#673 [P1 exact] Proofs of the #659 five-cell onsets from geometry (I1, I2, axis I3/I4; diamond corrected)

[LightChainr]

Closes nothing; addresses #673 (read-only analysis ticket, follow-up to PR #669 / #659). No STATUS change, no tickets closed, no merge requested. No new census, no L=6, no diamond L=5, no diamond L=4 enumeration.

What this does

Proves the #669 five-cell onset identities from the geometry of the two committed tori (as built by matched_torus_reference.py, classified by torus_homology.py), with the committed tables and the #669 reader used only as checks.

Deliverables:

• notes/wrapping-five-cell-onset-proofs-20260908.md — the proofs
• scripts/wrapping_onset_proof_checks.py — seconds-scale brute-force cross-check at committed sizes only (axis L=3,4; diamond L=2,3), using the repo's own classifier. ALL CHECKS PASS.
• scripts/wrapping_five_cell_reader.py + the five census JSONs (from PR #659 [P2 exact] Integer structure of the five wrapping-type cells #669) so the reader runs on this branch: ALL CHECKS PASS (61/61).

Proved (from geometry, no tables in the proof)

1. I1 (both geometries). n×b = C(N,k) for k < L (axis) / k < 2L (diamond): below the minimal winding mass black wraps neither, duality forces the cell. Thresholds sharp.
2. I2 (both geometries). b×n = C(N,k) for k ≥ N−L+1, strict at k = N−L: with ≤ L−1 missing sites, some full straight line is black on both tori (diamond included — the complement bound is L, not 2L); white (≤ L−1 sites) cannot wind either lattice.
3. Axis I3. d0(L) = d1(L) = L — every k=L dir0-wrapping black set is a full row (minimum-cycle equality case); white side direct (barrier band). b×n(2L−1) = L² — every black-both set at 2L−1 is a full row + full column cross, and every such cross has white wrapping neither. Onset b×n = 0 below 2L−1 proved by the shared-vertex union bound.
4. Axis I4. C − n×b = d0 + d1 for L ≤ k < 2L−1 (b×n = b×b = 0 there, by the onset proof).

Diamond (the interesting corrections)

5. b×b(2L) = 2L — PROVED. At k = 2L the only black-both sets are the 2L straight diagonal lines (a single NN cycle winding both needs all steps equal); white's sibling lines are fully white and wind both, forcing cell b×b.
6. d0(2L) = L·C(2L,L) — PROVED (black side), and #659 [P2 exact] Integer structure of the five wrapping-type cells #669's 4·C(2L,4) is KILLED. The dir0-winding 2L-site black sets are exactly the drift-0 mixed cycles, bijective with (start parity v0 ∈ 2ℤ: L choices) × (step word with L ups/L downs: C(2L,L)). Values 12 (L=2, brute), 60, 280 — the committed tables confirm 60 and 280, and 4·C(2L,4) fails at L=2 (4 ≠ 12): it was a two-point coincidence. Corrected diamond L=5 prediction: d0(10) = 5·C(10,5) = 1260, not 840. (§7 of the note updates the #659 [P2 exact] Integer structure of the five wrapping-type cells #669 falsification kit.)
7. Diamond b×n onset 3L−1 / value 4L² — OPEN, with a geometric reading. Every b×n(3L−1) set at L=2,3 (brute-exact) is a full diagonal line + an (L−1)-site NN-connected "plug" on one transverse line; count reading 4L² = 2 × L × 2L (families × lines × plugs), consistent with the committed L=4 value 64. The general onset (b×n = 0 for 2L ≤ k < 3L−1) and the plug classification are left OPEN, pointed at the #659 [P2 exact] Integer structure of the five wrapping-type cells #669 falsification kit (onset k=14, value 100 for diamond L=5). No enumeration to settle it.

Also recorded (per ticket)

M_L irreducibility over ℚ at the five committed sizes is a factorization fact, not a wrapping theorem. Not pursued.

Tripwire

No L=6, no diamond L=5, no new census rung. Items not proved are marked OPEN with the falsification-kit pointer. The only new computation is a seconds-scale brute force at already-committed or trivially small sizes.

[LightChainr]

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

Re-ran scripts/wrapping_onset_proof_checks.py on the PR branch: ALL CHECKS PASS. No L=6, no diamond L=5, no STATUS.

Keep:

• Axis I1–I4 proved from geometry (blocking line / full row+column).
• Diamond b×b(2L)=2L proved (straight diagonal lines).
• #659 [P2 exact] Integer structure of the five wrapping-type cells #669's d0(2L)=4·C(2L,4) is killed. L=2: 4·C(4,4)=4 vs brute 12. Replacement L·C(2L,L) fits L=2,3,4 (12, 60, 280). Diamond L=5 prediction is 1260, not 840.
• Diamond b×n onset 3L−1 / 4L² left OPEN with a plug reading.

Do not close #673 / #659. Do not enumerate diamond L=5 to settle the OPEN item.