[petalas]

Summary

Scoped constructive proof with exact finite corroboration. Reversing n distinct records in k stable-lane split/merge passes of capacities T_1,...,T_k is possible exactly when n <= product(T_i). For fixed T >= 2, the minimum depth is ceil(log_T n).

- Reversal has the decreasing-subsequence pass lower bound.
+ A mixed-radix route attains that bound for every n and heterogeneous capacities.
+ Only real records are routed, including incomplete virtual grids.

A pass may arbitrarily assign and interleave its lanes while preserving each lane's input order, with no record copies or deletions. The seven-record example needs exactly two three-lane passes. This adds attainment to Matarener's #3825 pass obstruction; James Reed's #3839 supplies the earlier finite composition census. Their calculations are cited and are not re-audited. No historical novelty claim for radix routing or the lower-bound method, no new kappa, and no selected-frontier change.

Package: research/reversal-stable-lane-tariff. Base: a5df48c55ea806d7952b65acc9c76d878ff09b13. Contribution head: 7a49d01908933e044acf96f040b15818ef748051. Priority revision 7, formal-interface-lemma card explicitly changed to this abstract layout side result. Advance CLAIM.

Evidence

Before, the cited pass model had a lower bound but no general attaining reversal route in its stated evidence. After, PROOF.md proves the exact criterion and verify.py constructs routes and independently checks actual lane streams and stage seams.

Local CPython 3.12.15 on macOS passed 11,114 routes, 43,237 stage checks, 341 insufficient-capacity rejections, 18 independent shortest-path minima and 12 malformed-input/route-mutation controls. Direct normal and optimized commands took 0.67 and 0.69 seconds respectively. Each registered command has a 120-second bound. These are observed check durations, not a performance comparison.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/reversal-stable-lane-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/reversal-stable-lane-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/reversal-stable-lane-tariff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/reversal-stable-lane-tariff --check-source-history

Any CPython 3.12 interpreter can replace the local path. Outputs include PASS_REVERSAL_STABLE_LANE_TARIFF, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The receipt is generated and hashes all six package files. Official export integrity and Git-free normal/optimized replays passed with identical output. The source-history check confirms the base is an ancestor of the publication commit. No external runtime inputs or dependencies are required.

The written proof supplies the universal quantifiers. Independent human and formal proof-kernel review are unclaimed. Physical splitting, merge control, movement, storage, returned state, reversible cleanup, frames, precision and a complete supplier invoice remain unpriced. Lean, Docker and the selected scientific cold replay were not run. The historical card baseline fails with KeyError 'bit_lower'; its current source-binding replacement passes. Neither orientation command certifies mathematics.

Related open work

No open PR studying this exact stable-lane reversal capacity was found in the overlap snapshot, refreshed live open list and handoff checks before committing. The snapshot covered 305 open PRs and 4,293 Discussion records. Sparse local notes were incomplete and supplemented by all-main README and merged-PR searches. Unpublished work and arbitrary code diffs are outside that coverage. This result does not count XOR subsequences or price the formal address-call alphabet.

Completion refresh of this account's open PRs created after the advance CLAIM covered #5145, #5146 and #5149 through #5155. Their declared objects are Neumann tails, guard-character sums, prime supply/products, Gaussian inverse edges, a projection recurrence, rounding guards and E8/P112 factorization floors. None studies stable-lane reversal capacity or reports the same quantity; no numerical comparison or additional computation was performed.

Open next questions

Can a named fixed-tape caller implement the attaining route with a fully priced controller and cleanup? Begin with the seven-record two-pass three-lane fixture. Lane selection, output merge control, head movement, returned state and reversible cleanup remain the blockers. No follow-on investigation is claimed here.

CI notes

Both hosted checks passed for 7a49d01908933e044acf96f040b15818ef748051 in CI run 39943320852: plan in 3m52s and the registered package replay in 1m14s. The completed run has conclusion success and its head SHA matches this contribution. All registered checks, direct normal/optimized replays, source-history, export-integrity and Git-free checks also passed locally. No local obligation was deferred to CI.

Merge Danger

Door: two-way.

Blast Radius: research.

The change adds one isolated six-file research package and an opt-in replay manifest. Shared code, predecessor evidence and selected artifacts are unchanged.

This contribution was produced by Nick Petalas. Credit Matarener, James Reed and their retained native/Walsh/compiler/coverage predecessors. Apache-2.0.