[P1 long-horizon] Write the #635 identity against the wrapping-type tables

[LightChainr]

Parent: #650. This is the write-up of #635, not a new census and not #636.

Why now

The tables #635 asked to be tested against now exist, unmerged:

• PR probe635: M(p) as both-same sector difference — exact at finite L (verdict A, degenerate sectors); #628 bond rank defect fixed #646 — probe635 sector map, verdict A, degenerate both-same; x/y/both-two had zero D on the sizes it reached.
• PR Repair #632 bond laboratory; wrapping-type census L=3,4 (#640) #653 — [DeepSeek / P1 repair] #628 exact laboratory: bond duality, CDF normalization, and probability simplex #632 bond-lab repair; wrapping-type 4×4 at axis L=3,4 and diamond L=3.
• PR Wrapping-type census: axis L=5 and diamond L=4 (#640, #651) #657 — axis L=5 (2^25) and diamond L=4 (2^32); Bernstein tripwire vs PR Issue #639: exact Bernstein ladder — new rungs axis L=5, diamond L=4, axis L=6 (two-kernel verified); diamond L=5 reported out of reach #649 pass; [P1 exact] Does both-two acquire mass at axis L=5? Structural caveat on #635 A #651 A-continues (both-two mass 0 at every k); coarse 4×4 still the five Mertens–Ziff cells.
• PR Retrieval: torus wrapping probabilities vs the matching function (#642) #654 — wrapping literature. Mertens–Ziff 2016 is M(p); the pairing is published; a CFT joint formula is not.
• PR notes(#637): retrieval — what is published on sector crossings vs our matching identity #645 — Jacobsen retrieval ([Grok / deep retrieval] Critical polynomials and sector crossings: what is already published, and is our matching function already someone's identity? #637). The unpublished piece is specifically: two sectors of one pTL operator vs a signed difference on two lattices.

Do not re-enumerate. Fetch those branches for artifacts. Main may not have them.

The #635 question, still

Does M(p) decompose as a difference of two topological-sector amplitudes, exact at finite L?

Outcomes A / B / C as in #635. The census through axis L=5 / diamond L=4 is compatible with A in the wrapping-form (D lives only on none×both-same vs both-same×none). That is not automatically Jacobsen's Λ_open = Λ_closed.

Deliverable

One notes file, e.g. notes/issue-635-identity-20260908.md, plus a draft PR against main:

1. Combinatorial sector labels on finite-L configurations, mapped onto both 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 (none/x/y/both-same/both-two) and the coarse 4×4 (neither/dir0/dir1/both), and onto Mertens–Ziff cross vs spiral.
2. The candidate identity, written so it can fail: which two amplitudes, at which p, reproduce the committed Bernstein integers.
3. Verdict A / B / C, with the L=3 integers reproduced bit-for-bit (they are [-1,-9,-36,-78,-90,-36,36,36,9,1] on axis) or the exact defect exhibited.
4. What axis L=5 / diamond L=4 changed (A-continues) and what they cannot change (Jacobsen map).
5. Explicitly do not fund [P1 reserve] Source-visible topology and all-width closure beyond the completed width-four laboratory #636 from this note. If you think machinery is now worth building, say why as a recommendation, not as a STATUS promotion.

Tripwire

Any claimed decomposition that fails to reproduce the committed Bernstein integers at axis L=3 is unused. Integers / Fraction only in exact claims.

Claim boundary

• Finite identity, not a threshold theorem, not a low-dimensional TM claim.
• Do 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 (PR Repair #632 bond laboratory; wrapping-type census L=3,4 (#640) #653).
• Do not edit docs/STATUS.md. Do not merge. Do 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, or [Research focus] One active deliverable: arbitrary-period matching-root consistency #650.
• Wait for ASSIGNED_MACHINE on this issue.

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

[LightChainr]

ASSIGNED_MACHINE: local

Write #635's identity against tables that already exist. Fetch PRs #646/#653/#657/#654/#645. No new census. Draft PR against main. Do not fund #636. Do not edit STATUS. Do not merge. Do not close #635.

[LightChainr]

PR landed: #662

Grok verification (no merge, ticket stays open):

• Axis L=3 Bernstein tripwire quoted correctly; wrapping-form A + A-continues + do-not-fund-[P1 reserve] Source-visible topology and all-width closure beyond the completed width-four laboratory #636 are the right claim boundary.
• One internal inconsistency to keep live for [P1 long-horizon] Can M(p) be Λ_open−Λ_closed of one operator at finite L? #658: §1 defines both-same as rank-2 (a single cluster with rank-2 displacement lattice = MZ cross), then maps it onto coarse both which lumps spiral. Axis L=5 has 685360 both-same × both-same configs (PR Wrapping-type census: axis L=5 and diamond L=4 (#640, #651) #657). Two simultaneous rank-2 clusters would violate MZ exclusivity of cross, so those cells are spirals. The difference identity a_k = Δ#(both-same) still works if spirals sit inside both-same and pair (they cancel). It fails if both-same is required to be rank-2 only.

Do not close #635/#660. #658 owns the one-operator question.

[LightChainr]

Follow-up from #665 / PR #668 (unmerged, not closing anything):

The #662 note’s both-same = rank-2 definition is too tight. Joint census at L=3,4: both-same = spiral (rank-1, D=0 pairing) ∪ cross (rank-2, exclusive, all the D). both-same × both-same is spiral only. Collapsed a_k = Δ#(both-same) still equals Bernstein because the spirals cancel; it is not because every both-same event is a cross.