040,3935,8,1008],[10068,5944,7,1008],[9460,5368,7,1008],[10482,8948,6,1008],[7953,6386,5,1008],[11107,11093,6,1008],[13717,11093,8,1008],[11870,7486,7,1008],[13530,13494,7,1008],[10987,10973,6,1008],[7975,6631,5,1008],[7587,7011,8,1008],[10137,6037,7,1008],[14702,10078,6,1008],[12540,9466,6,1008],[12059,10491,5,1008],[11108,11090,6,1008],[13714,11090,8,1008],[11858,7474,7,1008],[13521,13489,7,1008],[10977,10961,6,1008],[7970,6625,5,1008],[7599,7016,8,1008],[10143,6047,7,1008],[14696,10072,6,1008],[12543,9471,6,1008],[12056,10488,5,1008],[13743,11112,8,1008],[13983,13727,7,1008],[14952,11864,6,1008],[13551,13535,6,1008],[12104,10984,5,1008],[11976,7588,7,1008],[14756,10130,6,1008],[15140,14692,5,1008],[14098,12529,5,1008],[11111,11095,6,1008],[13719,11095,8,1008],[11868,7484,7,1008],[13531,13495,7,1008],[10986,10972,6,1008],[7973,6630,5,1008],[7596,7019,8,1008],[10134,6042,7,1008],[14694,10070,6,1008],[12531,9461,6,1008],[12051,10483,5,1008],[13740,11115,8,1008],[13974,13722,7,1008],[14950,11862,6,1008],[13539,13525,6,1008],[12099,10979,5,1008],[11979,7591,7,1008],[14765,10139,6,1008],[15145,14698,5,1008],[14110,12541,5,1008],[13739,11116,8,1008],[13981,13725,7,1008],[14956,11866,6,1008],[13550,13534,6,1008],[12106,10985,5,1008],[11975,7595,7,1008],[14763,10141,6,1008],[15148,14700,5,1008],[14109,12542,5,1008],[14016,13728,7,1008],[15520,13968,6,1008],[15168,14944,5,1008],[14208,13536,5,1008],[15040,11968,6,1008],[15648,14752,5,1008],[11119,11103,6,1008],[13720,11103,8,1008],[11860,7480,7,1008],[13524,13496,7,1008],[10978,10964,6,1008],[7969,6626,5,1008],[7592,7023,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diff --git a/research/five-cube-quadratic-code-seed44/CLAIM.json b/research/five-cube-quadratic-code-seed44/CLAIM.json
new file mode 100644
index 0000000000..8b5215a1ff
--- /dev/null
+++ b/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."
+}
diff --git a/research/five-cube-quadratic-code-seed44/NOTICE.md b/research/five-cube-quadratic-code-seed44/NOTICE.md
new file mode 100644
index 0000000000..7f63863c72
--- /dev/null
+++ b/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.
diff --git a/research/five-cube-quadratic-code-seed44/OVERLAP.json b/research/five-cube-quadratic-code-seed44/OVERLAP.json
new file mode 100644
index 0000000000..36095bfb7f
--- /dev/null
+++ b/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."
+  }
+}
diff --git a/research/five-cube-quadratic-code-seed44/PROOF.md b/research/five-cube-quadratic-code-seed44/PROOF.md
new file mode 100644
index 0000000000..268baf1182
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/PROOF.md
@@ -0,0 +1,169 @@
+# 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.
diff --git a/research/five-cube-quadratic-code-seed44/README.md b/research/five-cube-quadratic-code-seed44/README.md
new file mode 100644
index 0000000000..bce843dc04
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/README.md
@@ -0,0 +1,61 @@
+# 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.
diff --git a/research/five-cube-quadratic-code-seed44/SOURCES.json b/research/five-cube-quadratic-code-seed44/SOURCES.json
new file mode 100644
index 0000000000..90801e8ab8
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/SOURCES.json
@@ -0,0 +1,68 @@
+{
+  "base_commit": "1b59b624d21924f8e3537cdc1ac52d3be6eda2fe",
+  "license": "Apache-2.0",
+  "implementation_dependencies": "Python 3.10+ standard library only; no predecessor code imported",
+  "predecessor_pr_heads": {
+    "1046": "7cadcb0a2fc71fdedc0066704dec3fec58c2ca2c",
+    "958": "28d68c88f0184835d8c8a7266ab8d0dad3f18a8a"
+  },
+  "retained_files": [
+    {
+      "local_path": "sources/PR1046-PROOF.md",
+      "source_path": "research/five-cube-coded-copy-seed42/PROOF.md",
+      "source_commit": "1b59b624d21924f8e3537cdc1ac52d3be6eda2fe",
+      "pr": 1046,
+      "sha256": "f6ed4a359a5e4cc4b27d9ffb096ef34beab45dc2eddf976f2984dc544853408c",
+      "git_blob_sha1": "2f804f0d881477cb2274796d4e3027dbe7d827e4"
+    },
+    {
+      "local_path": "sources/PR1046-NOTICE.md",
+      "source_path": "research/five-cube-coded-copy-seed42/NOTICE.md",
+      "source_commit": "1b59b624d21924f8e3537cdc1ac52d3be6eda2fe",
+      "pr": 1046,
+      "sha256": "ab713342a1bb817f1f68c3a02bd248f66a6e96630cb13a8405225da104506ded",
+      "git_blob_sha1": "d1f31bc09eed8cbfe966b1adb87d83ff4f6e71db"
+    },
+    {
+      "local_path": "sources/PR958-PROOF.md",
+      "source_path": "research/five-cube-coded-copy-seed42/sources/PR958-PROOF.md",
+      "source_commit": "1b59b624d21924f8e3537cdc1ac52d3be6eda2fe",
+      "pr": 958,
+      "sha256": "df4437193b9e8f478673c5050735d5c67964c103e5e7c29ee124ae0f38b9b442",
+      "git_blob_sha1": "57d6abf19382012797a47d9769163e77397fed83"
+    },
+    {
+      "local_path": "sources/PR958-NOTICE.md",
+      "source_path": "research/five-cube-coded-copy-seed42/sources/PR958-NOTICE.md",
+      "source_commit": "1b59b624d21924f8e3537cdc1ac52d3be6eda2fe",
+      "pr": 958,
+      "sha256": "b91b5efc9fb5a237555c4830d5c8d7bff382b8169c1441116fc18d5843093396",
+      "git_blob_sha1": "7895480850cecf00072a11e2aa07bd1bd879bc8b"
+    },
+    {
+      "local_path": "sources/SELECTED-RESULT.json",
+      "source_path": "certificates/selected-result.json",
+      "source_commit": "1b59b624d21924f8e3537cdc1ac52d3be6eda2fe",
+      "pr": 1094,
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
+      "git_blob_sha1": "ae6a99f28aa468649eb66a361e764d769363bf11"
+    }
+  ],
+  "selected_context": {
+    "selected_kappa": "836330098538459/500000000000000000",
+    "selected_reviewed_head": "0fc329d0bec93ca7cd288eaea1295172e8789810",
+    "reproduce": "make frontier1094-verify",
+    "scope": "Context only; no selected supplier replay or modification by this package.",
+    "pins": {
+      "research/community-frontier1094-audit/SELECTED-SOURCE.json": "79b3de5db3dbcb9829789ba58e9e45758926031e2a7d80667a08429d369be53b",
+      "research/community-frontier1094-audit/ASSEMBLY.json": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045",
+      "docs/research/community-batch1099-review.md": "7e76d2c38e5fce30587835b7251df1430488ae4dd372611ef69a3f03f476b6cb"
+    }
+  },
+  "adjacent_pending_work": {
+    "pr": 1177,
+    "head": "e878ef36109bc6d7e21e62d03c22d8ec36f6f936",
+    "proof_sha256": "2c1d811929d1d70f0753433fd97b435729050724236d5e15e7940fd02952c6f1",
+    "scope": "At most three arbitrary support terms; inspected and credited, no code or theorem imported."
+  }
+}
diff --git a/research/five-cube-quadratic-code-seed44/VALIDATION.json b/research/five-cube-quadratic-code-seed44/VALIDATION.json
new file mode 100644
index 0000000000..3751e31ed9
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/VALIDATION.json
@@ -0,0 +1,63 @@
+{
+  "status": "complete local finite verification; hosted CI not claimed",
+  "base_commit": "1b59b624d21924f8e3537cdc1ac52d3be6eda2fe",
+  "python": "3.14.6 (main, Jun 10 2026, 10:03:53) [Clang 21.0.0 (clang-2100.0.123.102)]",
+  "platform": "macOS-27.0.1-arm64-arm-64bit-Mach-O",
+  "implementation_dependencies": "Python standard library only",
+  "commands": [
+    {
+      "command": "python3 -I -O -B research/five-cube-quadratic-code-seed44/verify.py --write research/five-cube-quadratic-code-seed44/CERTIFICATE.json",
+      "exit_code": 0,
+      "result": "complete deterministic certificate generated"
+    },
+    {
+      "command": "python3 -I -O -B research/five-cube-quadratic-code-seed44/independent.py",
+      "exit_code": 0,
+      "result": "65536 polynomials, 16383 transports, 3136 exact rational ranks, edge certificates and all-pairs attaining identities pass"
+    },
+    {
+      "command": "python3 -I -O -B research/five-cube-quadratic-code-seed44/test_package.py",
+      "exit_code": 0,
+      "result": "Six test groups pass from an isolated export of the exact staged files, including added zero-cost edge and equal-price wrong-decoder controls.",
+      "elapsed_seconds": 23.214
+    },
+    {
+      "command": "make selected-record-check",
+      "exit_code": 0,
+      "result": "selected record, exact kappa, original source inventory and proof supplement pass"
+    },
+    {
+      "command": "python3 -B -m unittest discover -s tests -p 'test_selected*.py'",
+      "exit_code": 0,
+      "result": "25 selected-record tests pass"
+    }
+  ],
+  "staged_export": {
+    "status": "PASS",
+    "export_method": "git checkout-index of all 16 package files into a fresh directory; no repository or predecessor implementation present",
+    "command": "python3 -I -O -B research/five-cube-quadratic-code-seed44/test_package.py",
+    "exit_code": 0,
+    "test_groups": 6,
+    "semantic_rejection_faults": 12,
+    "source_corruption_cli_rejections": 2,
+    "occupied_output_preservation": "PASS",
+    "read_only_file_relocation": "PASS",
+    "certificate_regeneration": "byte-identical",
+    "elapsed_seconds": 23.214,
+    "log": "evidence/package-test.txt"
+  },
+  "scope": "Complete finite coverage of the specified quadratic code domain; written graph reduction plus exact certificates. No formal multiplication theorem, human mathematical review, native supplier replay, physical/hosting/precision admission, new kappa, or hosted CI claimed.",
+  "source_pin_check": "All five retained files match exact base Git bytes and refreshed origin/main; all existing files untouched.",
+  "tested_mathematical_files_sha256": {
+    "verify.py": "f294554c6a741898b1bae36fea3a8abb7cccb41429caba14142c73ccd7f4033f",
+    "independent.py": "1fe8add4fcb792d5d1c5786e517115c4c3731972f26d374003075cd2d2d25e3a",
+    "test_package.py": "323a4dc40254ae7a694397fd1191985ea01cb5f44fc25adf633cf27c50c38514",
+    "CERTIFICATE.json": "7c29bd9a696544fca7012f457dd061dc6fb92b1a92e69668a43927039f704f0e",
+    "SOURCES.json": "18276d05299a9de66963fe86c6c4229e3381bd253b15540af26a98d9227f1e55"
+  },
+  "completed_checks_utc": "2026-10-13T20:22:53.528679+00:00",
+  "whitespace_check": {
+    "command": "git diff --cached --check",
+    "exit_code": 0
+  }
+}
diff --git a/research/five-cube-quadratic-code-seed44/evidence/package-test.txt b/research/five-cube-quadratic-code-seed44/evidence/package-test.txt
new file mode 100644
index 0000000000..97cd99cfed
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/evidence/package-test.txt
@@ -0,0 +1,11 @@
+test_actual_rank_corruption (__main__.PackageTests.test_actual_rank_corruption) ... ok
+test_complete_regeneration_and_independent_replay (__main__.PackageTests.test_complete_regeneration_and_independent_replay) ... ok
+test_coverage_and_symmetry_faults (__main__.PackageTests.test_coverage_and_symmetry_faults) ... ok
+test_exact_rank_controls (__main__.PackageTests.test_exact_rank_controls) ... ok
+test_potential_and_word_faults (__main__.PackageTests.test_potential_and_word_faults) ... ok
+test_relocation_sources_and_output_preservation (__main__.PackageTests.test_relocation_sources_and_output_preservation) ... ok
+
+----------------------------------------------------------------------
+Ran 6 tests in 23.214s
+
+OK
diff --git a/research/five-cube-quadratic-code-seed44/independent.py b/research/five-cube-quadratic-code-seed44/independent.py
new file mode 100644
index 0000000000..884a6ce9dc
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/independent.py
@@ -0,0 +1,295 @@
+#!/usr/bin/env python3
+"""Check the finite proof without importing the generator or running Dijkstra.
+
+Uses coordinate sets, Boolean polynomials, Fraction elimination and edge
+potentials. Hanson Sun (@bbMisk), substantial OpenAI Codex assistance.
+Apache-2.0. A separate implementation is not independent human review.
+"""
+
+import argparse
+from collections import Counter
+from fractions import Fraction
+import hashlib
+from itertools import combinations
+import json
+from pathlib import Path
+import sys
+
+DIRECTORY = Path(__file__).resolve().parent
+PAIRS = tuple(combinations(range(5), 2))
+MONOMIAL_ORDER = tuple((i,) for i in range(4)) + PAIRS
+
+
+def check(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def exact_rank(matrix):
+    if not matrix:
+        return 0
+    rows = [[Fraction(entry) for entry in row] for row in matrix]
+    pivot_row = 0
+    for column in range(len(rows[0])):
+        selected = next((k for k in range(pivot_row, len(rows)) if rows[k][column]), None)
+        if selected is None:
+            continue
+        rows[selected], rows[pivot_row] = rows[pivot_row], rows[selected]
+        scale = rows[pivot_row][column]
+        for j in range(column, len(rows[0])):
+            rows[pivot_row][j] /= scale
+        for k in range(pivot_row + 1, len(rows)):
+            factor = rows[k][column]
+            if factor:
+                for j in range(column, len(rows[0])):
+                    rows[k][j] -= factor * rows[pivot_row][j]
+        pivot_row += 1
+        if pivot_row == len(rows):
+            break
+    return pivot_row
+
+
+def polynomial(code):
+    return {term for bit, term in enumerate(MONOMIAL_ORDER) if code & (2**bit)}
+
+
+def transform(code, generator):
+    terms = polynomial(code)
+    if generator < 5:
+        # Substitute x_i+1 for x_i directly in the algebraic normal form.
+        terms ^= {tuple(j for j in term if j != generator)
+                  for term in tuple(terms) if generator in term}
+    else:
+        i = generator - 5
+        rename = {i: i + 1, i + 1: i}
+        terms = {tuple(sorted(rename.get(j, j) for j in term)) for term in terms}
+    terms.discard(())  # Quotient by the constant function.
+    if (4,) in terms:
+        terms ^= {(j,) for j in range(5)}  # Quotient by parity.
+    check(all(term in MONOMIAL_ORDER for term in terms), "transformation left quadratic domain")
+    return sum(2**MONOMIAL_ORDER.index(term) for term in terms)
+
+
+def geometry():
+    orientations = [tuple((address // (2**i)) % 2 for i in range(5))
+                    for address in range(32)]
+    ports = [{2 * i + orientation[i] for i in range(5)} for orientation in orientations]
+    gram = [[2 * (len(s & t) - 1) * (len(s & t) - 3) for t in ports] for s in ports]
+    parity = [sum(x) % 2 for x in orientations]
+    decoder = [[int(parity[i] == parity[j]) for j in range(32)] for i in range(32)]
+    return orientations, gram, parity, decoder
+
+
+def masks_for_domain(orientations, parity):
+    full_terms = ((),) + tuple((i,) for i in range(5)) + PAIRS
+    evaluations = [sum(int(all(x[i] for i in term)) * (2**address)
+                       for address, x in enumerate(orientations)) for term in full_terms]
+    full = (2**32) - 1
+    odd = sum(bit * (2**i) for i, bit in enumerate(parity))
+    all_masks = [0] * (2**16)
+    quotient_counts = Counter()
+    for coefficient in range(2**16):
+        if coefficient:
+            bit = coefficient.bit_length() - 1
+            all_masks[coefficient] = all_masks[coefficient - 2**bit] ^ evaluations[bit]
+        h = all_masks[coefficient]
+        quotient_counts[min(h, h ^ full, h ^ odd, h ^ full ^ odd)] += 1
+    check(len(set(all_masks)) == 65536, "full quadratic evaluation has a kernel")
+    check(len(quotient_counts) == 16384 and set(quotient_counts.values()) == {4},
+          "incomplete quotient domain")
+    quotient_masks = []
+    keys = set()
+    for code in range(16384):
+        h = 0
+        for term in polynomial(code):
+            h ^= evaluations[full_terms.index(term)]
+        key = min(h, h ^ full, h ^ odd, h ^ full ^ odd)
+        check(key in quotient_counts and key not in keys, "duplicate or absent quotient code")
+        keys.add(key)
+        quotient_masks.append(h)
+    return quotient_masks
+
+
+def check_isometries(orientations, gram, decoder):
+    for generator in range(9):
+        image = []
+        for x in orientations:
+            y = list(x)
+            if generator < 5:
+                y[generator] = 1 - y[generator]
+            else:
+                i = generator - 5
+                y[i], y[i + 1] = y[i + 1], y[i]
+            image.append(orientations.index(tuple(y)))
+        check(len(set(image)) == 32, "non-bijective proposed symmetry")
+        for x in range(32):
+            for y in range(32):
+                check(gram[x][y] == gram[image[x]][image[y]] and
+                      decoder[x][y] == decoder[image[x]][image[y]],
+                      "proposed symmetry does not preserve the model")
+
+
+def check_transports(certificate):
+    owners = {}
+    roots = set()
+    for child, parent, generator, root in certificate["transports"]:
+        check(type(child) is int and 1 <= child < 16384 and child not in owners,
+              "missing/duplicate/invalid transport node")
+        if parent == 0:
+            check(generator == -1 and root == child, "invalid transport root")
+            roots.add(root)
+        else:
+            check(parent in owners and owners[parent] == root and 0 <= generator < 9,
+                  "transport parent is not established")
+            check(transform(parent, generator) == child, "false polynomial symmetry transport")
+        owners[child] = root
+    check(set(owners) == set(range(1, 16384)), "quadratic coverage omitted")
+    check(len(roots) == 64, "wrong number of roots")
+    # Closure certifies that the connected transport classes are exactly orbits.
+    for code, root in owners.items():
+        for generator in range(9):
+            check(owners[transform(code, generator)] == root, "orbit not closed")
+    return roots, Counter(owners.values())
+
+
+def outer_bits(a, b):
+    rows = [(a // (2**i)) % 2 for i in range(3)]
+    columns = [(b // (2**i)) % 2 for i in range(3)]
+    value = 0
+    for i in range(3):
+        for j in range(3):
+            if rows[i] and columns[j]:
+                value += 2**(3 * i + j)
+    return value
+
+
+def check_potential(record, ranks, functions, decoder):
+    cap = record["rank_cap"]
+    check(type(cap) is int and cap in (15, 32), "unexpected cap")
+    generators = [(outer_bits(a, b), ranks[a - 1][b - 1])
+                  for a in range(1, 8) for b in range(1, 8)
+                  if ranks[a - 1][b - 1] <= cap]
+    reachable = {0}
+    for bits, _ in generators:
+        reachable |= {state ^ bits for state in tuple(reachable)}
+    potential = record["potential"]
+    check(len(potential) == 512 and potential[0] == 0, "bad potential origin/length")
+    for state in range(512):
+        if state not in reachable:
+            check(potential[state] is None, "false reachability")
+            continue
+        check(type(potential[state]) is int and potential[state] >= 0, "bad potential value")
+        for bits, cost in generators:
+            other = potential[state ^ bits]
+            check(type(other) is int and abs(other - potential[state]) <= cost,
+                  "edge potential inequality fails")
+    target = 2**0 + 2**4
+    check(record["minimum"] == potential[target], "target potential mismatch")
+    if target not in reachable:
+        check(record["word"] == [], "false unreachable target")
+        return
+    # Check the attaining decoder literally on all 1024 port pairs.
+    word = record["word"]
+    total = 0
+    for a, b in word:
+        check(type(a) is int and type(b) is int and 1 <= a < 8 and 1 <= b < 8,
+              "invalid factor in witness")
+        check(ranks[a - 1][b - 1] <= cap, "witness exceeds cap")
+        total += ranks[a - 1][b - 1]
+    check(total == record["minimum"], "witness price mismatch")
+    for x in range(32):
+        for y in range(32):
+            value = sum(functions[a][x] * functions[b][y] for a, b in word) % 2
+            check(value == decoder[x][y], "witness fails all-pairs decoder identity")
+
+
+def verify_data(certificate):
+    check(certificate["schema"] == "five-cube-quadratic-code-v1", "wrong schema")
+    check(certificate["basis"] == ["even", "odd", "h"], "wrong code basis")
+    check(certificate["quotient_monomials"] == [list(m) for m in MONOMIAL_ORDER],
+          "wrong polynomial order")
+    check(certificate["target_coefficient_matrix_bits"] == 17, "wrong matrix target")
+    check(certificate["transport_columns"] == ["child", "parent", "generator", "representative"],
+          "wrong transport schema")
+    orientations, gram, parity, decoder = geometry()
+    masks = masks_for_domain(orientations, parity)
+    check_isometries(orientations, gram, decoder)
+    roots, sizes = check_transports(certificate)
+    check(len(certificate["orbits"]) == 64, "wrong number of orbit records")
+    check({r["representative"] for r in certificate["orbits"]} == roots,
+          "orbit roots do not match transported coverage")
+    rank_checks, capped_histogram = 0, Counter()
+    for record in certificate["orbits"]:
+        code = record["representative"]
+        mask = masks[code]
+        check(record["mask"] == mask and record["orbit_size"] == sizes[code],
+              "wrong mask or orbit size")
+        nonlinear = any(len(m) == 2 for m in polynomial(code))
+        check(record["nonlinear"] is nonlinear, "wrong nonlinear classification")
+        basis = [[1 - p for p in parity], parity, [(mask // (2**x)) % 2 for x in range(32)]]
+        functions = [[sum(basis[i][x] for i in range(3) if a & (2**i)) % 2
+                      for x in range(32)] for a in range(8)]
+        check(len({tuple(f) for f in functions}) == 8, "evaluation basis is not independent")
+        ranks = []
+        for a in range(1, 8):
+            row = []
+            for b in range(1, 8):
+                matrix = [[gram[x][y] for y in range(32) if functions[b][y]]
+                          for x in range(32) if functions[a][x]]
+                row.append(exact_rank(matrix))
+                rank_checks += 1
+            ranks.append(row)
+        check(record["ranks"] == ranks, "rational rank table mismatch")
+        check(record["unrestricted"]["rank_cap"] == 32 and record["rank15"]["rank_cap"] == 15,
+              "wrong record cap")
+        check_potential(record["unrestricted"], ranks, functions, decoder)
+        check_potential(record["rank15"], ranks, functions, decoder)
+        check(record["unrestricted"]["minimum"] == 32, "copy floor not established")
+        minimum = record["rank15"]["minimum"]
+        check(not nonlinear or minimum is None or minimum >= 34, "nonlinear gap not established")
+        capped_histogram["unreachable" if minimum is None else str(minimum)] += sizes[code]
+    expected_summary = {
+        "quadratic_quotient_codes": 16383, "nonlinear_codes": 16368,
+        "orbit_count": 64, "nonlinear_orbits": 62,
+        "ordered_term_ranks": rank_checks, "coefficient_states_per_code": 512,
+        "unrestricted_transition_checks": 64 * 512 * 49,
+        "minimum_copy_rank_sum": 32,
+        "nonlinear_minimum_with_rank_cap15": 34,
+        "rank15_minimum_histogram_weighted_by_codes": dict(sorted(capped_histogram.items())),
+        "finite_rank34_attaining_codes": capped_histogram["34"], "no_new_kappa": True,
+    }
+    check(certificate["summary"] == expected_summary, "summary/coverage mismatch")
+    check(capped_histogram["34"] == 60, "sharp nonlinear rank-15 witness missing")
+    return expected_summary
+
+
+def check_source_files(certificate):
+    manifest_bytes = (DIRECTORY / "SOURCES.json").read_bytes()
+    check(hashlib.sha256(manifest_bytes).hexdigest() == certificate["source_manifest_sha256"],
+          "source manifest binding differs")
+    for record in json.loads(manifest_bytes)["retained_files"]:
+        path = (DIRECTORY / record["local_path"]).resolve()
+        check(DIRECTORY in path.parents, "source outside package")
+        data = path.read_bytes()
+        check(hashlib.sha256(data).hexdigest() == record["sha256"], "source content differs")
+        blob = ("blob " + str(len(data)) + "\0").encode() + data
+        check(hashlib.sha1(blob).hexdigest() == record["git_blob_sha1"], "source blob differs")
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--certificate", type=Path, default=DIRECTORY / "CERTIFICATE.json")
+    args = parser.parse_args()
+    certificate = json.loads(args.certificate.read_text())
+    check_source_files(certificate)
+    verify_data(certificate)
+    print("PASS_INDEPENDENT_QUADRATIC_CODES polynomials=65536 transports=16383 "
+          "rational_ranks=3136 all_pairs_witnesses=PASS no_new_kappa")
+
+
+if __name__ == "__main__":
+    try:
+        main()
+    except (OSError, ValueError, KeyError, TypeError, IndexError) as exc:
+        print("FAIL_INDEPENDENT_QUADRATIC_CODES: " + str(exc), file=sys.stderr)
+        raise SystemExit(1)
diff --git a/research/five-cube-quadratic-code-seed44/sources/PR1046-NOTICE.md b/research/five-cube-quadratic-code-seed44/sources/PR1046-NOTICE.md
new file mode 100644
index 0000000000..d1f31bc09e
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/sources/PR1046-NOTICE.md
@@ -0,0 +1,51 @@
+# Attribution and scope
+
+New finite theorem, witnesses, checkers and packaging prepared for the
+authenticated contributor **@martinxxx9**, with substantial OpenAI Codex
+assistance in mathematics, implementation, verification and writing.
+One delegated AI worker reviewed only this cube's argument and independently
+implemented its exact integer rank census and reduced-factor enumeration.
+This is not independent human peer review. Apache-2.0, 2026.
+
+The following original inputs are retained byte for byte, with SHA256 and
+Git-blob pins in `SOURCES.json`:
+
+- **Ross Richardson (@RossRichardson), PR958**, head
+  `28d68c88f0184835d8c8a7266ab8d0dad3f18a8a`: the natural five-port quadratic
+  Gram, source identity, separate-star accounting and explicit exclusion
+  of coded totals. Its proof and full attribution notice are preserved.
+  It was prepared with substantial OpenAI Codex assistance. This result
+  covers only its smallest cube, using a changed total basis.
+- **Joel Pulikkan (@GamingPuzzled), PR231**: higher-port XOR star identity
+  and the open five-port geometry/pricing question, with **icekylinx** and
+  **eumemic**'s paired-cube lineage. Its preserved README includes its
+  Anthropic Claude assistance disclosure.
+- **DreamingOfClouds, PR446**, original head
+  `309482849b6ddf34db998f53612d34c19f6e3233`: per-total pairing-rank copying,
+  frame and stage interfaces. The proof retained by PR958 is preserved
+  unchanged. Its credits include **Jacob Sussman**'s frame lemma and
+  **Zhihao Chen (@jacklightChen)**'s common-label/coordinate lineage.
+- **DreamingOfClouds, PR595 format axioms**, as retained by PR958: the
+  explicit deficit/pressure/positive-overhead premises. Only the stated
+  accounting implication is used; no new all-size audit is asserted.
+- **Jonathan Capalbo, Chafik Boukhalfa, Jeremy Kothe, Hao Yan,
+  DreamingOfClouds and eumemic**, and the cumulative contributors recorded
+  by the selected T37 review: the frozen benchmark context. Its selected
+  record is preserved as context, not re-certified by this package.
+
+All snapshots in `sources/` are read from the exact base commit; the
+PR446/format texts are the already vendored immutable originals in PR958.
+Original licenses, author notices and assistance disclosures are retained.
+The six source files are documentation/context, not imported implementations.
+
+Related pending work inspected: **Sam Sung (@the-missionary-company), PR669**,
+studies proper-star hierarchies and a different integral odd-diagonal copying
+obstruction. Our Gram has diagonal 16 and is outside that hypothesis.
+**PR765** concerns tensor E8 basis mixing. **PR912/888** concern nonlinear
+Hurwitz/Gaussian messages. **PR998** concerns a complex P112 single lift.
+Their code is not consumed. No literature-wide priority claim is made.
+
+The peak-eight result concerns **pairing rank**, not source span, physical
+frame width, native precision, circuit count, runtime or an improved kappa.
+Source-access, helper, frame/restoration and full recursive interfaces must
+still be implemented before either algebraic witness becomes a supplier.
diff --git a/research/five-cube-quadratic-code-seed44/sources/PR1046-PROOF.md b/research/five-cube-quadratic-code-seed44/sources/PR1046-PROOF.md
new file mode 100644
index 0000000000..2f804f0d88
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/sources/PR1046-PROOF.md
@@ -0,0 +1,215 @@
+# Affine recoding on the smallest quadratic five-port cube
+
+This is a finite algebraic theorem with a separately stated conditional
+accounting consequence. It is not an all-size multiplication theorem or a
+physical implementation of the displayed totals.
+
+## Object and exact scope
+
+Take five coordinate pairs and all 32 ports that select one coordinate from
+each pair. Identify a port with its orientation `x in F2^5`. If ports `x,y`
+have intersection `j=5-|x XOR y|`, use Ross Richardson's PR958 quadratic
+Gram matrix
+
+```
+G[x,y] = 2(j-1)(j-3).
+```
+
+The common labels are rational and may realize this Gram matrix in a larger
+space. The coordinate-star decoder matrix is binary:
+
+```
+B[x,y] = j mod 2 = 1 + p(x) + p(y),
+p(x) = x_0 + x_1 + x_2 + x_3 + x_4  (over F2).
+```
+
+The remaining source corrections have `j in {1,3}`. Hence their binary
+matrix plus `B` is exactly the identity. This is the inherited five-port
+decoder identity, not a new identity attributed to this contribution.
+
+The coordinate-star observation code on this particular cube is
+
+```
+C = { f(x)=a_0*x_0+...+a_4*x_4+c : a_i,c in F2 }.
+```
+
+It has dimension six: the star for coordinate `2i+1` evaluates to `x_i`
+and the other coordinate's star evaluates to `1+x_i`. In contrast, the
+decoder matrix `B` has binary rank two, with row and column space
+
+```
+U = span_F2 {1,p},   U\{0} = {1,p,1+p}.
+```
+
+This distinction is essential. Requiring six nonzero delivered terms from
+the observation rank would incorrectly exclude a two-total decoder.
+
+Allow **any finite, possibly redundant** binary rank-one factorization
+
+```
+B[x,y] = sum_(i=1..n) f_i(x)*g_i(y)  (over F2),
+0 != f_i,g_i in C.
+```
+
+Zero factors can be discarded. Invertible, nonsymmetric basis recodings
+and overcomplete decompositions are included. The equality is required on
+every one of the 1,024 source/target entries. It is not only a masked
+identity on the correction graph. Let `R_i=support(f_i)` and
+`S_i=support(g_i)`. Define the **algebraic pairing-rank objective**
+
+```
+c_i = rank_Q G[R_i,S_i],  C_total = sum_i c_i.
+```
+
+Under the inherited per-total copy-tree rule this objective is a lower
+bound on additive paid copying. A decomposition attaining the objective
+does not itself realize that paid cost: source access, gathers, endpoints,
+frames and restoration still need a construction.
+
+**Theorem.** Every such factorization has `C_total >= 32`. This is sharp.
+The minimum possible largest term rank is eight, also sharp. Both optima
+are attained simultaneously by the four-total factorization below.
+
+## Complete finite rank certificate
+
+Encode an affine form by a six-bit integer: bits 0 through 4 are its linear
+coefficients and bit 5 is its constant. The nonzero forms are exactly
+`1,...,63`. Both checkers explicitly enumerate all **3,969 ordered pairs**.
+`verify.py` uses elimination over the checked prime 65,521; a nonzero
+modular minor gives a rational rank lower bound. `independent.py` computes
+the same ranks by integer elimination with gcd normalization, without
+importing the modular checker. It confirms equality over Q everywhere.
+
+The complete rank table is retained in `CERTIFICATE.json`.
+
+| Rank | Ordered pairs | Maximum `|R intersect S|` |
+|---:|---:|---:|
+| 6 | 30 | 0 |
+| 8 | 480 | 8 |
+| 10 | 1,230 | 8 |
+| 11 | 1,530 | 16 |
+| 13 | 690 | 16 |
+| 16 | 9 | 32 |
+
+Every nonzero term has rank at least six. Every rank-six pair has disjoint
+supports. The nine pairs with both forms in `U\{0}={31,32,63}` have rank
+16. A pair with one form in this set and the other an arbitrary nonzero
+affine form has rank at least 11. The whole quadratic Gram has rational
+rank 16.
+
+These are exhaustive finite facts about the specified 32-port object,
+not an extrapolation to larger five-port families. The modular lower
+bounds alone suffice for the lower-bound argument; exact rational ranks
+are additionally needed for the sharp witnesses and reported census.
+
+## Lower bound, including arbitrary redundant factorizations
+
+Suppose `C_total < 32`. The six-per-term lower bound forces `n <= 5`.
+Let `A` and `D` have the evaluated reader and source forms as their
+columns, so `B=A*D^T` over F2. Write `r_A=rank A`, `r_D=rank D`.
+Both ranks are at least two and Sylvester's rank inequality gives
+
+```
+r_A+r_D <= n+rank(B) = n+2.
+```
+
+For one term, binary rank alone is impossible. For two terms, both factor
+spaces equal `U`; each term costs 16 and their sum is 32. For three terms,
+one factor space has dimension two and equals `U`. Every term has one
+nonzero form in `U`, so each costs at least 11 and the sum is at least 33.
+This reasoning does not require distinct factors or a minimal observation
+basis.
+
+For four and five terms use the diagonal. Because `B[x,x]=1` for every
+port, each port must occur in an odd, hence positive, number of the sets
+`R_i intersect S_i`. Therefore
+
+```
+sum_i |R_i intersect S_i| >= 32.
+```
+
+The census table implies that at total cost at most 31, four terms can
+cover at most 24 diagonal entries, counting repeated coverage, and five
+terms can cover none. One way to check the first assertion is the exact
+budget recurrence in `verify.py`: start with `{cost 0: coverage 0}` and
+add one independently chosen table type per term, retaining the largest
+coverage at each integer cost up to 31. This relaxation permits every
+type combination and ignores correlations between actual forms. Its
+maxima are 24 and zero. `independent.py` separately enumerates all six
+rank-type sequences of length four and five and obtains the same maxima.
+Both contradict the necessary diagonal coverage of 32.
+
+Six or more terms already cost at least 36. These cases exhaust every
+finite affine-code factorization and prove `C_total >= 32`.
+
+For the peak claim, a factorization whose term ranks were all below eight
+could use only rank-six terms. Their supports are disjoint, so their
+combined diagonal is zero. Thus at least one term has rank at least eight.
+
+## Sharp algebraic witnesses
+
+The two parity totals give
+
+```
+B[x,y] = p(x)*p(y) + (1+p(x))*(1+p(y)).
+```
+
+The two restricted Grams both have rank 16. This attains total 32 but
+has peak 16.
+
+Put `h(x)=x_0`, `q(x)=x_1+x_2+x_3+x_4`, so `p=h+q`. Then
+
+```
+B[x,y] = h(x)*q(y)
+       + q(x)*(1+h(y))
+       + (1+h(x))*(1+q(y))
+       + (1+q(x))*h(y).
+```
+
+The reader/source form pairs are `(1,30),(30,33),(33,62),(62,1)`.
+Each restricted Gram has rational rank eight. The sum is 32 and the
+peak is eight. Both checkers verify every source/target coefficient,
+including off-diagonal entries, and compute all four ranks exactly.
+This changes the total basis and pairing geometry, while leaving the
+copy-rank sum unchanged from the two-total optimum.
+
+As optional corroboration, `independent.py --exhaustive-factorizations`
+enumerates the reduced subspace cases that could possibly violate the
+floor: for `n<=5`, one factor span has dimension at most three and contains
+`U`; there are exactly one dimension-two and 15 dimension-three choices.
+Repeated reader forms can be merged without increasing cost, since XOR
+support is contained in the union and restricted Gram rank is subadditive
+under concatenation. After merging, enumerate distinct readers and every
+binary source assignment solving the full matrix identity. This secondary
+experiment is not needed by the simpler diagonal proof, and does not
+claim to enumerate factorizations with six or more terms.
+
+## Conditional complete-budget obstruction
+
+Assume the inherited per-replica stage accounting with `v=32`, stage count
+`s>=3`, additive per-total copy charges at least the pairing ranks, and
+
+```
+D <= (s-1)*v - s*C_total.
+```
+
+Then even the favorable complete budget that grants source access,
+gathers, decoder arithmetic, helpers, frame changes, restoration, idle
+work, precision and all other nonnegative expenses zero satisfies
+
+```
+D <= (s-1)*32-s*32 = -32.
+```
+
+The inherited pressure equation for a positive saving has nonnegative
+child terms `z_r*r*((m/r)^gamma-1)` for `0<r<m`, and nonnegative fallback
+only makes admission harder. It cannot equal a negative deficit. Thus
+affine recoding does not rescue this **smallest cube** in the stated
+stage/copy model. No larger-p theorem is claimed.
+
+Changed Gram pairings, non-affine source or reader supports, nonlinear or
+interactive messages, masked decoder identities, information retained
+across stages and a different copying ledger remain outside scope. A
+physical implementation of either witness remains unprovided. The selected
+construction's all-size/compiler, common-label/mutable-source, whole-call
+hosting, precision/tape, bit H1-H5 and analytic obligations are unchanged.
diff --git a/research/five-cube-quadratic-code-seed44/sources/PR958-NOTICE.md b/research/five-cube-quadratic-code-seed44/sources/PR958-NOTICE.md
new file mode 100644
index 0000000000..7895480850
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/sources/PR958-NOTICE.md
@@ -0,0 +1,53 @@
+# Attribution, license and assistance
+
+New proof, witness construction, checkers and packaging: Ross Richardson
+(@RossRichardson), 2026. Licensed under the repository's Apache-2.0 license.
+Substantial OpenAI Codex assistance was used for task selection, mathematics,
+implementation, verification and writing. No independent human peer review
+or proof-assistant verification is claimed.
+
+Predecessors and retained sources:
+
+- **Joel Pulikkan (@GamingPuzzled), PR231**, head
+  `3c11e696a2fb2c454585d72710de27cd6bd3be68`: the general k-port parity
+  identity and the explicit open geometry/pricing question for k=5. Its
+  incorporated README is retained unchanged from the exact base, including
+  the original Anthropic Claude assistance disclosure.
+- **icekylinx and eumemic**: the paired-cube design, bit decoder and local
+  construction lineage credited by PR231. We introduce no claim of priority
+  for the XOR star identity.
+- **DreamingOfClouds, PR446**, head
+  `309482849b6ddf34db998f53612d34c19f6e3233`: the content-frame, stage-rank,
+  pairing-rank copy and clique arguments. `sources/pr446-PROOF.md` is
+  retained verbatim from that commit. It was prepared with Anthropic
+  Claude assistance, as disclosed by the predecessor. Its frame lemma
+  credit belongs to **Jacob Sussman**; the preceding common-label and
+  compressed-window lineage includes **Zhihao Chen (@jacklightChen)**.
+- **DreamingOfClouds' format-limits package**, incorporated at the base:
+  the explicit pressure, deficit, window, active-bank and batching axioms.
+  Its README is retained verbatim, with its original credits and AI notice.
+  We use the named accounting as explicit premises and do not claim a new
+  audit of all its other results.
+- **Chafik Boukhalfa, Jeremy Kothe, Hao Yan, DreamingOfClouds and eumemic**,
+  and the earlier contributors in the selected review: the frozen PR890
+  conditional benchmark. The selected certificate is copied only as
+  context, not modified or re-certified by this contribution.
+
+Related work inspected for overlap:
+
+- **Sam Sung (@the-missionary-company), PR669** and its credited predecessors
+  study information obtainable from proper stars after puncturing. Our
+  unpunctured quadratic geometry and conditional cost obstruction are
+  distinct; no code or certificates from that PR are used.
+- **William Lawyer, PR868**, with the PR738/770 predecessors, closes the
+  weighted-clique Gaussian certificate class. It motivated rejecting that
+  candidate task; it is not a dependency of this proof.
+- PR201 is a complex three-stage paired-cube ceiling; PR620 concerns E8
+  decoder floors; PR912 concerns nonlinear adaptive Hurwitz routing; PR920
+  concerns complex parity frames. None supplies the five-port witnesses.
+
+Only the four text/JSON sources named in `SOURCES.json` are retained.
+Their SHA256 and Git blob identities, original paths and exact commit pins
+are checked. No predecessor implementation code is imported. Original
+sources retain their attribution and license; no literature-wide priority
+claim is made for the parity-clique or polynomial-feature techniques.
diff --git a/research/five-cube-quadratic-code-seed44/sources/PR958-PROOF.md b/research/five-cube-quadratic-code-seed44/sources/PR958-PROOF.md
new file mode 100644
index 0000000000..57d6abf193
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/sources/PR958-PROOF.md
@@ -0,0 +1,240 @@
+# Five-port quadratic decoder: a scoped format obstruction
+
+This is a written mathematical argument with exact finite rank certificates.
+It is **not** a proof-assistant verification, an unrestricted decoder lower
+bound, or a new integer-multiplication construction.
+
+## 1. Object and statement
+
+Let `p >= 5`. The ports are the `v = 32 binom(p,5)` five-element subsets of
+`{0,...,2p-1}` containing at most one element of each pair `{2i,2i+1}`.
+Write `j = |S intersect T|`. The five-port, level-one star identity over F2 is
+
+```
+x_T = sum_(a in T) z_a + sum_(S: |S intersect T| in {1,3}) x_S,
+z_a = sum_(S: a in S) x_S.
+```
+
+The coefficient of `x_S` on the right is `j + 1_{j in {1,3}} (mod 2)`.
+It is zero for `j = 0,...,4` and one for `j = 5`, which means `S = T`.
+Joel Pulikkan's PR231 establishes the parity context and explicitly leaves
+the geometry and cost of `k=5` open. We investigate **one** geometry:
+
+```
+G(S,T) = 2(j-1)(j-3).
+```
+
+The theorem concerns every rational common-label realization of this Gram
+matrix, with the displayed correction terms delivered drop-free and each
+of the `2p` coordinate-star totals gathered drop-free and copied separately
+to all its coordinate-star readers in each stage, under the accounting
+premises in section 4. The copy of one total may use a tree or split its
+readers; mixing different totals into a new coded message is outside scope.
+
+**Theorem.** In this class, no positive supplier saving is possible at
+`p=5,6,7`. For every `p >= 5` and every integer number of stages `s >= 3`,
+any positive supplier saving `gamma` satisfies
+
+```
+gamma < 125/76433 = 0.0016354192560804888... .
+```
+
+This is below the **frozen, maintainer-reviewed** selected
+`kappa = 1661957774647029/10^18` at base
+`47481c89fd30a43f8f882eb4b391e89c29273fd0`. Since the inherited assembly
+requires kappa below its bit supplier saving, this decoder class cannot
+improve that selected result, even if the other supplier is free. The
+number above is an upper bound on a proposed supplier class, **not a
+new certified kappa**.
+
+## 2. Why a new geometry is needed
+
+Fix five coordinate pairs. Among their 32 orientations, take the 16 whose
+number of odd coordinates is even. Two distinct such ports differ in two
+or four coordinates, hence have intersection three or one. The decoder
+requires a drop-free correction between every two distinct ports in this
+set. Under the common-label frame rule, their nonisotropic rank-one labels
+must be pairwise orthogonal. They are therefore linearly independent.
+
+In contrast, affine functions of the five orientation bits span a space
+of dimension at most six. Any linear transformation of the original
+ten incidence coordinates, even with a constant coordinate and an
+arbitrary symmetric rational bilinear form, still has label span at most
+six on this cube. Thus the old incidence-linear geometry cannot realize
+this correction pattern. This is not merely a failure of a symmetric
+choice of bilinear form.
+
+A quadratic lift does realize the required zeros. Use feature coordinates
+`1`, `1_{a in S}`, and `1_{{a,b} subset S}` for pairs of coordinates from
+distinct coordinate pairs. Give the three groups diagonal form weights
+`6`, `-6`, and `4`, respectively. The resulting rational symmetric form is
+nondegenerate on this feature space, and
+
+```
+<u_S,u_T> = 6 - 6j + 4 binom(j,2) = 2(j-1)(j-3).
+```
+
+In particular the diagonal is 16 and both correction classes are
+orthogonal. The form is indefinite; positive definiteness is not claimed
+or needed for the retained rational frame model. Up to scaling, this is
+the unique nonzero polynomial in `j` of degree at most two that vanishes
+at one and three. The 16-orientation clique attains rank 16 inside one
+cube for this lift. The rest of this note shows why algebraic admissibility
+alone does not make the lift useful.
+
+## 3. Finite rank certificates and the infinite tail
+
+Let `N_T` be all ports intersecting target `T` in one or three coordinates.
+Let `P_a` be all ports containing coordinate `a`. The two needed ranks are:
+
+- `d_p <= rank G[N_T,N_T]`, a lower bound on the bilinear rank of the
+  drop-free corrector space;
+- `r_p <= rank G[P_a,P_a]`, a lower bound on the source/reader pairing rank
+  for the separate total `z_a`.
+
+Permuting coordinate pairs and swapping the two coordinates of any pair
+preserves both the port family and `G`. This group acts transitively on
+targets and on coordinates. A witness at the canonical target
+`{0,2,4,6,8}` and coordinate `0` therefore applies to **every** target or
+star at that `p`.
+
+| p | v | d_p | r_p | c_p = 2p r_p |
+|---|---:|---:|---:|---:|
+| 5 | 32 | 15 | 11 | 110 |
+| 6 | 192 | 50 | 35 | 420 |
+| 7 | 672 | 83 | 60 | 840 |
+| 8 | 1,792 | 111 | 84 | 1,344 |
+| 9 | 4,032 | 143 | 112 | 2,016 |
+| 10 | 8,064 | 179 | 144 | 2,880 |
+| 11 | 14,784 | 219 | 170 | 3,740 |
+| 12 | 25,344 | 263 | 210 | 5,040 |
+| 13 | 41,184 | 311 | not needed | granted zero in tail |
+
+`WITNESSES.json` contains the row and column ports of 17 square minors,
+their dimensions, and their nonzero determinants modulo **65521**. Each
+entry is an integer. A determinant nonzero modulo a prime is a nonzero
+integer determinant, so each minor proves the stated **rational lower
+bound**. This implication is exact; it is not a probabilistic modular-rank
+estimate and needs no determinant height bound. The row and column sets
+may differ; both lie in the same stated neighborhood or star.
+
+The proof needs only these lower bounds. It does not assume that the
+listed ranks are maximal. In particular it does **not** extrapolate the
+ordinary star values to 180 at `p=11` or 220 at `p=12`: discovery found
+degenerate pairings there, and the actual witnesses certify 170 and 210.
+No all-parameter rank formula or unverified numerical rank is a premise.
+
+For **every `p >= 13`**, fix any target and choose eight additional
+coordinate pairs. The 13 selected pairs contain an isometric copy of the
+entire `p=13` configuration, including its corrector minor. Consequently
+every target has corrector rank at least **311**. This embedding, together
+with the finite certificate, proves the all-p tail statement. It is not
+an inference from failure to find a better candidate in a finite search.
+
+## 4. Explicit accounting premises and optimistic budget
+
+We retain the following common-label, compressed-window accounting
+premises. They specialize DreamingOfClouds' PR446 Theorems B/C and the
+format-limit axioms `(P)`, `(Dfc)`, `(Win)`, `(Sch)`, `(Bat)` to the present
+decoder. The retained sources are pinned in `SOURCES.json`. We do not
+claim to prove their all-size machine realization here.
+
+1. There are `s >= 3` stages, one active source/target bank pair per stage,
+   rank-one data labels, drop-free delivery of the displayed correctors,
+   and a common stage climb `a`. Corrector rank forces `a >= d_p`.
+   The retained fixed-window `(Win)` accounting requires ambient dimension
+   `m >= 1+s*a`. This last relation is an explicit format premise; the
+   rank witnesses alone would only force `m >= 1+s*d_p`.
+2. At each stage, each of the `2p` totals is gathered **drop-free** from
+   `P_a` and delivered to `P_a` through its own rooted copy tree. The
+   pairing-rank drop bound is paid **additively
+   for these separate totals**, so per-stage copy charge `c >= c_p`.
+3. Per replica, the pressure deficit obeys
+   `D <= (s-1)v - s*c`. For paid recursive children of ranks `0<r<m`,
+   the saving satisfies the pressure equation
+   `sum_r z_r r((m/r)^gamma - 1) = D`. A nonnegative fallback term only
+   makes admission harder.
+4. The two active data climbs per port per stage have total mass `2sva`.
+   Each child doing this active work has rank at most `a`, as in the
+   fixed-window batching premise. Thus their pressure alone is at least
+   `2sva((m/a)^gamma - 1)`. A child cannot discharge this mass twice.
+
+These are restrictions on the format, not assertions about every possible
+integer-multiplication algorithm. They are compatible with granting all
+of the following unpriced work zero additional expense:
+
+| Work | Optimistic treatment |
+|---|---|
+| Source access, gathers and decoder arithmetic | zero |
+| Helpers, frame adapters, private scratch | zero |
+| Restoration and dirty compensation | zero |
+| Idle climbs, hosting, bridges and stage changes | zero |
+| Copy-related pressure beyond its deficit charge | zero |
+| All other recursive children and their fallback charges | zero |
+| Precision, tape, row, and outer-assembly overhead | zero |
+
+Required active recursive mass and the separate-star copy deficit are the
+only charges retained. They are **lower bounds**, not a complete proposed
+supplier invoice. Any additional nonnegative cost strengthens the
+obstruction. A mechanism invalidating a premise must be priced afresh.
+
+For `gamma>0`, `exp(x)-1 >= x` and `m/a > s >= 3` give
+
+```
+D >= 2sva((m/a)^gamma - 1)
+  >  2sv d_p gamma log(3).
+```
+
+Combining with the deficit bound and discarding the favorable `-1/s`
+term yields
+
+```
+gamma < (1 - c_p/v) / (2 d_p log(3)).
+```
+
+If `c_p >= v`, the deficit is negative at every positive `s`; this
+settles `p=5,6,7` before implementing any supplier.
+
+For a rational bound, the first six positive terms of
+
+```
+log(3) = 2 sum_(i>=0) (1/2)^(2i+1)/(2i+1)
+```
+
+already exceed `549/500`. The verifier checks that strict inequality
+using `Fraction`, with no logarithm library or floating point. Substitution
+gives the following strict supplier upper bounds:
+
+| p | rational upper bound |
+|---|---|
+| 8 | `125/121878` |
+| 9 | `125/78507` |
+| 10 | **`125/76433`** |
+| 11 | `31375/20198808` |
+| 12 | `5875/4235352` |
+
+For `p >= 13`, grant **even the copies** zero cost and use `d_p >= 311`:
+
+```
+gamma < 1/(2*311*log(3)) < 250/170739 < 125/76433.
+```
+
+All comparisons, including the strict gap to the frozen selected kappa,
+are rational and reproduced by the checker. No stage-count cutoff is
+used: `s>=3` was eliminated algebraically. This completes the theorem.
+
+## 5. Boundaries
+
+This closes the **specified natural quadratic Gram / separate coordinate
+star / compressed-window accounting** class. It does not close other
+quadratic label pairings, asymmetric labels, coded combinations of stars,
+punctured ports, higher-star decoders, retained cross-stage information,
+paid non-drop-free gathers or corrections, or a different bank schedule.
+It makes no claim that a stronger kappa
+exists in those classes.
+
+There is no circuit, hosted profile, all-size compiler or assembly
+certificate here. Realization, common-label/mutable-source validity,
+whole-call hosting, restoration, precision/tape, bit H1–H5 and analytic
+transfer remain inherited obligations for any actual construction.
+Finite exact rank verification does not discharge them.
diff --git a/research/five-cube-quadratic-code-seed44/sources/SELECTED-RESULT.json b/research/five-cube-quadratic-code-seed44/sources/SELECTED-RESULT.json
new file mode 100644
index 0000000000..ae6a99f28a
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/sources/SELECTED-RESULT.json
@@ -0,0 +1,222 @@
+{
+  "status": "Maintainer-reviewed conditional finite construction; see the integration receipt for publication and PR resolution status",
+  "kappa": "836330098538459/500000000000000000",
+  "decimal": "0.001672660197076918",
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+  ],
+  "source_manifest": "research/community-frontier1094-audit/SELECTED-SOURCE.json",
+  "certificate": "research/community-frontier1094-audit/ASSEMBLY.json",
+  "certificate_sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045",
+  "proof": "docs/research/community-batch1099-review.md",
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+    "docs/research/community-frontier799-review.md": "bd91c10c67f2a873fd456b95cbb43f715a43dac0e0a66938737d591c6b754052",
+    "scripts/verify_frontier799.py": "19fe1b32079095160dc00fe2f7df75e0fcbc82cff37f7159b755c8a63bd5ce99",
+    "scripts/verify_frontier799_bindings.py": "43897b3c99212d6134870898a1840f32898aa1b3679b299e654901568ad7e19b",
+    "research/gauge44-frames-x/PROOF-NOTE-FRAMES-X4.md": "39c50420ebb6e8813706a49b2cf759f277c4afac9f6780fc0a7ac6a5fb86e036",
+    "docs/research/community-frontier890-review.md": "cd03b7c2c6c09db2cfa89ae40133ae44e1deb6e32ec5ddd23aacecc9c50f2fea",
+    "scripts/verify_frontier890.py": "88ea38dbef62dd7c0dd46e212618520cc89213edddb2032800e5509993958175",
+    "scripts/verify_replay_coverage.py": "babe5941d7e6dac4248613ef01f950ee7bb6199e7355021aea4cf854fd1b91c5",
+    "scripts/verify_frontier741_bridge_coverage.py": "9a14f26762b6126eb54ed43db90371773374fe98733ada80ee871a74afa3b836",
+    "certificates/replay-coverage/pr741-sw4-v1.json": "7acd5033bc78e3f473718269edaa267c2785d84fbe95ccf6815a33dcf58c2756",
+    "research/complex-t21-owners/PROOF.md": "78721b32fceb34b21b8e77c95381c38c60276e72e6d9c2e9c6262cb347457def",
+    "research/complex-t21-owner-bound/PROOF.md": "2aa0aa6ec1d87402138564c38749b5ca0f058c16817efdf58a2685d2d90cb417",
+    "research/gauge44-barch/PROOF-NOTE-BARCH.md": "c24f211b09358b25d6cf47dd18e4bbd8bcf4ecf279e6f1c6c90613cfa230818d",
+    "research/complex-double-replica-owners/PROOF.md": "db99599e3966bec253c577c23915e2a2899c4c4e8568044efc11c7918b137bbc",
+    "docs/research/community-batch952-review.md": "04db59dfc2c31453229b6061bbf59547e71ce3abdc0ce0f6779144bb0dde127b",
+    "scripts/verify_frontier942.py": "27c6713532d9f53f55dfff4884d55a2d9ae74b68156e37a44596c0418fbed5b7",
+    "research/complex-t37-owners/PROOF.md": "09b39ad0d5f1f59f54dcff15c16640f7a7acf27e6597660acaf957b95c7e8510",
+    "research/complex-t37-owners/ALL-T-PROOF.md": "4c0c71ccd0afb850538af1b467e5208b7eb3edd0260fded84e02a8c01dfc868b",
+    ".github/workflows/t37-owners.yml": "c13acc2902c007237103072b7ec10caf5ca21c1b319d27934f194af2f7d87026",
+    "docs/research/community-batch1047-review.md": "a6271faf5db0afe3ff7d24a34a3be7a959cdc33ee37784572fba014ee5358787",
+    "scripts/verify_frontier1042.py": "700fdcaeb6e8f57f6e63c71c2dd1652fc5b183e8d706cd092779bdc0d271e854",
+    "docs/research/community-batch1099-review.md": "7e76d2c38e5fce30587835b7251df1430488ae4dd372611ef69a3f03f476b6cb",
+    "scripts/verify_frontier1094.py": "c7e20f59968ed5736fb859735de4de207de3418083e37012310ebcded946b949",
+    "research/community-batch1099-audit/check_frontier1094.py": "2608ef213260a43d09a40b3429742f7a5f280861b54acae66400a7c29a123f51"
+  },
+  "review": "docs/research/community-batch1099-review.md",
+  "validation": "docs/research/community-batch1099-integration.json",
+  "reproduce": "make frontier1094-verify",
+  "scope": "Finite composition under inherited all-size/compiler, common-label/mutable-source, whole-call hosting, precision/tape, analytic and bit H1-H5 assumptions. Not a full Lean multiplication theorem.",
+  "previous_kappa": "1672188119205421/1000000000000000000",
+  "previous_record": "certificates/checkpoints/frontier1042.json",
+  "audited_source_head": "0fc329d0bec93ca7cd288eaea1295172e8789810",
+  "reviewed_dependency_heads": {
+    "588": "6d0e53b66c29abd17981881a09eaff41ac862d85",
+    "543": "2065fe4f72a914d907f1046603f5a36f17018535",
+    "568": "71c99ac935a33bb7b59a1b4a7af8725a939b0b90",
+    "569": "1424da81263a5305b35e4cf74ef686c275f03f3f",
+    "571": "b7bdc5a8de18fcf418cfb9c7d83a2b1097a5e4a3",
+    "575": "eac6f7e378ab622dac50a2fd4b7ef6eab014de44",
+    "583": "89be768f0157f5a3d8e4db69cf1748768605eac9",
+    "586": "4846fee5b06af770e9b3a10c36200f673d656725",
+    "591": "f622547244bf6d895a571a7b035142ba66279806",
+    "592": "1f471c445c06a45a32e2c47e8d74c8264b68356b",
+    "593": "fdb6dcc6d7bc29764c5b6b4d32b7fa050acf74a4",
+    "596": "7732dd5e46863ff294b5e443661cb9d99abf4e01",
+    "599": "bc13c8a3e000442606bd2dbe8583a17649ba6c32",
+    "601": "529ab7dcb59d5ea7a5798e9e06cf5b3aee8a1550",
+    "604": "1b795c5d7dea746ba310572585c0ef067574b4a6",
+    "615": "bf8a49a34350599763cced8592db81dc62c2c1c0",
+    "617": "904d62855723c00f8acef9a287a10768598fdeb2",
+    "622": "f69477f4f6d3c7eb1f740b166881bffed4ffa9f6",
+    "623": "e39a615ca601ce5d62abeb24f3fcdb8644ca1005",
+    "626": "660de893d27bb7abd1d5b01b1640207528a5c851",
+    "589": "a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46",
+    "741": "e516b1463606840a15f80aac1995d64ce0092e89",
+    "799": "80e6463157cfc80b85af09e7e2f14748fe2dac60",
+    "858": "ae2e25fee91540c83ceee5a60e3b07422ce21c7d",
+    "869": "73e86f69bf9a391e3c334999f7bbe3825dc42759",
+    "870": "b734850dc348800942f0ffd1f0723fdec7539b2b",
+    "890": "4b3bd8cbe8e3e81c44161017c9b72c905a8040a8",
+    "942": "1960cfa919a0945d0ca4cb14ea96237000037903",
+    "1004": "1974578f82a30158dd5fcdcdf226b83df8e7196f",
+    "1005": "779e80aef469bc3a28b19f1adde199871a9a22e3",
+    "1018": "eac7547cb191f875d5f5ea5f9bbb1d2b8c51c4ef",
+    "1022": "58618bcbfe88a252467755fc283c04c3db5ca4fd",
+    "1028": "be4bd89cb4483dcee43257200cd27601f938f13d",
+    "1031": "e900534f0beda892c726723524203b97471fb9a3",
+    "1042": "7e8ef087c02a77038a28a337a7ee3a77039cac0d",
+    "1094": "0fc329d0bec93ca7cd288eaea1295172e8789810",
+    "1087": "589d1cedf5ef5f231182f4ec86408616311a8af0",
+    "1082": "0c2e294ec204e83846d27344f829d15ed9ee7cc6",
+    "1074": "052afc128954f7b6a2b3c409eb4d72248e4020f2",
+    "1068": "cc71375eae4e8754da0f5d940cb519d66625fb2e",
+    "1065": "387d935e354a931d636e2c5e6ee63a6910f78a1b",
+    "1063": "d9377d32e0c97e41338e704c3960fd6f2fa73806",
+    "1061": "223f581afb1d63d5f8476a69040617e11bcd50e4",
+    "1056": "d0201efbd68c4acc9832b84bcc986c8781b83c4f",
+    "1055": "a8ea308976f1709eef0a328463f010cf51469277",
+    "1053": "6f92ce8884929c9fd4730c86a28e2e3735de02af",
+    "1050": "c87242f846cfab8c0b97fd889b5fecb0eacef939",
+    "987": "5142511af0437572aca71b35d1987cd2b47bb5e3"
+  }
+}
diff --git a/research/five-cube-quadratic-code-seed44/test_package.py b/research/five-cube-quadratic-code-seed44/test_package.py
new file mode 100644
index 0000000000..5ad5d5dda7
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/test_package.py
@@ -0,0 +1,160 @@
+#!/usr/bin/env python3
+"""Bounded mathematical and packaging controls. Apache-2.0; AI-assisted."""
+
+import copy
+from fractions import Fraction
+import importlib.util
+import json
+from pathlib import Path
+import shutil
+import subprocess
+import sys
+import tempfile
+import unittest
+
+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
+
+
+primary = load("verify")
+oracle = load("independent")
+
+
+class PackageTests(unittest.TestCase):
+    @classmethod
+    def setUpClass(cls):
+        cls.certificate = json.loads((HERE / "CERTIFICATE.json").read_text())
+
+    def test_complete_regeneration_and_independent_replay(self):
+        generated = primary.compute_certificate()
+        self.assertEqual(primary.canonical_bytes(generated), (HERE / "CERTIFICATE.json").read_bytes())
+        oracle.check_source_files(self.certificate)
+        result = oracle.verify_data(self.certificate)
+        self.assertEqual(result["ordered_term_ranks"], 3136)
+
+    def test_exact_rank_controls(self):
+        cases = [([], 0), ([[]], 0), ([[0, 0], [0, 0]], 0), ([[2]], 1),
+                 ([[1, 1], [1, -1]], 2), ([[1, 2], [2, 4]], 1),
+                 ([[0, 2, 0], [0, 0, 3]], 2), ([[0, 0], [1, 2]], 1)]
+        for matrix, rank in cases:
+            with self.subTest(matrix=matrix):
+                self.assertEqual(primary.rational_rank(matrix), rank)
+                self.assertEqual(oracle.exact_rank(matrix), rank)
+        self.assertEqual(oracle.exact_rank([[Fraction(1, 2), 1], [1, 2]]), 1)
+        # The changed all-ones Gram has a two-total cost 2, so the floor is
+        # specific to the inherited quadratic Gram, not every common label.
+        self.assertEqual(2 * oracle.exact_rank([[1] * 16 for _ in range(16)]), 2)
+
+    def test_coverage_and_symmetry_faults(self):
+        bad = copy.deepcopy(self.certificate)
+        bad["transports"].pop()
+        with self.assertRaisesRegex(ValueError, "coverage omitted"):
+            oracle.check_transports(bad)
+        bad = copy.deepcopy(self.certificate)
+        edge = next(row for row in bad["transports"] if row[1] != 0)
+        edge[2] = next(g for g in range(9) if oracle.transform(edge[1], g) != edge[0])
+        with self.assertRaisesRegex(ValueError, "false polynomial"):
+            oracle.check_transports(bad)
+        bad = copy.deepcopy(self.certificate)
+        bad["transports"].append(bad["transports"][-1])
+        with self.assertRaisesRegex(ValueError, "duplicate"):
+            oracle.check_transports(bad)
+
+    def test_potential_and_word_faults(self):
+        r = self.certificate["orbits"][0]
+        functions = [[(f >> x) & 1 for x in range(32)]
+                     for f in primary.factor_masks(r["mask"])]
+        decoder = oracle.geometry()[3]
+        mutations = []
+        wrong_target = copy.deepcopy(r["unrestricted"])
+        wrong_target["potential"][17] += 1
+        mutations.append(wrong_target)
+        false_unreachable = copy.deepcopy(r["unrestricted"])
+        false_unreachable["potential"][17] = None
+        mutations.append(false_unreachable)
+        missing_term = copy.deepcopy(r["unrestricted"])
+        missing_term["word"].pop()
+        mutations.append(missing_term)
+        same_price_wrong_map = copy.deepcopy(r["unrestricted"])
+        same_price_wrong_map["word"] = [[1, 1], [1, 2]]
+        mutations.append(same_price_wrong_map)
+        free_decoder = copy.deepcopy(r["unrestricted"])
+        free_decoder["minimum"] = 0
+        mutations.append(free_decoder)
+        wrong_cap = copy.deepcopy(r["unrestricted"])
+        wrong_cap["rank_cap"] = 15
+        mutations.append(wrong_cap)
+        for receipt in mutations:
+            with self.subTest(receipt=receipt["rank_cap"]):
+                with self.assertRaises(ValueError):
+                    oracle.check_potential(receipt, r["ranks"], functions, decoder)
+        # An incorrect coefficient target must not be accepted by a decoder
+        # that happens to have the right total rank.
+        bad = copy.deepcopy(self.certificate)
+        bad["target_coefficient_matrix_bits"] ^= 1
+        with self.assertRaisesRegex(ValueError, "wrong matrix target"):
+            oracle.verify_data(bad)
+        # Zero-cost transitions must impose equality on the potential.
+        weights = [[16] * 7 for _ in range(7)]
+        weights[0][0] = 0
+        free_edge = primary.shortest_paths(weights, 32)
+        self.assertEqual(free_edge["minimum"], 16)
+        self.assertEqual(free_edge["potential"][1], 0)
+        primary.validate_graph_certificate(weights, free_edge)
+        free_edge["potential"][1] = 1
+        with self.assertRaisesRegex(ValueError, "potential violates"):
+            primary.validate_graph_certificate(weights, free_edge)
+        with self.assertRaisesRegex(ValueError, "edge potential"):
+            oracle.check_potential(free_edge, weights, functions, decoder)
+
+    def test_actual_rank_corruption(self):
+        bad = copy.deepcopy(self.certificate)
+        bad["orbits"][0]["ranks"][0][0] = 0
+        with self.assertRaisesRegex(ValueError, "rational rank table mismatch"):
+            oracle.verify_data(bad)
+
+    def test_relocation_sources_and_output_preservation(self):
+        with tempfile.TemporaryDirectory(prefix="quadratic package ") as directory:
+            root = Path(directory)
+            package = root / "standalone copy"
+            shutil.copytree(HERE, package, ignore=shutil.ignore_patterns("__pycache__"))
+            for file in package.rglob("*"):
+                if file.is_file():
+                    file.chmod(0o444)
+            for script, marker in [("verify.py", "PASS_FIVE_CUBE_QUADRATIC_CODES"),
+                                   ("independent.py", "PASS_INDEPENDENT_QUADRATIC_CODES")]:
+                result = subprocess.run([sys.executable, "-I", "-O", "-B", str(package / script)],
+                                        cwd=root, text=True, capture_output=True, timeout=120)
+                self.assertEqual(result.returncode, 0, result.stdout + result.stderr)
+                self.assertIn(marker, result.stdout)
+            output = root / "occupied.json"
+            output.write_text("preserve me\n")
+            result = subprocess.run([sys.executable, "-I", "-O", "-B", str(package / "verify.py"),
+                                     "--write", str(output)], cwd=root, text=True,
+                                    capture_output=True, timeout=120)
+            self.assertNotEqual(result.returncode, 0)
+            self.assertEqual(output.read_text(), "preserve me\n")
+            fresh = root / "fresh.json"
+            result = subprocess.run([sys.executable, "-I", "-O", "-B", str(package / "verify.py"),
+                                     "--write", str(fresh)], cwd=root, text=True,
+                                    capture_output=True, timeout=120)
+            self.assertEqual(result.returncode, 0, result.stderr)
+            self.assertEqual(fresh.read_bytes(), (HERE / "CERTIFICATE.json").read_bytes())
+            source = package / "sources" / "PR1046-PROOF.md"
+            source.chmod(0o644)
+            source.write_bytes(source.read_bytes() + b"\ncorrupted\n")
+            for script in ["verify.py", "independent.py"]:
+                result = subprocess.run([sys.executable, "-I", "-O", "-B", str(package / script)],
+                                        cwd=root, text=True, capture_output=True, timeout=120)
+                self.assertNotEqual(result.returncode, 0)
+                self.assertIn("source", result.stderr.lower())
+
+
+if __name__ == "__main__":
+    unittest.main(verbosity=2)
diff --git a/research/five-cube-quadratic-code-seed44/verify.py b/research/five-cube-quadratic-code-seed44/verify.py
new file mode 100644
index 0000000000..ac9cdfaca0
--- /dev/null
+++ b/research/five-cube-quadratic-code-seed44/verify.py
@@ -0,0 +1,295 @@
+#!/usr/bin/env python3
+"""Exact finite certificate for quadratic-augmented five-cube parity codes.
+
+Hanson Sun (@bbMisk), with substantial OpenAI Codex assistance.
+Apache-2.0. No predecessor implementation is imported.
+"""
+
+import argparse
+from collections import Counter
+from functools import reduce
+import hashlib
+from heapq import heappop, heappush
+from itertools import combinations
+import json
+from math import gcd
+from pathlib import Path
+import sys
+
+HERE = Path(__file__).resolve().parent
+FULL = (1 << 32) - 1
+PARITY = sum((x.bit_count() % 2) << x for x in range(32))
+MONOMIALS = [(i,) for i in range(4)] + list(combinations(range(5), 2))
+TARGET = (1 << 0) | (1 << 4)  # E E^t + O O^t, in basis (E,O,h).
+INFINITY = 10**6  # Above 511 edges times the maximum possible rank 32.
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def canonical_bytes(value):
+    return (json.dumps(value, sort_keys=True, separators=(",", ":")) + "\n").encode()
+
+
+def check_sources():
+    raw = (HERE / "SOURCES.json").read_bytes()
+    manifest = json.loads(raw)
+    for record in manifest["retained_files"]:
+        path = HERE / record["local_path"]
+        require(path.resolve().is_relative_to(HERE), "source path escapes package")
+        data = path.read_bytes()
+        require(hashlib.sha256(data).hexdigest() == record["sha256"],
+                "source SHA256 mismatch: " + record["local_path"])
+        blob = b"blob " + str(len(data)).encode() + b"\0" + data
+        require(hashlib.sha1(blob).hexdigest() == record["git_blob_sha1"],
+                "source Git blob mismatch: " + record["local_path"])
+    return hashlib.sha256(raw).hexdigest()
+
+
+def gram_matrix():
+    return [[2 * (4 - (x ^ y).bit_count()) * (2 - (x ^ y).bit_count())
+             for y in range(32)] for x in range(32)]
+
+
+def canonical_mask(mask):
+    # Unique representative modulo <1,p>: constant and x4 coefficient zero.
+    if mask & 1:
+        mask ^= FULL
+    if (mask >> 16) & 1:
+        mask ^= PARITY
+    return mask
+
+
+def code_masks():
+    monomials = [sum(int(all((x >> i) & 1 for i in term)) << x
+                     for x in range(32)) for term in MONOMIALS]
+    masks = [0] * (1 << 14)
+    for c in range(1, len(masks)):
+        low = c & -c
+        masks[c] = masks[c ^ low] ^ monomials[low.bit_length() - 1]
+    require(len(set(masks)) == 1 << 14, "quadratic quotient is not injective")
+    return masks
+
+
+def cube_generators():
+    # Pullbacks x -> x+e_i, then x_i <-> x_(i+1).
+    generators = [[x ^ (1 << i) for x in range(32)] for i in range(5)]
+    for i in range(4):
+        generators.append([
+            x ^ (((x >> i ^ x >> (i + 1)) & 1) * ((1 << i) | (1 << (i + 1))))
+            for x in range(32)
+        ])
+    return generators
+
+
+def permute_mask(mask, permutation):
+    return sum(((mask >> x) & 1) << y for y, x in enumerate(permutation))
+
+
+def orbit_certificate(masks):
+    lookup = {mask: c for c, mask in enumerate(masks)}
+    remaining = set(range(1, len(masks)))
+    records, orbits = [], []
+    generators = cube_generators()
+    while remaining:
+        representative = min(remaining)
+        queue = [representative]
+        remaining.remove(representative)
+        records.append([representative, 0, -1, representative])
+        for parent in queue:
+            for index, permutation in enumerate(generators):
+                image = canonical_mask(permute_mask(masks[parent], permutation))
+                child = lookup[image]
+                if child in remaining:
+                    remaining.remove(child)
+                    queue.append(child)
+                    records.append([child, parent, index, representative])
+        orbits.append((representative, len(queue)))
+    require(len(records) == 16383 and len(orbits) == 64, "orbit coverage changed")
+    return orbits, records
+
+
+def rational_rank(rows):
+    """Integer row elimination; gcd normalization preserves rational row span."""
+    matrix = [list(row) for row in rows]
+    if not matrix:
+        return 0
+    rank = 0
+    for column in range(len(matrix[0])):
+        pivot = next((i for i in range(rank, len(matrix)) if matrix[i][column]), None)
+        if pivot is None:
+            continue
+        matrix[rank], matrix[pivot] = matrix[pivot], matrix[rank]
+        for i in range(rank + 1, len(matrix)):
+            entry = matrix[i][column]
+            if entry:
+                common = gcd(matrix[rank][column], entry)
+                left, right = matrix[rank][column] // common, entry // common
+                matrix[i] = [left * x - right * y
+                             for x, y in zip(matrix[i], matrix[rank])]
+                divisor = reduce(gcd, matrix[i], 0)
+                if divisor:
+                    matrix[i] = [x // divisor for x in matrix[i]]
+        rank += 1
+        if rank == len(matrix):
+            break
+    return rank
+
+
+def factor_masks(mask):
+    basis = [FULL ^ PARITY, PARITY, mask]
+    return [reduce(int.__xor__, (basis[i] for i in range(3) if c >> i & 1), 0)
+            for c in range(8)]
+
+
+def term_ranks(mask, gram):
+    functions = factor_masks(mask)
+    supports = [[x for x in range(32) if f >> x & 1] for f in functions]
+    weights = [[0] * 7 for _ in range(7)]
+    for a in range(1, 8):
+        for b in range(a, 8):
+            cost = rational_rank([[gram[x][y] for y in supports[b]] for x in supports[a]])
+            weights[a - 1][b - 1] = weights[b - 1][a - 1] = cost
+    return weights
+
+
+def dyad(a, b):
+    return sum(((a >> i & 1) & (b >> j & 1)) << (3 * i + j)
+               for i in range(3) for j in range(3))
+
+
+def shortest_paths(weights, rank_cap):
+    terms = [(dyad(a, b), weights[a - 1][b - 1], a, b)
+             for a in range(1, 8) for b in range(1, 8)
+             if weights[a - 1][b - 1] <= rank_cap]
+    distance = [INFINITY] * 512
+    predecessor = {}
+    distance[0] = 0
+    heap = [(0, 0)]
+    while heap:
+        cost, state = heappop(heap)
+        if cost != distance[state]:
+            continue
+        for step, weight, a, b in terms:
+            target = state ^ step
+            if cost + weight < distance[target]:
+                distance[target] = cost + weight
+                predecessor[target] = (state, a, b)
+                heappush(heap, (cost + weight, target))
+    word = []
+    if distance[TARGET] != INFINITY:
+        state = TARGET
+        while state:
+            state, a, b = predecessor[state]
+            word.append([a, b])
+        word.reverse()
+    return {
+        "rank_cap": rank_cap,
+        "minimum": None if distance[TARGET] == INFINITY else distance[TARGET],
+        "potential": [None if d == INFINITY else d for d in distance],
+        "word": word,
+    }
+
+
+def validate_graph_certificate(weights, receipt):
+    cap, potential = receipt["rank_cap"], receipt["potential"]
+    require(len(potential) == 512 and potential[0] == 0, "invalid potential origin/size")
+    require(all(d is None or type(d) is int and 0 <= d < INFINITY for d in potential),
+            "invalid potential value")
+    for state, distance in enumerate(potential):
+        if distance is None:
+            continue
+        for a in range(1, 8):
+            for b in range(1, 8):
+                cost = weights[a - 1][b - 1]
+                if cost <= cap:
+                    other = potential[state ^ dyad(a, b)]
+                    require(other is not None and other <= distance + cost,
+                            "potential violates transition/closed reachability")
+    require(potential[TARGET] == receipt["minimum"], "wrong target potential")
+    if receipt["minimum"] is None:
+        require(not receipt["word"], "unreachable target has a word")
+        return
+    state, cost = 0, 0
+    for a, b in receipt["word"]:
+        require(type(a) is int and type(b) is int and 1 <= a <= 7 and 1 <= b <= 7,
+                "invalid word factor")
+        weight = weights[a - 1][b - 1]
+        require(weight <= cap, "word exceeds rank cap")
+        state ^= dyad(a, b)
+        cost += weight
+    require(state == TARGET and cost == receipt["minimum"], "invalid attaining word")
+
+
+def compute_certificate():
+    source_hash = check_sources()
+    gram = gram_matrix()
+    require(rational_rank(gram) == 16, "wrong five-cube Gram")
+    masks = code_masks()
+    orbits, transports = orbit_certificate(masks)
+    results = []
+    capped_histogram = Counter()
+    for representative, size in orbits:
+        weights = term_ranks(masks[representative], gram)
+        unrestricted = shortest_paths(weights, 32)
+        capped = shortest_paths(weights, 15)
+        validate_graph_certificate(weights, unrestricted)
+        validate_graph_certificate(weights, capped)
+        require(unrestricted["minimum"] == 32, "quadratic copy floor changed")
+        nonlinear = representative >= 16
+        require(not nonlinear or capped["minimum"] is None or capped["minimum"] >= 34,
+                "nonlinear peak-rank gap changed")
+        key = "unreachable" if capped["minimum"] is None else str(capped["minimum"])
+        capped_histogram[key] += size
+        results.append({
+            "representative": representative, "mask": masks[representative],
+            "orbit_size": size, "nonlinear": nonlinear, "ranks": weights,
+            "unrestricted": unrestricted, "rank15": capped,
+        })
+    return {
+        "schema": "five-cube-quadratic-code-v1",
+        "source_manifest_sha256": source_hash,
+        "basis": ["even", "odd", "h"],
+        "quotient_monomials": [list(m) for m in MONOMIALS],
+        "target_coefficient_matrix_bits": TARGET,
+        "transport_columns": ["child", "parent", "generator", "representative"],
+        "transports": transports,
+        "orbits": results,
+        "summary": {
+            "quadratic_quotient_codes": 16383, "nonlinear_codes": 16368,
+            "orbit_count": 64, "nonlinear_orbits": 62,
+            "ordered_term_ranks": 64 * 49, "coefficient_states_per_code": 512,
+            "unrestricted_transition_checks": 64 * 512 * 49,
+            "minimum_copy_rank_sum": 32,
+            "nonlinear_minimum_with_rank_cap15": 34,
+            "rank15_minimum_histogram_weighted_by_codes": dict(sorted(capped_histogram.items())),
+            "finite_rank34_attaining_codes": capped_histogram["34"],
+            "no_new_kappa": True,
+        },
+    }
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--write", type=Path, help="write regenerated certificate to a new path")
+    parser.add_argument("--certificate", type=Path, default=HERE / "CERTIFICATE.json")
+    args = parser.parse_args()
+    result = compute_certificate()
+    encoded = canonical_bytes(result)
+    if args.write:
+        with args.write.open("xb") as stream:
+            stream.write(encoded)
+    else:
+        require(args.certificate.read_bytes() == encoded, "certificate differs from exact regeneration")
+    print("PASS_FIVE_CUBE_QUADRATIC_CODES codes=16383 orbits=64 ranks=3136 "
+          "copy_floor=32 nonlinear_rank15_floor=34 no_new_kappa")
+
+
+if __name__ == "__main__":
+    try:
+        main()
+    except (OSError, ValueError, KeyError, TypeError) as exc:
+        print("FAIL_FIVE_CUBE_QUADRATIC_CODES: " + str(exc), file=sys.stderr)
+        raise SystemExit(1)