[DreamingOfClouds]

Result

No κ claim and no word. Every k-block pattern on the ports of P104 (zero blocks on orthogonal pairs, invertible diagonal blocks) has rank at least 9k over every field. So the window satisfies w(P104; F) ≥ 9 for every field F, and w(P104; F) = 9 exactly in characteristic 2 (the Gram matrix has rank 9). P112 and P120 are recovered with no weights.

• Every orthogonal 8-set of P104 has minimum degree 5 in its edge graph (P112: 6, P120: 7); the K-type counts are recomputed.
• A second, independent proof through a star inequality reaches the same conclusion.
• Corollaries: H_f(G_{K_n}) = n+1 for n ≥ 5 over every field, and H_f(Petersen; F) ≥ 9/2 over every field. The Petersen statement was not checked against the literature.
• w(V; F) depends only on the characteristic of F.
• Combined with Exhibit a 104 by 13 factorisation of the P104 window over F_5. #7014 and Port windows are a fractional Haemers bound: w >= 53/6 on P104 (no new κ) #6933 Theorem 4: w(P104) = 9 in characteristic 2 and 13 otherwise (P112: 9 and 14; P120: 9 and 15).
• Recorded kissat rows (F_2, no DRAT proofs) on Petersen and G_{K_5} sub-instances agree with the theorem, with same-size controls. A k = 6, rank 53 run on 30 ports was undecided at 20 minutes and is not used.

Why

The proof of #6933 Theorem 3 bounds the rank of X_TT by g_T(c) + Y_c and then adds it to g_T(c) + g_T(d), which counts g_T(c) twice. Its Step 2 containment already puts the old part of col X_TT inside col γ_T(c). Using the containment itself gives the sharper edge inequality

k ≤ g_T(c) + g_T(d) + min(Y_c, Y_d),

and summing it against the star inequality gives j ≥ k on every orthogonal 8-set, with no weights to tune.

What it closes

The interval [53/6, 9] that #6933 left open for w(P104; F_2), and the question whether Theorem 3 extends to the floor 9. No window below 9 exists in the block-mask model, so a fractional window saves nothing on P104; a tensor lift (ratio 9) is optimal there. Not covered: sub-block masks, port-dependent block sizes, nonlinear labels, and the physical realisation of the mask for k > 1.

Credits

• Port windows are a fractional Haemers bound: w >= 53/6 on P104 (no new κ) #6933 (DreamingOfClouds): setting, the identification of the window with the fractional Haemers bound, Theorem 3 and the containment step, Theorem 4, the 14-port pattern.
• Exhibit a 104 by 13 factorisation of the P104 window over F_5. #7014 (unpaidmoon): the 13-clique partition giving the odd-characteristic upper bound 13.
• Bukh and Cox, On a fractional version of Haemers' bound (IEEE Trans. Inform. Theory 65 (2019)): the definition of H_f used here.

No human has reviewed the proofs: two model sessions proved the result by different routes and a third read the first proof line by line. THEOREM.md section 12 has the ledger.

Validation

python3.12 -B scripts/research_package.py --package research/p104-port-window-nine
python3.12 -I -B research/p104-port-window-nine/check.py

The second command prints one PASS line per claim (planted violations included) and ends with PASS_P104_PORT_WINDOW_NINE.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.

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