[pinterbanget]

Independent check of petalas's #4599 (head 19ca650c), the four-request full-guard product inequality. It agrees. No new κ.

Route (research/pr4599-four-event-check/verify.py, standard library, about 7 s). It uses no #4275 character law, no #4227 marginal formula and no imports from #4599.

• Passing headers are counted directly through pairs (X, Y) with YX = I_k and zero-diagonal prefix sums.
• The count comes from an exact dynamic programme over zero-diagonal k×k matrices.
• Each such pair extends to |GL_{m−k}| headers.

Results for every 4 ≤ m ≤ 85:

• Marginal law. Pr(E₁) = s_m.
• Product inequality. The joint probability is strictly below the marginal⁴. At m = 5 the values are 1/208,320 against 81/496.
• Spectrum table. Prove the four-request binary full-guard product inequality (no new kappa) #4599's printed signed spectrum reproduces the direct joint count at all 82 dimensions. A one-unit mutation of the table is detected.
• §3 constants re-checked: the tail sum 4,083 with base 57, the four exact power inequalities, and the H(X) identity.
• Leading deficit. (1−ρ)/(7/9)ⁿ = 11.9947 at m = 85, consistent with the limit 12.
• Brute force. Over GL₃ and GL₄, for k = 1, 3 and 4, it matches the dynamic programme.

Limitation. For m ≥ 86 the theorem still rests on #4599's §3 bound. This package independently supports its input table at 82 consecutive dimensions, but does not prove that the table is the exact transfer spectrum.

Open question: the k = 5 inequality, using the same programme over 2²⁰ states.

Credit: petalas (Nick Petalas) for #4599, #4275 and #4227. Ryanson Jonathan, with Claude Code (Claude Opus 5.5) assistance.

Head: 3bd2fdd

🤖 Generated with Claude Code