DONE: second-order obstruction for PR4953 seven-call address-twist pattern (#5003)

[krzysztofwojt]

DONE / 2026-10-23T06:24Z / priority revision 7. Claim released.

PR #5003, exact head 3ee15e813f921c547763e57c3e2578de403ab321, supplies a written second-order obstruction for PR4953's fixed seven-call J/P_s/D_s pattern. It covers all call orders and per-register S3 × S3 twists, and generalizes to arbitrary odd-denominator rank-one frames and PBQ conjugations in that same pattern. Any diagonal endpoint must have target ranks at most one modulo two, so three I_3 targets are impossible. This closes the requested bounded second-order step. The full two-sided model remains open; no new κ, native supplier, or scientific acceptance is claimed.

The new condition tests arbitrary dirty degree-one inputs modulo I^3. Off-diagonal cancellation forces equal shared/private transports, cancelling their diagonal first-order contribution. For the exact predecessor assignment, input [E00]+[0] in role zero leaves residual t00² + t00·t11 in role one, in every order. Polynomial squares are retained.

Evidence is author-checked with substantial OpenAI Codex assistance. Normal/optimized direct and registered replay pass: 11,664 reduced cancellation cases, 511 nonzero linear-form maps, 36 permutation twists, two general PBQ fixtures, 110 separate sparse-oracle comparisons and a 165-vector direct positive control. All five deliberate verifier corruptions fail in both modes. Committed-byte/source-history checks and fresh read-only Git-free export replay pass, with unchanged bytes and identical reports. Hosted CI run 39910640060 passed its plan and sole-package replay jobs at the exact published head. The quantified result comes from the written proof, not finite sampling of general twists.

Reproduce from the repository or official export with python3.12 -I -W error -B research/bit-address-twist-second-order/verify.py (about one second); repeat with -O. Exact source pins, controls, commands, publication/export evidence and limitations are in the package and PR.

Next useful task: a different first-order-feasible seven/eight-call projector pattern, with at least three contributions in an off-diagonal block, tested on dirty degree-one inputs before full synthesis. No such candidate is supplied or reserved. Extra mixing/helpers, even denominators and native twist pricing remain outside this result; repeating this fixed pattern is unnecessary.

Predecessor credit: JoyboyBrian / PR4953; Cinna Davis / 1337isnot1337 / PR2110; Ali Romman / PR468, eumemic / PR288 and huxint / PR203. Prepared by Krzysztof Wójt / krzysztofwojt with substantial OpenAI Codex assistance and bounded Luna support. No independent human or proof-assistant review. Apache-2.0.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "bit-new-register-model",
  "scope": "Second-order obstruction for the fixed PR4953 seven-call J/P_s/D_s pattern under per-role two-sided address twists; full two-sided model remains open",
  "base": "bee95ce88f9f19802f0613f92838895cd0e14a1f",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5003"
}

Original claim and scope history

CLAIM / 2026-10-23T06:00Z / priority revision 7

Taking the specific second-order follow-up in PR #4953, handoff: can its seven-call J, P_0, P_1, P_2, D_0, D_1, D_2 pattern survive modulo the cube of the augmentation ideal when each call has arbitrary per-role S3 × S3 address twists?

The intended result is a written second-order obstruction, with an independently implemented exact truncated-polynomial/operator check. The key additional test is action on arbitrary degree-one dirty inputs, not only on constants. I will retain the exact projector pattern and count all seven calls, allow every call order, and distinguish this pattern from arbitrary seven/eight-call words. If the proof supports arbitrary odd-denominator PBQ twists and rank-one frames in this same pattern, I will state and check that extension explicitly. No nine-call lower bound for the full two-sided model, native implementation, or new kappa is claimed.

Base: fresh canonical main bee95ce88f9f19802f0613f92838895cd0e14a1f; predecessor #4953 head aa468fa11ed5ca04b389021d35d39f70bc211dfe. The unchanged predecessor verifier passes locally. Value/overlap check: complete cached pagination of 305 open PRs, 4,300 closed PRs, 4,091 Discussion records and 6,690 local notes, followed by live open-PR and recently updated Discussion checks immediately before this claim. The explicit second-order handoff remains unanswered in that coverage; this is not a repeat of its 5,040 exact order checks. Unpublished work and arbitrary diffs are outside that check.

Expected artifact: one scoped successor package and focused PR, or a falsifying counterexample. Stop after the fixed-pattern proof/check, without a broader alphabet census. LOCAL-ONLY until publication. Claim expires 2026-10-23T08:00:00Z.

Credit: JoyboyBrian for the changed twist model and seven-call relaxation witness; Cinna Davis / 1337isnot1337 (#2110) for the odd-denominator square-zero method; Ali Romman (#468), eumemic (#288) and huxint (#203) for the retained register-model lineage. Prepared by Krzysztof Wojt / krzysztofwojt with substantial OpenAI Codex assistance. Apache-2.0.