[petalas]

Summary

Finite certificate with a scoped combinatorial proof: the layout 3,2,1,6,5,4,9,8,7,0 has decreasing-subsequence width four but requires exactly three stable two-lane passes. Ten records is the smallest possible obstruction to sufficiency of the two-pass width-product bound four.

width-product necessity: width <= 2*2
new exact boundary: every qualifying layout through 9 records is routable
                    an explicit 10-record layout needs 3 passes

Each pass permits arbitrary assignment to two stable lanes and arbitrary order-preserving interleaving. A forced four-cycle rules out every intermediate order for the displayed layout. Three literal passes attain it. This is an abstract routing result; no new kappa or selected-frontier change is claimed.

The contribution was produced by Nick Petalas. Credit Matarener #3825 for the pass-model context and James Reed / jwr404 #3839 for the earlier forward-composition evidence. Mark Karpilovskij's published Theorem 15 and Question 16 already establish the broad failure of product-width sufficiency; that noninclusion is prior work. The new repository contribution is the explicit smallest-order certificate, four-cycle routing proof and standalone checker. No historical priority claim for the example or standard methods is intended. Predecessor scientific suites were not replayed as the deliverable. Older reversal and Gray-layout packages are background; their constructions and code are not the main step.

Evidence

Before, the cited width-product necessity and forward-composition evidence did not provide this explicit minimum-order inverse-composition certificate. The new checker establishes complete set equality at order nine, an explicit obstruction at ten, and an optimal route:

• All 362,880 nine-record permutations; 4,862 independently matched one-pass factors; all 23,639,044 ordered factor pairs; exactly 261,808 two-pass layouts, equal to the complete width-at-most-four set. Restriction settles all smaller sizes.
• All 16,796 ten-record first factors exclude a legal second factor for this endpoint. Independently, all 262,144 normalized lane assignments reduce to 216 forced ordering cycles. Directed-graph cycle checks agree.
• Three literal two-lane passes, every one-record deletion, six rejecting route controls and a feasible width-four reversal control pass.

Warning-strict direct normal and optimized local commands pass in 7.57 and 7.55 seconds, with identical generated source-hash-bound receipts. Both registered commands pass together in 15.20 seconds. Metadata/source pins, AST compilation, function-size bounds, whitespace, exact committed bytes/modes/source ancestry, official export integrity and read-only Git-free optimized replay pass. These durations describe bounded check execution, not a speedup benchmark.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/two-pass-width-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/two-pass-width-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/two-pass-width-obstruction
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/two-pass-width-obstruction --check-source-history --check-modes

Any clean CPython 3.12 installation can replace the displayed interpreter. Standard library only. Expected markers are PASS_TWO_PASS_WIDTH_OBSTRUCTION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. No generated receipt is handwritten or committed. The advance CLAIM and package source manifest record scope and provenance.

Limitations

Minimum order rests on exhaustive finite computation. The four-cycle exclusion and three-pass route have written proofs. Independent human and formal proof-kernel review are unclaimed. No complete forbidden-pattern basis, arbitrary-layout recognition algorithm, actual physical caller, controller, movement/stock/precision bill, reversible cleanup, native supplier or complete multiplication theorem is supplied. More lanes or a different routing model can escape the obstruction. Selected scientific cold replay, predecessor scientific suites, Lean, Docker and a separate static type/lint analyzer are unrun. The selected #7000 frontier and five assumptions are unchanged.

Open next questions

• Determine whether further minimal forbidden patterns give a useful constructive two-pass recognition test. Start with one named layout and the displayed four-cycle; a complete basis is missing.
• Bind a current producer/consumer layout containing this record-order pattern, then price its legal split/merge controller, movement and cleanup. No physical caller is bound here.

Related open work

• Audit uniform Beneš compiler with charged sequential arena scans #4192 lowers an encoded-permutation Beneš compiler with explicit sequential arena scans. Its switching-network and tape invoices differ from this unrestricted stable-lane pass alphabet; no same-instance numerical quantity is compared.
• #7247 counts contiguous source blocks in a rounded tensor-image layout. That block-copy quantity differs from stable-lane pass depth; no same-instance numerical quantity is compared.

The pre-DONE refresh read all ten own-account open PRs created since the CLAIM, including this PR. The other nine are #7247 through #7252 and #7254 through #7256. None computes this two-pass inverse-composition quantity or supplies a competing value for the ten-record obstruction. No additional scientific calculations were run to compare other contributions. Public open/merged searches and main-tree README searches supplement the sparse overlap cache; unpublished work remains outside coverage.

CI notes

CI did not certify this package. At head 17e0213f31725315c476792fd282bd201ff4c4ac, run 43024227699, plan job 130534726487, failed in Onboarding helper tests. I read the completed checks once and confirmed the actual job log. It reports exactly one failure among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

The AssertionError says that `make frontier7000-verify` was not found in the CI verification document. Canonical base e54523517d0f03e58352e647c403a5af3e525a57 selects that command in certificates/selected-result.json, while the current section of docs/ci-verification.md still describes #6030 and make frontier6030-verify. Both files and the failing test are byte-identical between that canonical base and this PR head. This is an inherited repository-level navigation failure. The dependent package replay job was skipped after planning failed.

I also read the failed-job logs of all eight following runs and confirmed in each the same sole failing test, missing-command AssertionError, 760 tests and 24 skipped: 42905094314, 42903313790, #7247 / 43023514635, #7248 / 43023633785, #7249 / 43023693429, #7250 / 43023829856, #7251 / 43024009248, and #7252 / 43024030495.

Local checks listed above pass at the unchanged head. Hosted package evidence remains absent. No fix, retrigger commit, push, shared-file edit or job rerun was made during closeout; the repository repair is left to maintainers.

Merge Danger

Door: two-way.

Blast Radius: research.

Six files in one isolated package with opt-in replay. Shared scripts, predecessor evidence and selected artifacts are unchanged.