[unpaidmoon]

Change and mathematical scope

Scoped factorisation. No new κ and no selected-frontier change. The selected record stays κ = 781451/250000000.

The odd graph on the 104 ports has a partition into 13 cliques of size 8. The 104×13 indicator matrix W of that partition satisfies M = W W^T with M_ii = 1 and M_ij = 0 on every F_2-orthogonal pair. One row from each clique is an identity minor, so rank M = 13 over every field, including F_5. This is an explicit 1-block pattern, not a restricted ansatz and not a failed search.

Consequences, with the lower bound cited rather than replayed:

• Proved here: w_1(P104; F) ≤ 13, hence w(P104; F) ≤ 13, for every field F. Over F_5 this is a 104×13 factorisation.
• Accepting Port windows are a fractional Haemers bound: w >= 53/6 on P104 (no new κ) #6933, Theorem 4: w(P104; F_5) = 13, and w(P104; F) = 13 for every field of characteristic other than 2.
• The 16 omitted weight-3-mod-4 vectors are two further 8-cliques. Adjoining either gives a 112×14 factorisation, and adjoining both gives a 120×15 factorisation. Accepting Theorem 4, w = 14 on P112 and w = 15 on P120 in every characteristic other than 2.
• Characteristic 2 stays where Port windows are a fractional Haemers bound: w >= 53/6 on P104 (no new κ) #6933 left it: w(P104; F_2) ∈ [53/6, 9], and w = 9 on P112 and P120. This witness only recovers the weaker ceiling 13 there.

The rank-36 ceiling of #6946 and the characteristic-3 class-constant argument in discussion comment 19507187 are not replayed. The same 0-1 matrix has rank 13 in characteristic 3; that is a separate statement from those arguments.

Task and overlap

Base commit: 7c49fe32c55de02c2c873655c804b16e4c3fef0a.

Card complex-rank-shape-decoder, priority revision 8. Scope status: changed. The card asks whether a decoder or coframe change improves the signed moment. The assigned question is open question 3 of #6933, a 104×13 factorisation over F_5. No moment is computed.

Balanced overlap queries (find_overlap.py --all-terms, cache 2026-11-10, 895 open PRs and 6953 discussion records, complete pagination): P104 + F_5 + factorisation; F_5 + rank-13; w(P104 + 13; 8-clique + P104. No open PR already gives a 104×13 factorisation.

• Port windows are a fractional Haemers bound: w >= 53/6 on P104 (no new κ) #6933 proves w ≥ 13 for char ≠ 2 and leaves the odd-character upper bound open. Its port list is vendored byte-for-byte.
• Bound the odd-character port window by 36. #6946 and discussion comment 19492602 prove w ≤ 36 off characteristic 3, and ≤ 38 in characteristic 3. This witness replaces that upper bound by 13.
• Discussion comment 19507187 gives a characteristic-3 rank-36 pattern for class-constant matrices and states that the F_5 rank-13 factorisation was not touched.
• Local 8-clique notes use pairwise non-orthogonal lines or a decoder column-space rank. That is the opposite zero pattern.

Validation

CPython 3.12, standard library, from the repository root:

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

Observed PASS_P104_F5_RANK13 rank13=13 fields=2,3,5,7 p112=14 p120=15 orth_pairs_split=2496, PASS_REGISTERED_RESEARCH_PACKAGE with outcome completed, and PASS_RESEARCH_COMMIT_BYTES.

The checker pins the port-list sha256, checks the partition and the mask, checks an identity minor, and rejects two mutants: an orthogonal pair forced into one frame, and two frames assigned the same coordinate. Equality w = 13 uses #6933 Theorem 4, which is not replayed.

Credit

Port integers: DreamingOfClouds, PR #6933, Apache-2.0. Previous odd-character ceilings: #6946 and discussion comments 19492602 and 19507187.

Prepared by unpaidmoon with substantial Cursor Grok assistance.

Open next questions

The odd-character values 13, 14 and 15 are the witness plus the cited Theorem 4. The gap that would affect a paid profile is physical: can the 13 frame-indicator rows be hosted as a legal P104 label word whose complete signed bill beats the selected profile? First step: host one frame indicator and charge every new monomial before touching the other twelve. #6933 does not derive a physical realisation of the block mask, and this package does not supply one.

Workflow feedback (optional)

A balanced --all-terms overlap query separates this factorisation from the many open PRs that only mention P104. Link: #1200