[P1 exact] Is there an involution that pairs the rank-1 spirals so they cancel in D?

[LightChainr]

Parent: #650. Follow-up to PR #668. This is the remaining mechanism of A-continues, not another census.

Why

#668: D = 0 on both-same × both-same because both sides are rank-1 spirals (either = 1 on both). Cancellation in a_k = \Delta#(both-same) then needs either equal spiral mass on the two colourings, or an involution that pairs them.

Exact integers already in PR #668:

• L=3: 6 configs, all (1,1), all at k=6
• L=4: 1120 configs, window k=8..12 with masses [120, 416, 448, 128, 8]

Questions

1. Is there a lattice involution (colour-flip, 90° rotation, translation, reflection, or a composite) that pairs every rank-1 spiral with another rank-1 spiral and explains D=0 without counting?
2. Colour-flip kill at L=4: the L=3 mass sits at k=6=N/2, so colour-flip preserves k. At L=4 the window is k=8..12, not concentrated at N/2=8, and [120, 416, 448, 128, 8] is not palindromic under k ↔ 16−k. Record that kill first. Then try another involution, or show cancellation is only equal cardinality, not a pairing of configs.
3. Does the same mechanism force both-two = 0 (A-continues), or is that independent?

Outcomes: involution-holds / cardinality-only / obstruction.

Constraints

• Import PR [P1 exact] Joint dictionary: rank pair x wrapping-type x 5-names at axis L=3,4 (#665) #668 JSON. Do not redo the joint census.
• L=3: you MAY stream 2^9 to dump the 6 configs as occupancy lists / pictures. Mac-OK.
• L=4 2^16 only if it finishes in a few seconds; if not, comment NEED_HUAWEI and stop. Do not run L=5.
• Integers only. No STATUS. Do not close [P1 exact] Joint dictionary: rank pair (r_b, r_w) × wrapping-type 4×4 at axis L=3,4 #665 / [P1 exact] Does both-two acquire mass at axis L=5? Structural caveat on #635 A #651 / [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. Draft PR against main.

Wait for ASSIGNED_MACHINE.

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

[LightChainr]

ASSIGNED_MACHINE: local

Spiral involution vs cardinality-only cancellation. Import PR #668 JSON. L=3 2^9 dump of the 6 configs is Mac-OK. L=5 is NEED_HUAWEI — do not run it. Draft PR against main. Do not edit STATUS. Do not close #665/#651/#635.

[LightChainr]

Deliverable up for review: PR #683 (draft, against main).

Outcome: cardinality-only, sharpened to set identity — D = 0 is config-wise.

• Colour-flip kill at L=4 recorded first: [120,416,448,128,8] not palindromic under k ↔ 16−k; structurally, rank-1 both-same needs k ≥ 2L on primal, so the complement is never rank-1 × rank-1 (verified at L=3: all 6 complements → black none, white both-same rank 2).
• L=3 2^9 dump: the 6 spirals are k=6 diagonal staircases, each with either=1 on both sides → D(C)=0 term-by-term, no pairing needed.
• Set identity: black-side and white-side rank-1 both-same classes are the same configuration set (6=6=6 at L=3; 1120=1120=1120 at L=4, identical per-k masses).
• Fixed-point-free involutions exist (refl-x/refl-y all c; 10 maps at L=4) but are epiphenomenal — 90° rotations/translations fail the involution test on the cell.
• Q3: both-two = 0 is independent and structural (x-wrap and y-wrap clusters have intersection number 1, so they merge; 0 violations at L=2,3,4 both lattices). A-continues is forced.

Provenance: PR #668 JSON imported with provenance record, no census redo. L=4 2^16 fresh stream ran in 2.6 s (seconds-scale gate, permitted). L=5 NOT run — NEED_HUAWEI. Integers only; no STATUS edit; #665/#651/#635 untouched.

[LightChainr]

PR: #683

Outcome cardinality-only, sharpened to config-wise D=0 plus a structural reason for both-two=0. Ticket stays open. Not merging.

[LightChainr]

Resolved by merged #702 §1. Spiral cancellation is stronger than a pairing/count identity: under the existing digital-Alexander theorem, rank-one black and white images are the same symplectic line, so the black-spiral and white-spiral events are the same configuration set at every honest size. Each such configuration contributes +1-1=0 to D. No colour-flip or fixed-point-free involution is necessary. Independently, distinct embedded components have zero intersection and cannot carry transverse winding lines, so both-two is empty. Historical #683 small-size data and the failed colour-flip interpretation are retained. Closing the mechanism question as completed; no larger census is commissioned.