Rewrite the #635 identity after #668: both-same is not rank-2 (#672)

[LightChainr]

Deliverable for #672 (parent #650; follow-up to PR #662 from #660 and PR #668 from #665). Draft — do not merge.

What

One new file: notes/issue-635-identity-rewrite-20260908.md. No new census; consumes the unmerged artifacts of PRs #662, #668, #646, #653, #657.

The four ticket questions

1. Wrapping-form A′ written in the joint (rank pair × wrap × 5-name) language of [P1 exact] Joint dictionary: rank pair x wrapping-type x 5-names at axis L=3,4 (#665) #668. The D-carrying cells are named as exclusive crosses: neither × both = (0,2) (D=−1, 259 at L=3 / 36 559 at L=4) and both × neither = (2,0) (D=+1, 91 / 9 045). both-same × both-same is exactly (1,1) — pure spiral, D=0 (6 at L=3, 1 120 at L=4).
2. Spiral-cancellation lemma: a_k = Δ#(both-same) = Δ#(exclusive cross) still holds because D never fires on a (1,1) rank pair. Proved from the [P1 exact] Joint dictionary: rank pair x wrapping-type x 5-names at axis L=3,4 (#665) #668 integers at axis L=3,4 (bit-for-bit Bernstein collapse [-1,-9,-36,-78,-90,-36,36,36,9,1] / the committed L=4 rung); marked conjecture for all L (Conjectures 3.1/3.2), with the structural half (r_b + r_w = 2 forbids rank-2 × rank-2 at any L) separated from the genuinely per-L wrap-cell support claim.
3. A-continues ([P1 exact] Does both-two acquire mass at axis L=5? Structural caveat on #635 A #651) survives the rewrite. It is a statement about the 5-name/coarse projection, which discards exactly the rank column the wrong dictionary misidentified; the verdict letter and the Bernstein collapse are rank-blind. Axis L=5 / diamond L=4 facts (PR Wrapping-type census: axis L=5 and diamond L=4 (#640, #651) #657) are unchanged under A′.
4. [P1 reserve] Source-visible topology and all-width closure beyond the completed width-four laboratory #636 not funded — the rewrite strengthens notes(#635/#660): the #635 identity against the wrapping-type tables #662's §5: the sector amplitudes are not rank-pure ([P1 exact] Joint dictionary: rank pair x wrapping-type x 5-names at axis L=3,4 (#665) #668), so any future sector-level TM reading must first split both-same by rank, an extra cost the old dictionary hid.

Boundaries

• Integers only in exact artifacts; all integers quoted from committed PR artifacts, nothing re-enumerated. Axis L=5 joint census NEED_HUAWEI — not run.
• No docs/STATUS.md edit; 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 long-horizon] Write the #635 identity against the wrapping-type tables #660/[P1 exact] Joint dictionary: rank pair (r_b, r_w) × wrapping-type 4×4 at axis L=3,4 #665/[P1 long-horizon] Rewrite the #635 identity after #668: both-same is not rank-2 #672; draft, not for merge without review.

Related: #635, #636 (not funded), #650, #651, #660, #665; PRs #645, #646, #649, #653, #654, #657, #662, #668.

[LightChainr]

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

Notes-only, no new census, no L=5 joint, no STATUS, #636 not funded. The four #672 questions are answered:

1. A′ names D-carrying cells as exclusive crosses (neither×both)=(0,2) and (both×neither)=(2,0).
2. Spiral-cancellation lemma: D=0 on (1,1) / both-same×both-same is arithmetic; a_k = Δ#(both-same) still equals Bernstein because spirals cancel. Exact at L=3,4 from [P1 exact] Joint dictionary: rank pair x wrapping-type x 5-names at axis L=3,4 (#665) #668 integers; Conjectures 3.1/3.2 for all L, with the structural half (r_b+r_w=2 forbids rank-2×rank-2) separated from wrap-cell support.
3. A-continues survives: it lives on the 5-name/coarse projection, which discards the rank column the old dictionary got wrong.
4. [P1 reserve] Source-visible topology and all-width closure beyond the completed width-four laboratory #636 still not funded; the rewrite makes a future sector TM more expensive (must split both-same by rank).

Related: PR #675's black-only collapse D = 1{black cross} − 1{black none} is the same exclusive-cross statement written on one colour. This note did not have to re-derive it; the two dictionaries should be read together.

Do not close #672 / #635 / #660 / #665.