[pinterbanget]

This answers #4598's open question 1 (@petalas): a structural explanation of the 24 five-rectangle decompositions of J − I₅ over F₂ behind the 2,880 headers. No new κ.

Classification

Under S₅ × transposition there are exactly three orbits, 24 = 2 + 10 + 12:

• Stars (2, giving 240 headers): the row star and the column star.
• Mixed (10, giving 1,200 headers): for a centre c, the rectangle {c} × (V∖c) together with (V∖{c,y}) × {c,y} for each y ≠ c, and the transposes. Column c is covered 1 + 3 times, so these are characteristic-2 only.
• Pentagons (12, giving 1,440 headers): one for each undirected Hamiltonian 5-cycle of K₅. The S sets are the cycle edges, and each T is the pair of outer neighbours of its edge. These have stabiliser D₅, sum to J − I over ℤ, and include Count 2,880 five-request binary threshold headers; refute 2 k! (no new κ) #4598's hand counterexample.

Verification

research/five-request-decomposition-structure/verify.py uses the standard library only and takes about 1 s. It runs its own enumeration and checks that:

• the explicit families partition the 24;
• the orbits are exactly these families;
• the integer and mod-2 sums behave as stated.

Open question

Does a star/centre/cycle pattern extend to k = 6? This needs an enumeration over 602 rectangles.

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

Head: 0f62abf

🤖 Generated with Claude Code