notes(#635/#660): the #635 identity against the wrapping-type tables

[LightChainr]

Write-up deliverable for #660 (parent #650). One new file: notes/issue-635-identity-20260908.md. No new census; consumes the unmerged artifacts of PR #646, #653, #657, #654, #645.

The five deliverable bullets

1. Sector labels — the probe635: M(p) as both-same sector difference — exact at finite L (verdict A, degenerate sectors); #628 bond rank defect fixed #646 5-name labels (none/x/y/both-same/both-two, combinatorial on finite-L configurations via cluster displacement lattices) mapped onto the Repair #632 bond laboratory; wrapping-type census L=3,4 (#640) #653 coarse 4×4 (neither/dir0/dir1/both, which folds both-same and both-two into both) and onto Mertens–Ziff cross vs spiral (via the PR Retrieval: torus wrapping probabilities vs the matching function (#642) #654 identification: the repository's x=either difference equals the cross difference by MZ (20)).
2. Candidate identity, written so it can fail — a_k = #(black ∈ both-same, k) − #(white ∈ both-same, k); acceptance gate is the committed axis L=3 Bernstein integers, re-run this session and reproduced bit-for-bit: [-1,-9,-36,-78,-90,-36,36,36,9,1] (scripts/exact_matching_polynomial.py --geometry axis --L 3). The two amplitude rows sum to the reference exactly (PR probe635: M(p) as both-same sector difference — exact at finite L (verdict A, degenerate sectors); #628 bond rank defect fixed #646 script re-run); M(1/2) = −21/64, u_c = 43/128 re-confirmed by Fraction arithmetic.
3. Verdict A at finite L (wrapping-form) on every size reached: axis L=3,4,5; diamond L=2,3,4. Not B (no finite-L defect exists to exhibit), not C (MZ 2016 proves the wrapping-form at every finite L).
4. What axis L=5 / diamond L=4 changed — degeneracy caveat upgraded to A-continues (both-two mass 0 at every k on 2²⁵ and 2³²; coarse 4×4 still the five MZ cells) — and what they cannot change: the Jacobsen map (PR notes(#637): retrieval — what is published on sector crossings vs our matching identity #645), which is an m→∞-first eigenvalue statement about one pTL operator, not reachable by more enumeration; the crux (matching-lattice content vs open/closed sectors of one operator) is stated as the open gap.
5. [P1 reserve] Source-visible topology and all-width closure beyond the completed width-four laboratory #636 not funded — the wrapping-form of A is published (MZ 2016); the unpublished piece is the sector identification, and degeneracy, if exact, would make sector machinery carry less structure than its cost. Recommendation recorded in the note only, not as a STATUS promotion.

Boundaries

• Tripwire honored: integers/Fraction only in exact claims.
• Does not cite #622 probe: invariant shape of the threshold law — S defined, W1–W5 adjudicated, g separated #628's 118133 bond dual_fail as physics (repaired compound artifact, PR Repair #632 bond laboratory; wrapping-type census L=3,4 (#640) #653/probe635: M(p) as both-same sector difference — exact at finite L (verdict A, degenerate sectors); #628 bond rank defect fixed #646).
• Does not edit docs/STATUS.md; does not close [P1 long-horizon] Is M(p) the difference of two topological sector amplitudes? Prove or kill the map before anyone builds a transfer matrix #635, [P1 exact] Wrapping-type-resolved census: the exact table #635 has to test any sector decomposition against #640, [P1 exact] Does both-two acquire mass at axis L=5? Structural caveat on #635 A #651, [Research focus] One active deliverable: arbitrary-period matching-root consistency #650; draft, not merged.

Related: #635, #637, #640, #642, #646, #651, #653, #654, #657, #660.

[LightChainr]

Verification comment, not a merge request.

The L=3 Bernstein gate and the A / A-continues / not-#636 boundary are in good shape.

Please keep this distinction explicit in a follow-up commit if you touch the note: both-same as rank-2 (cross) vs both-same as coarse both (cross ∪ spiral). The 685360 both-same × both-same configs at axis L=5 are the spiral pairing; they are why the difference Δ#(both-same) can still equal collapsed D even though not every both-same event is a cross. Ticket stays open.