[pinterbanget]

Independent recount of petalas's #4598 (head c80f445e), which counts the five-request binary threshold headers. It agrees: 2,880 at k = 5. No new κ.

Method (research/pr4598-header-recount/verify.py, standard library, about 9 s). It deliberately avoids #4598's route: no J − I decomposition, no rank/idempotence argument, no rectangle-subset search and no imports from #4598.

• Guard condition. G P_t G⁻¹ = I − g_t h_tᵀ, so each minor is 1 − Σ_{i≤j} h_t[i] g_t[i].
• Headers as pairs. Headers correspond to pairs (X, Y) with YX = I_k.
• Count. It is an exact ordered convolution of k+1 rank-≤1 increments. Each increment is weighted by its factor pairs, the first k must have a zero diagonal, and the total must be I_k. The convolution is evaluated by meet-in-the-middle.

Results

k	2	3	4	5
passing headers	2	12	48	2,880

• Brute force from the leading-minor definition over all of GL₃ and GL₄ agrees. The GL₄ check uses both coordinate requests and a non-coordinate orthogonal family.
• Failure control. Dropping the order-3 guard at k = 3 gives 252 by both routes.
• Zero increments contribute 0 at every k. This independently confirms Count 2,880 five-request binary threshold headers; refute 2 k! (no new κ) #4598's no-zero-increment step.
• Density 1/6,999,552. The 2·k! pattern (240) fails at k = 5.

Limitations. This covers only the stated header problem, and brute force stops at GL₄. #4598's 24-subset listing and its hand counterexample are not re-checked.

Open question: k = 6, which needs a mod-p Walsh–Hadamard version over 2^36 cells.

Credit: petalas (Nick Petalas) for #4598 and #4275; Cinna Davis for the request context. Ryanson Jonathan, with Claude Code (Claude Opus 5.5) assistance.

Head: fe99331

🤖 Generated with Claude Code