Exhibit a 104 by 13 factorisation of the P104 window over F_5.

research/p104-f5-rank13-factor/NOTICE

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+The file data/ports_p104.json is byte-identical to research/p104-port-window-haemers/data/ports_p104.json in PR #6933 (DreamingOfClouds), commit a967c521074c2c7450255ae237fda9c439ee1b35. That package is Apache-2.0.
+
+Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.
+
+The clique partition, factorisation check and this note were prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

research/p104-f5-rank13-factor/README.md

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+# A 104 by 13 factorisation of the P104 window over F_5
+
+**No new κ.** The selected record stays κ = `781451/250000000`.
+
+## Scope and evidence
+
+Original card: `complex-rank-shape-decoder`. Scope status: **changed**. The card asks whether a decoder or coframe change improves the signed moment. This package answers a different question, open question 3 of [#6933](https://github.com/CrocSwap/integer-mult-bounds/pull/6933): one 104×13 factorisation of the odd-character P104 window over `F_5`. No moment is computed.
+
+**Witness.** The odd graph on the 104 ports has a partition into 13 cliques of size 8. The cliques are `data/frames.json`. Send each port to the standard basis vector of its clique in `F^13`, and let `W` be the resulting 104×13 matrix over `F_5`. Then `M = W W^T` satisfies
+
+- `M_ii = 1` for every port,
+- `M_ij = 0` whenever distinct ports `i, j` are orthogonal over `F_2`.
+
+So `M` is a 1-block pattern of rank 13. One row from each clique is an identity minor, so the rank is 13 over every field, including `F_5`. This is an unrestricted factorisation: the matrix is explicit, and the check does not depend on a gauge, a degree bound, or a constant value on the edges.
+
+**What that implies.** In the notation of #6933, `w_1(P104; F) ≤ 13` and therefore `w(P104; F) ≤ 13` for every field `F`. Over `F_5` the same matrix is the required 104×13 factorisation. Accepting #6933, Theorem 4, without replaying it, `w(P104; F) ≥ 13` whenever `char F ≠ 2`. The two bounds give
+
+`w(P104; F_5) = 13`,
+
+and the same equality for every field of characteristic other than 2. The lower bound is cited. The upper bound is the witness in this package.
+
+The 16 weight-3-mod-4 vectors omitted from P104 form two further odd-graph 8-cliques (`excluded_odd_cliques` in the same file). Adjoining either clique gives a 112×14 factorisation, and adjoining both gives a 120×15 factorisation, by the same rule. Accepting Theorem 4 again, `w = 14` on P112 and `w = 15` on P120 for every field of characteristic other than 2.
+
+Characteristic 2 is unchanged. #6933 already has `w(P104; F_2) ∈ [53/6, 9]` and `w = 9` on P112 and P120. The witness gives only the weaker ceiling 13 there. The rank-36 ceiling of [#6946](https://github.com/CrocSwap/integer-mult-bounds/pull/6946) and the characteristic-3 class-constant argument in [discussion comment 19507187](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19507187) are not replayed. The same 0-1 matrix has rank 13 in characteristic 3, which is a separate statement from those arguments.
+
+## Overlap
+
+Balanced `find_overlap.py --all-terms` queries against open PRs and discussions, cache refreshed 2026-11-10 with complete pagination (895 open PR records, 6953 discussion records):
+
+- `P104`, `F_5`, `factorisation`
+- `F_5`, `rank-13`
+- `w(P104`, `13`
+- `8-clique`, `P104`
+
+No open PR exhibits a 104×13 factorisation. The hits are #6933 (lower bound 13, odd-character upper bound left open), #6946 and [comment 19492602](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19492602) (upper bound 36 off characteristic 3, and 38 in characteristic 3), and [comment 19507187](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19507187), which states that the `F_5` rank-13 factorisation was not touched. Local notes that mention an 8-clique of E8 lines use the opposite relation, pairwise non-orthogonal lines, or a decoder column-space rank. They are not this zero pattern.
+
+## Reproduce
+
+From the repository root, CPython 3.12, standard library only:
+
+```sh
+python3 -B research/p104-f5-rank13-factor/verify.py
+python3 -I -O -B research/p104-f5-rank13-factor/verify.py
+python3 -B scripts/research_package.py --package research/p104-f5-rank13-factor
+python3 -B scripts/check_research_commit.py --package research/p104-f5-rank13-factor --check-source-history
+```
+
+The port file is byte-identical to #6933's `data/ports_p104.json` at `a967c521074c2c7450255ae237fda9c439ee1b35`, sha256 `691c7b7d212112115b1bfc76db89f90d304ec448dc8c4a0c43d9a5954317146c`. The verifier checks that hash, the partition, the mask, the ranks, and two mutants (an orthogonal pair forced into one frame, and two frames given the same coordinate).
+
+## Open next questions
+
+The odd-character window values 13, 14 and 15 are settled by this witness together with the cited Theorem 4. The remaining gap that would affect a paid profile is physical:
+
+Can the 13 frame-indicator rows be hosted as a legal physical P104 label word whose complete signed bill beats the selected profile? The mask rank is not a supplier price. First step: host one frame indicator in the existing P104 label interface and charge every new monomial before touching the other twelve frames. Blocked by the fact that #6933 does not derive a physical realisation of the block mask, and this package does not supply one.
+
+## Credit
+
+Port integers: DreamingOfClouds, PR #6933, Apache-2.0. Odd-character ceilings that this witness improves: #6946 and discussion comments 19492602 and 19507187.
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

research/p104-f5-rank13-factor/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    }
+  ]
+}

research/p104-f5-rank13-factor/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 8,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "original_card": {
+    "id": "complex-rank-shape-decoder",
+    "question": "Can a legal decoder or coframe change improve the full signed moment, including changed helper targets?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+      "docs/research/recursive-invoice-preflight.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "A source-bound arbitrary-input identity and legal producer/consumer boundary with the complete paid and productive target invoice, including restoration, ownership, native alignment and extra charges. Price under the actual current model; historical ceiling.py assumptions must be established before use. A favorable optimistic relaxation keeps a route open but is not a completed supplier. A scoped local result remains useful; expansion requires the family-level budget gate.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Price one complete changed itinerary with both paid and productive histograms. Check #5693/#5698 for overlap and their exact hypotheses before another coframe search. Use the signed invoice preflight where its boundary contract agrees; an inconclusive concave comparison is not a power crossing. Stop if an optimistic complete budget fails at the chosen target or native delivery remains unspecified.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "coframe",
+      "signed moment",
+      "5693",
+      "5698"
+    ],
+    "existing_checks": [
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Exhibit one 104 by 13 factorisation of the P104 odd-graph window over F_5, and record the window bound that factorisation proves.",
+  "scope_change_reason": "The catalogue card asks whether a decoder or coframe change improves the signed moment. The assigned question is open question 3 of PR #6933: the odd-character value of w(P104). A signed-moment search would not decide that factorisation, and this package computes no moment.",
+  "bottleneck_link": "Side result on the port-window mask. It does not change the selected paid profile or the signed-moment bill. A kappa effect would still need a physical word that realises the mask.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop when a checked 104 by 13 factorisation over F_5 is in hand. Do not rerun the rank-36 ceiling, the characteristic-3 class-constant argument, or characteristic 2.",
+  "outcome": "completed",
+  "evidence": "verify.py checks the stored 13-clique partition, the 0-1 factorisation, rank 13 over F_2, F_3, F_5 and F_7, and failure controls. PASS_P104_F5_RANK13.",
+  "author": "unpaidmoon",
+  "ai_assistance": "Cursor Grok provided substantial assistance with the search, the verifier and the write-up.",
+  "open_questions": [
+    {
+      "question": "Can the 13 frame-indicator rows be hosted as a legal physical P104 label word whose complete signed bill beats the selected profile?",
+      "why_it_matters": "The mask rank is not a supplier price. The selected frontier still decides kappa.",
+      "first_step": "Host one frame indicator in the existing P104 label interface and charge every new monomial before touching the other twelve frames.",
+      "blocked_by": "PR #6933 defines the window by the block mask and does not derive a physical realisation. This package does not supply one."
+    }
+  ]
+}

research/p104-f5-rank13-factor/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": ["{python}", "-B", "{package}/verify.py"],
+      "timeout_seconds": 60,
+      "checks": "Port-list sha256, the 13-clique partition, the 0-1 factorisation mask, rank 13 over F_2, F_3, F_5 and F_7, the identity minor, both P112 extensions and the P120 extension, plus orthogonal-pair and merged-frame failure controls."
+    }
+  ]
+}

research/p104-f5-rank13-factor/data/frames.json

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+{
+  "description": "Disjoint 8-cliques of the odd graph. frames partition the 104 P104 ports. excluded_odd_cliques are the two 8-cliques in the 120 weight-3-mod-4 vectors that P104 omits; together with frames they partition those 120 vectors. Integers are F_2^9 ports, bit i coordinate i.",
+  "frames": [
+    [
+      11,
+      21,
+      38,
+      56,
+      140,
+      146,
+      161,
+      191
+    ],
+    [
+      13,
+      26,
+      131,
+      148,
+      262,
+      273,
+      392,
+      415
+    ],
+    [
+      14,
+      69,
+      164,
+      239,
+      276,
+      351,
+      446,
+      501
+    ],
+    [
+      19,
+      41,
+      74,
+      112,
+      280,
+      290,
+      321,
+      379
+    ],
+    [
+      22,
+      37,
+      196,
+      247,
+      268,
+      319,
+      478,
+      493
+    ],
+    [
+      28,
+      49,
+      133,
+      168,
+      265,
+      292,
+      400,
+      445
+    ],
+    [
+      35,
+      81,
+      176,
+      194,
+      274,
+      352,
+      385,
+      499
+    ],
+    [
+      44,
+      84,
+      152,
+      224,
+      304,
+      328,
+      388,
+      508
+    ],
+    [
+      50,
+      70,
+      138,
+      254,
+      259,
+      375,
+      443,
+      463
+    ],
+    [
+      73,
+      98,
+      208,
+      251,
+      324,
+      367,
+      477,
+      502
+    ],
+    [
+      88,
+      100,
+      193,
+      253,
+      322,
+      382,
+      475,
+      487
+    ],
+    [
+      134,
+      145,
+      200,
+      223,
+      416,
+      439,
+      494,
+      505
+    ],
+    [
+      261,
+      296,
+      336,
+      381,
+      386,
+      431,
+      471,
+      506
+    ]
+  ],
+  "excluded_odd_cliques": [
+    [
+      7,
+      25,
+      42,
+      52,
+      76,
+      82,
+      97,
+      127
+    ],
+    [
+      67,
+      104,
+      137,
+      162,
+      266,
+      289,
+      448,
+      491
+    ]
+  ]
+}