[petalas]

Summary

A binary depth-six dirty-chain connector first admits a commuting retiming that cannot expose cancelling equal calls at 22 complete calls, exactly two above the known optimum. This answers Open next question 2 of #4748.

1234656564321234565432
   46 becomes 64
1236456564321234565432

Claim type: scoped construction and proof, with finite certificates. Both words restore every dirty intermediate and add B6 B5 B4 B3 B2 B1 x to the dirty target for arbitrary compatible maps in characteristic two. Each has a noncommuting blocker between every consecutive equal-call pair, so no sequence of commuting swaps can expose a cancellation. This excludes a redundant inserted cancelling pair.

The upper bound and blocker proof are new. The lower bound cites #4748 at fc09e5b783989d769e6ae6fffeb7d23b30177c86 for the absence of commuting neighbors in every shortest 20-call word, plus the elementary even-multiplicity constraint. No predecessor verifier is imported or replayed. Yifan Kang's Proposition 3.1 supplies the known path minimum and bridge construction. No new kappa or selected-frontier change is claimed.

Only research/depth6-chain-retiming/ is added. Produced by Nick Petalas. Credit Nick Petalas #4748/#4249/#4263/#4281, Ryanson Jonathan #4418 for separate depth-five residues, Yifan Kang and the Nash et al. construction cited there, and Ross Richardson's #943/#934 connector lineage. Advance CLAIM.

Evidence

Before: #4748 left the extra-call cost of genuine commuting retiming open. After: two explicit commutation-reduced 22-call words and a proof of the exact two-call premium.

Local CPython 3.12.15, macOS arm64, standard library only:

• Direct integer interval coefficients and separate 7-by-7 endpoint products certify both new words.
• 32 enumerated blocker certificates prove that commuting swaps cannot expose a first cancellation.
• All 16,384 binary scalar dirty map/state cases, 1,792 two-coordinate dirty replays, 1,536 exhaustive two-map local-relation comparisons and ten rejection controls pass.
• Both registered warning-strict normal/optimized commands pass. Timed direct runs took 0.099 s and 0.122 s against 120-second limits.
• Registered package replay, five-file exact committed-byte/source-history preflight, committed export integrity, and normal/optimized Git-free isolated export replays pass at ecd0e3512ea3f108112ea4c127dd4ced2369b7a0. Receipts are generated from all five package hashes.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/depth6-chain-retiming/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/depth6-chain-retiming/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/depth6-chain-retiming
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/depth6-chain-retiming --check-source-history

Expected marker:

PASS_DEPTH6_CHAIN_RETIMING calls=22 extra=2 blockers=32 scalar=16384 matrix=1792 relation=1536 controls=10

Any CPython 3.12 interpreter can replace the local path. Every registered mathematical check ran locally.

Limitations and unrun obligations

The exact minimum depends on the cited #4748 finite classification; it is not independently re-established here. This is one characteristic, depth and equal-price alphabet, with a fully specified cancellation rule. There is no length-22 census or all-depth theorem. Independent human and formal proof-kernel review are unclaimed.

No physical consumer, native preparation, routing, dispatch, precision, tape/storage, recursive supplier admission or complete multiplication theorem is supplied. No Lean, Docker or selected-construction cold replay was run. Bounded Python allocation and diagnostic limits are documented in the README.

The research baseline remains b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. Canonical main was refreshed to 1f87c3578ae546c8f903f7f4d83dc2e2fb99650b; priorities, catalogue, family status and selected record were unchanged. Pre-commit overlap covered 426 open PRs and 3,682 Discussion records with complete reported pagination, relevant PR bodies, and targeted main/merged searches. Sparse local-note coverage and unpublished work remain outside that check.

Open next questions

1. Count all commutation-reduced length-22 schedules and their commuting-equivalence classes, beginning with a bounded near-shortest-path search.
2. Replace the inherited 20-call finite classification with structural chronological constraints.
3. Name a real six-stage consumer and price its complete physical boundary before treating this retiming freedom as recursive savings.

Related open work

• Classify all ten shortest binary depth-six chain schedules (no new κ) #4748 studies the same depth-six binary connector at the optimum. It supplies the lower-bound classification; this contribution answers its second open question. Numeric length overlap agrees: the cited bridge has 20 letters, each new witness has 22, and 22-20=2. Its ten-schedule count is credited and not recomputed.
• Construction: three-stage persistent prefix shear chain (no new kappa) #1930 studies persistent prefix outputs with unrestored intermediates. Its output contract differs from this restored-intermediate chain; its three-call quantity has no shared numerical value here.

Merged #4600 concerns balanced depth-four row obstructions over general moduli, a different depth and quantity. No overlapping new count is asserted.

Related open work refresh

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22T19:16:42Z were refreshed. None of the other six new PRs studies the depth-six connector or its retiming threshold:

• Count rank-one residue lifts over local algebras (no new κ) #4770 counts rank-one residue lifts over local algebras. Its target counts do not overlap these chain schedules.
• Prove the all-field one-way corank-one header formula (no new κ) #4769 proves a one-way corank-one header formula. Header probabilities differ from chain call counts; no shared numerical claim is asserted.
• Prove all-field negative correlation for corank-one pairs (no new κ) #4767 proves corank-one negative correlation. Its probability inequality does not overlap this retiming threshold.
• Count square-zero radical targets over every finite field (no new κ) #4766 counts square-zero radical reachability over finite fields. Its algebraic targets and census totals differ from these chain words.
• Prove a cube-zero radical target has exact cost three (no new κ) #4764 gives a cube-zero radical target with right-vector cost three. Its paid-factor alphabet and target differ from this six-stage chain.
• Prove unbounded two-role right-vector cost over F2[t] (no new κ) #4763 proves unbounded polynomial right-vector costs. Those factor lengths use a different alphabet and target, with no numerical schedule-count overlap.

The existing #4748 and #1930 relationships and the checked 20/22 length comparison remain as recorded above. No conflicting overlapping number was identified.

CI notes

All 2 hosted checks passed on ecd0e3512ea3f108112ea4c127dd4ced2369b7a0 in Registered research packages run 39839452472. plan passed in 2m54s and the registered research/depth6-chain-retiming replay passed in 1m14s. Hosted CI certifies the registered package checks within their stated scope; it does not supply independent human or formal proof-kernel acceptance. All registered checks, committed-byte/source-history, export-integrity and Git-free replays also passed locally.

Merge danger

Door: two-way. Blast radius: package. The isolated addition changes no shared code, frozen evidence, predecessor package or selected-frontier file.