[P1 long-horizon] Rewrite the #635 identity after #668: both-same is not rank-2

[LightChainr]

Parent: #650. Follow-up to PR #662 (from #660) and PR #668 (from #665). Notes-first. Mac-only.

Why this ticket exists

PR #662 wrote wrapping-form A treating both-same as the rank-2 (cross) class, then mapped it onto coarse both. PR #668's joint census killed that identification:

• both-same = rank-1 spiral ⊎ rank-2 cross
• both-same × both-same is exactly (r_b, r_w) = (1,1) (6 at L=3, 1120 at L=4)
• rank-2 lives only in exclusive-cross cells: both × neither = (2,0) and neither × both = (0,2)
• collapsed a_k = \Delta#(both-same) still equals Bernstein because spirals cancel, not because every both-same event is a cross

#662 is therefore a correct numerical identity with a wrong dictionary. Rewrite the identity so a later reader cannot confuse the two.

Questions

1. Write wrapping-form A' 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, with D-carrying cells named as exclusive crosses.
2. State the spiral-cancellation lemma that makes \Delta#(both-same) = a_k still true. Prove it from the L=3,4 integers, or mark it a conjecture for all L.
3. Does A-continues ([P1 exact] Does both-two acquire mass at axis L=5? Structural caveat on #635 A #651, axis L=5 / diamond L=4, both-two = 0) survive the rewrite, or did it secretly depend on the wrong dictionary?
4. Do not fund [P1 proof / bounded CPU] Observable spectral remainder and width uniformity after the exact n=2 bridge #636. If the rewritten identity still does not need a transfer matrix, say so.

Constraints

• Read PR notes(#635/#660): the #635 identity against the wrapping-type tables #662, [P1 exact] Joint dictionary: rank pair x wrapping-type x 5-names at axis L=3,4 (#665) #668, probe635: M(p) as both-same sector difference — exact at finite L (verdict A, degenerate sectors); #628 bond rank defect fixed #646, Repair #632 bond laboratory; wrapping-type census L=3,4 (#640) #653, Wrapping-type census: axis L=5 and diamond L=4 (#640, #651) #657 notes/JSON. Do not re-enumerate L=4. Dumping the 6 L=3 spiral configs (2^9) is Mac-OK if you need occupancy lists.
• Axis L=5 joint census is NEED_HUAWEI; do not run it. If you need that rung, comment NEED_HUAWEI and stop.
• Integers only in exact artifacts. No docs/STATUS.md, no 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 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.

Deliverable

notes/issue-635-identity-rewrite-YYYYMMDD.md + draft PR against main.

Wait for ASSIGNED_MACHINE.

Related: #635, #660, #665, #651, PRs #662, #668, #646.

[LightChainr]

ASSIGNED_MACHINE: local

Notes-first rewrite of wrapping-form A after the #668 dictionary. No Huawei. No L=5 joint census. Draft PR against main. Do not edit STATUS. Do not merge. Do not close #635/#660/#665.

[LightChainr]

Read PR #675 (notes/sector-obstruction-20260908.md) before rewriting A. Claimed finite-L dictionary is now M_L = A^{cross}_{NN} − A^{none}_{NN} with white labels a deterministic function of black labels. Combine that with PR #668 (spirals = both-same×both-same = (1,1)). Still do not fund #636. Still no L=5 joint census on the Mac.

[LightChainr]

Draft PR: #676

Notes-first rewrite of the #635 identity in the #668 joint language (notes/issue-635-identity-rewrite-20260908.md):

• A′: D lives exactly on the exclusive-cross cells neither × both = (0,2) (−1) and both × neither = (2,0) (+1); both-same × both-same is (1,1) pure spiral, D = 0 (6 at L=3, 1 120 at L=4).
• Spiral-cancellation lemma: 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 via the bit-for-bit Bernstein collapse; conjecture for all L (Conjectures 3.1/3.2, structural vs per-L parts separated).
• A-continues survives: the 5-name/coarse projection is rank-blind; nothing in [P1 exact] Does both-two acquire mass at axis L=5? Structural caveat on #635 A #651's axis L=5 / diamond L=4 verdict depended on the wrong dictionary.
• [P1 proof / bounded CPU] Observable spectral remainder and width uniformity after the exact n=2 bridge #636 not funded; axis L=5 joint census NEED_HUAWEI — not run.

Integers only; no docs/STATUS.md edit; #635/#660/#665 left open.

[LightChainr]

PR: #676

A′ rewrite landed as notes. Ticket stays open. Not merging. Local clone removed after the branch was on origin at 0bb6ca5.