[P1 exact] Spiral cancellation in both-same x both-same is config-wise: cardinality-only via set identity (#674)

[LightChainr]

Closes nothing; deliverable for #674 (parent #650). Draft — do not merge.

Outcome

cardinality-only, sharpened: the equal cardinality is set identity, and D = 0 on both-same × both-same is config-wise — no involution pairing is needed.

1. Colour-flip kill at L=4 (recorded first, as instructed). [120, 416, 448, 128, 8] over k = 8..12 is not palindromic under k ↔ 16−k (mirror masses at k′=7,6,5,4 are all 0). Structural form: a rank-1 both-same spiral needs k ≥ 2L on the primal lattice (min k brute-forced: 6 at L=3, 8 at L=4), so the complement — with black/white and primal/matching roles preserved — is never rank-1 × rank-1. Verified at L=3: all 6 complements land on black none / white both-same rank 2. The L=3 "survival" of colour-flip was a window-position accident.

2. L=3 2^9 dump (Mac-OK). The 6 question-cell configs are k=6 diagonal staircases (occupancy lists in results/issue674-spiral-involution/axis-L3-spiral-dump.json). In every one, either = 1 on both sides: each config carries its own +1 and −1, D(C) = 0 term-by-term.

3. Set identity, not equal counts. Black-side rank-1 both-same and white-side rank-1 both-same are the same configuration set: 6 = 6 = 6 at L=3; 1120 = 1120 = 1120 at L=4 with identical per-k masses. Conjecture (mechanism): matching-lattice self-duality of the diagonal-winding event — verified as identity at L=3,4 only.

4. Fixed-point-free involutions exist but are epiphenomenal. All 33 (L=3) / 56 (L=4) candidate torus self-maps tested: refl-x(2c−x) and refl-y(2c−y) for every c are fixed-point-free k-preserving involutions on the cell (orbits e.g. {0,2},{1,3},{4,5} at L=3; 10 such maps at L=4 incl. translations (0,2),(2,0)). 90° rotations and translations are not involutions on the cell. None is needed for D = 0.

5. Q3: both-two = 0 is independent and structural. An x-wrapping and a y-wrapping cluster have homology-intersection number 1, so they cannot be disjoint — they merge into a single both-same cluster. Exhaustively verified (0 violations): L=2 primal 5/matching 7, L=3 primal 97/matching 265, L=4 primal 10165/matching 37679 configs with x-wrap ∧ y-wrap, all both-same. So A-continues is forced by the mechanism, independent of §3.

Provenance / budgets

• PR [P1 exact] Joint dictionary: rank pair x wrapping-type x 5-names at axis L=3,4 (#665) #668 joint census imported, not recomputed (imported-issue668-*.json + provenance record); ticket integers re-derived bit-for-bit from the import.
• L=4 2^16 fresh stream: 2.6 s — seconds-scale, permitted by the ticket gate.
• L=5: NOT run — NEED_HUAWEI.
• Script scripts/issue674_spiral_involution.py; test tests/test_issue674_spiral_involution.py (axis L=2,3, direct-run, 5/5 pass); notes notes/issue-674-spiral-involution-20260908.md.

Integers only. No docs/STATUS.md edit; no ticket closed. Related: #635, #651, #665, #650, PRs #668, #662.

[LightChainr]

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

Tripwires from #674 hold: colour-flip kill recorded first (L=4 masses not palindromic); L=3 2^9 dump of the 6 staircases; L=4 2^16 in 2.6s (Mac-OK); L=5 not run (NEED_HUAWEI); #668 imported not recomputed; no STATUS.

Science I will keep:

• D=0 is config-wise on both-same × both-same because either=1 on both sides — arithmetic, not a pairing.
• Colour-flip is dead at L=4 and does not even preserve rank-1×rank-1 at L=3 (complements land on black none / white rank-2).
• Q3 is the real theorem: algebraic intersection 1 ⇒ an x-cluster and a y-cluster cannot be disjoint ⇒ both-two=0 structurally, independent of the spiral cell. That is A-continues as a mechanism, checked exhaustively at L=2,3,4.

Caveats:

• axis-L4-spiral-actions.json is ~574KB (56 maps × 1120 configs). Fine as an artifact; not needed to believe D=0.
• scripts/probe635_sector_decomposition.py is vendored into this PR (it may already live on [P1 exact] Joint dictionary: rank pair x wrapping-type x 5-names at axis L=3,4 (#665) #668's branch). Harmless if identical.
• Set identity of black vs white rank-1 both-same at L=3,4 is the census form of [P1 exact] Prove PR #675 Fact 2: white wrapping label is a function of the black label #677 Fact 2's spiral row. Leave [P1 exact] Prove PR #675 Fact 2: white wrapping label is a function of the black label #677 to say whether that row is a theorem.

Do not close #674 / #651 / #665.