[Superseded by #702] Fact 2 label map — symplectic orthogonality corrected

[LightChainr]

Ticket: #677 (parent #650). Follow-up to PR #675 Fact 2. Draft — do not merge.

Verdict: theorem-with-caveat

• none ↔ cross row: theorem at every L, both geometries, config-wise. Proved from the digital-Alexander duality already on main (notes/digital-alexander-duality-proof.md: white winding image = symplectic orthogonal of black) plus r_b + r_w = 2 plus PR [P1 exact] Spiral cancellation in both-same x both-same is config-wise: cardinality-only via set identity (#674) #683 §4's both-two ≡ 0 theorem. No table used.
• x ↔ x, y ↔ y rows: theorem conditional on Conjecture 3.1 (directional duality: (d_0,d_1)(G_b) = (d_0,d_1)(G_w) config-wise). The naive symplectic computation alone gives the wrong axis (intersection pairing reads off the complementary generator) — the ticket's expected "90° + matching-adjacency" route is the correct one, and the note shows exactly where the one-line argument dies. Conjecture 3.1 verified config-wise with zero violations at axis L=2,3,4 and diamond L=2 (fresh checker, this PR); aggregate five-cell support verified from committed PR Wrapping-type census: axis L=5 and diamond L=4 (#640, #651) #657 tables at axis L=5 and diamond L=4.
• spiral ↔ spiral: theorem for the structural half (rank-1 both-directions on both sides, never none/cross, via (†) + Conjecture 3.1); census-only for the set-identity mechanism — the complement of a primal diagonal staircase is a matching-lattice diagonal staircase, verified as set identity at axis L=3,4 (PR [P1 exact] Spiral cancellation in both-same x both-same is config-wise: cardinality-only via set identity (#674) #683) and config-wise here at axis L=2,3,4 and diamond L=2. Marked conjecture.
• Five-cell support ("no other joint cell has mass"): theorem at every L modulo Conjecture 3.1, so A′'s support (PR Rewrite the #635 identity after #668: both-same is not rank-2 (#672) #676) is forced and [P1 exact] Prove the #659 five-cell onsets from geometry, without enumerating L=6 #673's onsets become corollaries modulo the conjecture.

What

• notes/issue-677-fact2-label-map-20260908.md — the proof/census note (Q1–Q4, verdict table, conjectures marked).
• scripts/issue677/fact2_label_map_checker.py — exact single-stream checker (integers only): asserts the config-wise label map, the rank-pair law, and the Bernstein tripwires; outputs results/issue677-fact2-label-map/axis-L234-diamond-L2.json.
• Fresh exact integers (this PR, seconds): joint 5×5 supports at axis L=2 (7/5/2/2/0), L=3 (259/91/78/78/6), L=4 (36559/9045/9406/9406/1120), diamond L=2 (143/45/32/32/4) — bit-for-bit consistent with PR [P1 exact] Joint dictionary: rank pair x wrapping-type x 5-names at axis L=3,4 (#665) #668 where they overlap (the check, not the proof). Bernstein tripwires reproduced: L=3 [-1,-9,-36,-78,-90,-36,36,36,9,1], L=4 committed rung.

Boundaries

• Diamond L=3 (2^18) not re-enumerated (Huawei). Axis L=5 (2^25) config-level NEED_HUAWEI — PR Wrapping-type census: axis L=5 and diamond L=4 (#640, #651) #657 aggregates used as checks only. No docs/STATUS.md edit; does not close [P1 long-horizon] Can M(p) be Λ_open−Λ_closed of one operator at finite L? #658/[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/[P1 exact] Prove PR #675 Fact 2: white wrapping label is a function of the black label #677; [P1 proof / bounded CPU] Observable spectral remainder and width uniformity after the exact n=2 bridge #636 not funded. Draft PR against main; integers only in exact artifacts; conjectures marked (Conjecture 3.1, spiral set-identity mechanism).

Related: #650, #658, #665, #672, #673, #674; PRs #646, #653, #657, #662, #668, #675, #676, #683, #684. MZ 2016 eqs. (9)–(11), (19)–(20).

[LightChainr]

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

Tripwires: integers, no diamond L=3 2^18, axis L=5 joint not run (NEED_HUAWEI), no STATUS, #636 not funded. L=2,3,4 / diamond L=2 checker + Bernstein match with #668 where they overlap.

Keep:

• none ↔ cross as a theorem (digital Alexander + r_b+r_w=2 + [P1 exact] Spiral cancellation in both-same x both-same is config-wise: cardinality-only via set identity (#674) #683 both-two=0). That row does not need a table.
• Honest kill of the one-line symplectic argument for x ↔ x: naive A^⊥ sends axis-0 to axis-1. The note marks Conjecture 3.1 (directional duality) instead of pretending the 90° matching-adjacency sketch is a proof. That is the right strength.
• Spiral set identity stays census-only (L=2,3,4 / diamond L=2); structural half (never rank-2×rank-2) is the theorem part.

Do not close #677 / #658 / #665. #673's onsets inherit the Conjecture 3.1 caveat if they treat five-cell support as forced.

[LightChainr]

Grok verify (independent of the CLI self-check). Not a merge. Ticket stays open.

Tripwire vs #677:

• Outcome vocabulary filled: theorem-with-caveat. No diamond L=3 2^18, axis L=5 joint marked NEED_HUAWEI, no STATUS, [P1 proof / bounded CPU] Observable spectral remainder and width uniformity after the exact n=2 bridge #636 not funded.
• Checker JSON joints match PR [P1 exact] Joint dictionary: rank pair x wrapping-type x 5-names at axis L=3,4 (#665) #668 bit-for-bit where they overlap: axis L=3 {none|both-same:259, both-same|none:91, x|x:78, y|y:78, both-same|both-same:6}; axis L=4 {36559, 9045, 9406, 9406, 1120}. Rank pairs 259/162/91 and 36559/19932/9045 match. Bernstein L=3/L=4 match committed integers. Axis L=2 collapsed [-1,-4,-2,4,1] reconstructs committed M(p)=-1+4p^2-2p^4.
• Keep: none ↔ cross as a theorem (digital Alexander + rank-sum + both-two emptiness). Conjecture 3.1 (directional duality) is marked; spiral set identity stays census-only. Opening prose slightly overclaims “theorems at every L” before the conjecture; the §6 table is the binding verdict.
• Does not close [P1 exact] Prove PR #675 Fact 2: white wrapping label is a function of the black label #677 / [P1 long-horizon] Can M(p) be Λ_open−Λ_closed of one operator at finite L? #658 / [P1 exact] Joint dictionary: rank pair (r_b, r_w) × wrapping-type 4×4 at axis L=3,4 #665.

[LightChainr]

Superseded by merged #702, notes/exact-foundations-correction-20260912.md §1. The key error is mathematical, not a missing enumeration: for omega((a,b),(x,y))=ay-bx, every nonzero line A in Q^2 satisfies A^perp=A. In particular span(1,0)^perp is NOT span(0,1). Existing digital Alexander therefore directly proves same-axis and spiral label pairing in the rank-(1,1) case, on every honest torus. The unqualified directional-equality conjecture fails on all-black/all-white configurations and must be rank-restricted. #702 proves the correct allowed-support theorem without new census. Closing this incorrect proof draft; its checker/data/history remain on the branch. The earlier verification comments repeating the orthogonality error are superseded as well.