[P1 exact] Prove the #659 five-cell onsets from geometry, without enumerating L=6

[LightChainr]

Parent: #650. Follow-up to PR #669 / #659. No new census. No L=6.

Why

#669 extracted exact integer identities from the committed 4×4 tables and left most of them empirical. Upgrade the cheap ones to combinatorial proofs, and isolate the two-point diamond conjectures.

The reader scripts/wrapping_five_cell_reader.py on PR #669 re-verifies every identity as integers. Use that as the check, not as the proof.

Prove or kill (in this order)

1. I1 (low-k binomial). Axis: n×b(k) = C(N,k) for k < L. Diamond: for k < 2L. The blocking-line reading is already in the #659 [P2 exact] Integer structure of the five wrapping-type cells #669 note; write it as a proof that does not mention the tables except as a check.
2. I2 (high-k binomial). b×n(k) = C(N,k) for k ≥ N−L+1 on both geometries.
3. Axis I3. d0(L) = L (the L full rows) and b×n(2L−1) = L² (one full row plus one full column).
4. Axis I4. Pre-both-wrap deficit C(N,k) − n×b = d0 + d1 for L ≤ k < 2L−1.
5. Diamond I3. d0(2L) = 4·C(2L,4), b×b(2L) = 2L, b×n onset 3L−1 with value 4L². These rest on two sizes. Give a geometric reading or explicitly leave them as two-point conjectures. Do not enumerate diamond L=5 (2^50 is not a ticket).

Also record: M_L irreducible over ℚ at the five committed sizes is a factorization fact, not a wrapping theorem, unless you find a reason it must be. Do not spend the ticket on Galois theory.

Tripwire

A "proof" that uses an unstated fact from one L is not a proof. If you cannot prove an item, write OPEN and point at the #669 falsification kit. Do not enumerate the next size.

No Huawei. No STATUS. Do not close #659 / #640. Draft PR against main.

Wait for ASSIGNED_MACHINE.

Related: #659, #640, #651, PRs #669, #653, #657.

[LightChainr]

ASSIGNED_MACHINE: local

Combinatorial proofs of the #669 onsets. No new census, no L=6, no Huawei. Draft PR against main. Do not edit STATUS. Do not close #659/#640.

[LightChainr]

Draft PR: #692

• I1, I2, axis I3, axis I4: PROVED from geometry (no tables in the proof; reader + tables as checks only). Includes sharpness of both binomial thresholds and the direct white-side barrier arguments on the axis.
• Diamond b×b(2L) = 2L: PROVED (straight diagonal lines; sibling-line white argument).
• Diamond d0(2L): corrected and proved (black side) as L·C(2L,L) — PR #659 [P2 exact] Integer structure of the five wrapping-type cells #669's 4·C(2L,4) fit is killed by the L=2 point (12 ≠ 4); it was a two-point coincidence. Corrected diamond L=5 prediction: 1260, not 840.
• Diamond b×n onset 3L−1/4L²: OPEN per tripwire, but with a geometric reading obtained: full diagonal line + (L−1)-site plug, count 2 × L × 2L, brute-exact at L=2,3, consistent with committed L=4 (64). Left pointed at the #659 [P2 exact] Integer structure of the five wrapping-type cells #669 falsification kit; no enumeration.
• M_L irreducibility recorded as factorization fact, not pursued.

No new census, no L=6, no diamond L=5, no STATUS edits, nothing closed. Seconds-scale brute-force cross-check (scripts/wrapping_onset_proof_checks.py, axis L=3,4 + diamond L=2,3) passes; #669 reader passes 61/61 on this branch.

[LightChainr]

PR: #692

Axis onsets proved; diamond 4·C(2L,4) killed in favour of L·C(2L,L). Ticket stays open. Not merging. #680 waits until this PID exits.

[LightChainr]

Diamond rank-2 onset and equality classification are theorems in stacked draft #710 (also see notes/two-cycle-core-onsets-20260912.md). Do not re-dispatch a 2^50 diamond L=5 census.

The #692 union-size lower bound had the inequality in the wrong direction; replace with the two-cycle-core lemma (wedge or theta; dumbbell excluded). Axis: onset 2L-1, L^2 crosses. Diamond: onset 3L-1, 4L^2 minimizers (2L-line + straight L-1 plug). L=5 is k=14, count 100 by proof. Independent exhaustive L=2,3: zero below onset, counts match. Preserve original notes/history; no new duplicate issue. This does not classify every higher-mass wrapping cell.