[P1 exact] Prove PR #675 Fact 2: white wrapping label is a function of the black label

[LightChainr]

Parent: #650. PR #675 Fact 2 (claimed exact at axis L=2,3,4 and diamond L=2,3):

black none   ⇒ white cross
black cross  ⇒ white none
black x      ⇒ white x
black y      ⇒ white y
black spiral ⇒ white spiral

No other joint cell has mass. The none ↔ cross half is Mertens–Ziff's torus pairing at the event level; the x ↔ x, y ↔ y, spiral ↔ spiral rows are the new claim.

Why

If this is a theorem, A′'s five-cell support is forced (not empirical), and #673's onset proofs become corollaries of a label map plus a blocking-line count. If it is only a small-L census, say so and give the first L that could kill a row.

Questions

1. Prove none ↔ cross from digital Alexander + r_b + r_w = 2 + the definition of cross (rank-2). This should not need a table.
2. Prove or kill x ↔ x and y ↔ y: a single-axis black wrap forces a single-axis white wrap on the same axis. A 90° argument plus matching adjacency is the expected route.
3. Prove or kill spiral ↔ spiral. This is the row that makes both-same × both-same exist and carry D = 0.
4. Use axis L=3,4 JSON (PR [P1 exact] Joint dictionary: rank pair x wrapping-type x 5-names at axis L=3,4 (#665) #668) and the coarse tables (PR 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) as checks, not as the proof. Diamond L=3 2^18 is Huawei; do not re-enumerate it. If you need axis L=5 joint rank, comment NEED_HUAWEI and stop.

Outcomes: theorem / theorem-with-caveat / census-only.

Constraints

No STATUS. Do not close #658 / #665 / #672. Do not fund #636. Draft PR against main.

Wait for ASSIGNED_MACHINE.

Related: #658, #665, #672, #673, PRs #668, #675, #676, MZ 2016 eqs. (9)–(11), (19)–(20).

[LightChainr]

ASSIGNED_MACHINE: local

Prove or kill Fact 2 from geometry. PR #668 JSON is the check, not the proof. Do not re-enumerate diamond L=3. Axis L=5 joint is NEED_HUAWEI. Draft PR against main. No STATUS. Do not close #658/#665/#672.

[LightChainr]

First CLI hit --max-turns 80 with no PR. Retrying locally with 120 turns. Still no Huawei, no L=5 joint census, draft against main.

[LightChainr]

PR: #690

Verdict theorem-with-caveat. Ticket stays open. Not merging. No new dispatch until #673/#679 (and this PID) exit.

[LightChainr]

Draft PR: #690

Verdict: theorem-with-caveat.

• Q1 none ↔ cross: theorem, every L, both geometries, config-wise, no table. Digital-Alexander duality (main: white winding image = symplectic orthogonal of black) + r_b + r_w = 2 + the both-two ≡ 0 intersection theorem (PR [P1 exact] Spiral cancellation in both-same x both-same is config-wise: cardinality-only via set identity (#674) #683 §4) close the row completely, label level included.
• Q2 x ↔ x, y ↔ y: theorem conditional on Conjecture 3.1 (directional duality (d_0,d_1)(G_b) = (d_0,d_1)(G_w) config-wise). Note documents exactly where the one-line symplectic argument fails (it reads off the complementary generator) and why the 90° + matching-adjacency route is required. Conjecture verified config-wise, zero violations: axis L=2,3,4, diamond L=2 (fresh exact checker in the PR); aggregate five-cell support verified from PR Wrapping-type census: axis L=5 and diamond L=4 (#640, #651) #657 at axis L=5 and diamond L=4.
• Q3 spiral ↔ spiral: theorem for the structural half, census-only for the set-identity mechanism (complement of a primal diagonal staircase is a matching-lattice staircase — verified axis L=2,3,4, diamond L=2 here; set identity at L=3,4 per PR [P1 exact] Spiral cancellation in both-same x both-same is config-wise: cardinality-only via set identity (#674) #683). Axis L=5 config-level NEED_HUAWEI.
• Five-cell support is a theorem at every L modulo Conjecture 3.1, so A′ (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.

Artifacts: one notes file (notes/issue-677-fact2-label-map-20260908.md), exact checker + JSON (axis L=2,3,4 + diamond L=2, Bernstein tripwires reproduced bit-for-bit). Diamond L=3 (2^18) not re-enumerated; no STATUS edit; no tickets closed; #636 not funded.

[LightChainr]

Completed by merged #702 (notes/exact-foundations-correction-20260912.md §1). For the existing digital-Alexander theorem's honest-torus scope, C=A^perp and every nonzero line in the symplectic plane is its own orthogonal. Therefore the rank-(1,1) images have the same rational line: x↔x, y↔y, spiral↔spiral, configurationwise at all sizes. Rank zero/two give none↔cross. This is an allowed-support theorem, not positive mass of all five cells at every size, and not equality of integer subgroups. The added direction conjecture in #690 is unnecessary; its unqualified all-configuration version is false at all-black/all-white. No larger census is needed for this question. Closing as completed under the owner's project-control delegation; data and source branches retained.