[pinterbanget]

This answers k = 5 of Open next question 2 in #4275. Take five pairwise orthogonal rank-one requests on F₂ᵐ and a uniform header in GL_m(F₂). Then the five full-guard events are strictly below the product of their marginals for every m ≥ 5, with deficit 1 − ρ_m = 20(7/9)^{m−1} + O((19/21)^{m−1}(7/9)^{m−1}). No new κ.

Claim type: a scoped counting proof with an exact finite component, following petalas's #4599 method at k = 4. PROOF.md has the argument.

Evidence

research/five-event-guard-correlation/verify.py uses the standard library only and runs in about 5 s.

• Exact signed spectrum over all 2²⁰ characters of the zero-diagonal 5×5 binary matrices. It has 62 bases, all positive: N_n = 243ⁿ − 20·189ⁿ + 60·171ⁿ − 40·165ⁿ + …. The second base is Prove the full-guard header law for k corank-one requests with orthogonal kernels: binary threshold m ≥ k+1 (no new κ) #4275's 7·3^{k−2} with multiplicity k(k−1).
• Cross-checks of the table that do not use characters:
  • direct meet-in-the-middle counts at n = 4, 5, 6;
  • the 2,880 complete headers at m = 6 (Count 2,880 five-request binary threshold headers; refute 2 k! (no new κ) #4598, Independently recount #4598's binary threshold headers by exact pair convolution: 2/12/48/2,880 for k=2..5, brute-force-checked on GL_3/GL_4 (no new κ) #4605);
  • the marginal law at k = 1;
  • the pair reduction against a brute force over GL₃ and GL₄;
  • a k = 4 positive control;
  • detection of a one-unit table mutation.
• Coverage:
  • for 5 ≤ m ≤ 118, exact rational comparisons;
  • for m ≥ 118, a uniform bound from exact integer inequalities (1,048,555·171¹¹⁷ ≤ 9·189¹¹⁷; H(X) − 1 < 85/X ≤ (7/9)ⁿ);
  • so no interval is left unchecked.
• Values. ρ₆ = 0.0056, ρ₂₀ = 0.881 and ρ₁₁₈ = 1 − 3×10⁻¹². The deficit ratio is 19.9995 at m = 118.

Limitations

• Scope. Only k = 5 is covered: orthogonal kernels, F₂ and complete equally weighted headers.
• Not claimed. A producer, selection, emission, dispatch, stock, precision, a supplier or a κ change. There has been no proof-kernel review.

Open next questions

• All k ≥ 4. First step: the k = 6 spectrum over 2³⁰ characters, which needs a vectorised or symmetry-reduced transform.

Credit: petalas (Nick Petalas) for #4275, #4599 and #4598. Ryanson Jonathan, with Claude Code (Claude Opus 5.5) assistance.

Head: 00f4eeb

Merge Danger

Door: two-way; this is an isolated research package. Blast radius: local.

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