[petalas]

Summary

Scoped written proof with exact finite corroboration: the n-bit reflected Gray address permutation and its inverse need exactly F_(n+1) stable lanes in one pass, with F_0=0 and F_1=1. At n=16 the exact requirement is 1,597 lanes. The matching lane assignment works for arbitrary record payloads.

G_n = G_(n-1) followed by 2^(n-1) + reverse(G_(n-1))
L_n = L_(n-1) + D_(n-1)
D_n = max(L_(n-1), D_(n-1))
one-pass lanes = D_n = F_(n+1)

Any sequence of stable passes has lane-capacity product at least F_(n+1). The map's n-1 adjacent binary shears therefore do not imply a small one-pass lane requirement. Multi-pass optimality is left open.

This contribution was produced by Nick Petalas. Credit Matarener / #3825 for the stable-pass layout model and bit-free supplier question, James Reed / #3839 for permutation-composition context, and the retained native/Walsh/compiler/coverage lineage. The older #4858 XOR-subsequence and #5148 reversal results remain distinct context. Reflected Gray order, Fibonacci numbers, chain coloring and patience sorting are classical tools; no global historical-priority claim is made.

Only six files in research/gray-layout-lane-number/ are added. Advance CLAIM. Base main is 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8. The selected frontier is unchanged.

Evidence

The cited pass-model evidence prices XOR layouts. This package adds the exact Gray-layout lane count, an attaining split/merge and a matching decreasing witness.

Local CPython 3.12.15 normal and optimized checks pass with identical generated receipts:

• 34 Gray/inverse orders through n=16; 262,142 routed record occurrences.
• 22 independent quadratic subsequence checks through n=10 and eight exhaustive increasing-subset partition checks through n=3.
• Seven literal adjacent-shear compositions and 14 boundary/semantic controls.
• Three disposable optimized-mode mutations reject a shifted Fibonacci index, incorrect Gray map and reversed stream.

Each registered command was timed locally at about 0.3 seconds, below its 120-second limit. These are check durations, not performance comparisons. Metadata/registered replay, in-memory compilation, whitespace, exact committed bytes, source history, executable modes, official export integrity and read-only Git-free normal/optimized replay pass at head 375746a027a23b5c6036577619c9406f550b6b2e. All seven exported files remain unchanged. No predecessor scientific replay was run.

From the repository root, with the standard library only:

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

Other machines may substitute their CPython 3.12 path. Expected markers are PASS_GRAY_LAYOUT_LANE_NUMBER, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Stdout receipts are generated and bound to all six package files by SHA256.

Limitations and unrun obligations

The universal claim rests on PROOF.md, with finite corroboration. The contract permits arbitrary stable stream splitting and interleaving, with no copies, deletions or within-stream reversal. It does not price physical lane allocation, instruction preparation, merge control, head movement, restored scratch or a selected native caller. Extra memory channels or another routing model can change the answer. All five selected framework assumptions remain.

There is no new kappa, unconditional multitape lower bound, supplier admission or complete multiplication theorem. Lean, Docker, native and selected cold replays were unrun. Independent human and formal proof-kernel review are unclaimed. No registered local check is deferred to CI.

Open next questions

• Determine optimal fixed-capacity multi-pass depth for this Gray layout. Begin with small exact two-lane factorizations; the product bound alone does not give an attaining factorization.
• Bind a concrete native producer/consumer to this order and price split assignment, merge control, storage, movement and restored state. A caller occurrence and physical stream implementation are missing.

Related open work

No open PR computing this exact Gray-layout stable-lane quantity was found in the inspected live and cached coverage. The cached search covered 369 open PRs and 6,284 Discussion records, with incomplete local notes; targeted main/merged searches and fresh claims supplemented it. Non-atomic snapshots and unpublished work remain limits.

• Prove the exact weighted tariff for multilinear dirty-input queries (no κ) #6393 uses a reflected Gray walk to schedule multilinear dirty-input evaluations. It has a different operation contract and objective; it does not compute this lane count.

• PR6427 uses a reflected fixed-weight Gray listing for restored borrowed-anchor parity tours. Its row-ADD tariff is a different quantity from this stable-lane layout count; no common numerical instance is compared.

• PR6431 proves Fibonacci coefficient growth inside restoring signed-unit shear words. It bounds intermediate integer magnitudes rather than record-order lanes; no numerical disagreement is implied by the different indices.

The completion refresh inspected all 15 open account contributions created since the advance CLAIM at 2026-11-02T22:13:51Z, including this PR. The fourteen others are #6418 through #6422 and #6424 through #6432. The two newly related scopes are distinguished above. No additional computations were run to check another contribution or compare unrelated numbers.

CI notes

Both hosted checks passed at exact head 375746a027a23b5c6036577619c9406f550b6b2e in Registered research packages run 41536483613: plan and registered package replay.

The completed check list was read once. Separate run metadata confirms completed status, success and the matching head. Hosted replay executes the registered warning-strict normal and optimized checks. All recorded local scientific, committed-byte/source-history/mode and export checks also passed. CI certifies execution of the finite checks; independent human and formal proof-kernel acceptance remain unclaimed. No failed or waived check remains.

Merge danger

Door: two-way. Blast radius: research package. This isolated addition can be reverted independently; no shared code, frozen predecessor, workflow or selected-result file changes.