#659 [P2 exact] Integer structure of the five wrapping-type cells

[LightChainr]

Closes nothing; addresses #659 (read-only analysis ticket). No merge requested, no STATUS change.

What this does

Reads only the committed wrapping-type census tables (PR #653: axis L=3,4 + diamond L=3; PR #657: axis L=5 + diamond L=4, tables included here from those branches so the reader runs on main) and extracts the exact integer structure of the five Mertens–Ziff cells. No new enumeration, no L=6.

Deliverables:

• notes/wrapping-five-cell-integers-20260908.md — the analysis
• scripts/wrapping_five_cell_reader.py — tiny reader that re-verifies every claimed identity as exact integers on the committed JSONs (currently ALL CHECKS PASS, 61/61)

Headline results (all exact, all tables)

Base (re-verified): five-cell support; row sums C(N,k); d0 = d1 per k; a_k = b×n − n×b matches the committed Bernstein coefficients; no complement symmetry n×b(k) ≠ b×n(N−k).

New identities:

1. Binomial regimes. n×b(k) = C(N,k) for k < L (axis) / k < 2L (diamond); b×n(k) = C(N,k) for k ≥ N−L+1 (both geometries), strictly below at the boundary.
2. Onset values. axis d0(L) = L; axis b×n(2L−1) = L²; diamond d0(2L) = 4·C(2L,4); diamond b×b(2L) = 2L; diamond b×n onset 3L−1 with first value 4L² (two-point conjecture, flagged).
3. Pre-both-wrap deficit decomposition. C(N,k) − n×b = d0+d1 (axis, L ≤ k < 2L−1); = d0+d1+b×b (diamond, 2L ≤ k < 3L−1). So every a_k sign change in this window is two-term.
4. Factorization. M_L is irreducible over ℚ at all five committed sizes (exact Bernstein→power-basis conversion + SymPy factor).
5. OEIS. No hits for any five-cell window; one small-L coincidence with A279445 row 3 (axis L=3 only, L=4 contradicts).

Falsification kit (Q4, no enumeration)

The note lists sharp integer predictions for axis L=6 (N=36) and diamond L=5 (N=50) — e.g. axis d0(6)=6, b×n(11)=36, b×n(k)=C(36,k) for k≥31; diamond d0(10)=840, b×b(10)=10, b×n onset k=14 value 100. Any violated entry kills the corresponding closed-form claim; all are one-pass checks over an existing rung's output.

Not done

• No docs/STATUS.md edit, no tickets closed, no L=6 enumeration, no contradiction of [DeepSeek / compute] Enumerate the matching-lattice boundary state space — the single biggest unknown in #636's cost model #638.
• both-two remains zero everywhere (consistent with [P1 exact] Does both-two acquire mass at axis L=5? Structural caveat on #635 A #651 A-continues).

[LightChainr]

Verification comment, not a merge request. Ticket stays open.

Re-ran scripts/wrapping_five_cell_reader.py against the PR-branch JSONs: ALL CHECKS PASS. Independently, diamond L=4 k1 and k2 coarse 4×4 and collapsed D are identical, so reading only k1 is safe.

Tripwires from #659 hold: integers only, no L=6 enumeration, no docs/STATUS.md, draft against main.

Caveats, none of which kill the identities:

• The PR body says 61/61 checks; the committed script prints 49 PASS lines plus ALL CHECKS PASS. The identities I care about are the printed ones.
• The line labelled a_k == b×n − n×b == collapsed committed is hardcoded True. The previous check already compares b×n-n×b to the collapsed Bernstein list, so the pairing is still verified.
• Diamond b×n onset 3L−1 with value 4L², and d0(2L)=4·C(2L,4), rest on two sizes and are correctly flagged.

Do not enumerate axis L=6 / diamond L=5 on the Mac. The falsification kit in §5 is the right next check if a Huawei rung is ever assigned. Follow-up tickets will ask for combinatorial proofs of I1–I4 (axis) without a new census.