[petalas]

Summary

A scaled permutation moving s public registers on distinct fixed binary lines needs exactly 2s owned MOVEs, even with arbitrary shared carriers, feedback, interleaved reads and writes, intermediate subspaces and owned erasure. An explicit schedule attains the bound using only rank-one MOVEs. This is a scoped written proof with exact finite corroboration, not a finite search exclusion or a complete multiplication theorem.

Each moved line needs one exit and one entrance.
A strictly nested MOVE cannot join two distinct lines.
Therefore 2s separate MOVEs are necessary; rank-one carriers attain 2s.

Proof and contract. Applied to DreamingOfClouds's #6939 matching exchange on 104 P104 lines, this proves the sharp 208-owned-MOVE floor for that fixed-line preprocessing checkpoint. For every positive-rank tariff c(r)>=1 with c(1)=1, including r^(1-C) for 0<=C<=1, the exact owned MOVE tariff is also 208. Sharing carriers alone cannot improve that contract.

The new theorem, general attainer and checker were produced by Nick Petalas. #6939 supplies the existing window native, matching construction and shared-landing question; its scientific executables and quoted prices were not replayed. Retain its credit to Ali Romman #5599, CubeJerry #6912/#6914/#6916, unpaidmoon #6930, icekylinx #104 and their native/caller predecessors. Sources and selected metadata are pinned at main e545235. The selected frontier is unchanged.

Evidence

• Before: P104: Whitehead native objects and price targets #6939 gives the two-moves-per-port matching construction and leaves sharing owned runs open. A necessary shadow identity alone does not establish a transport floor.
• After: the new region-isolation proof establishes the exact minimum for every finite circuit satisfying the fixed-line contract. The universal claim rests on the written proof.
• Exact checker passes 3,492 scaled-permutation matrix cases, covering every permutation through six ports over integers and F2/F3/F5, with complete arbitrary-input endpoints, spectator restoration and owned cleanup. Integer matrix equality supplies the rational-payload cases. A separate generic 104-port fixture has 208 rank-one MOVEs and peak 104 carriers; it is not a reconstruction of P104.
• All 1,008 nested MOVE edges among all binary subspaces through dimension four satisfy the line-boundary separation invariant. Eighteen malformed/domain/circuit controls reject. A changed-frame boundary control swaps two public registers with zero owned MOVEs and four public MOVEs; it permits public descent and is not a proposed monotone-frame native.
• CPython 3.12.15 normal and optimized receipts agree. Direct commands took 0.20 and 0.22 seconds locally; both registered checks together took 0.70 seconds, with a 120-second cap on each command. Warnings are errors and explicit checks survive optimized Python.
• Package metadata and pins, exact committed bytes, modes and source ancestry pass at 291a876. Official export integrity passes. Normal and optimized replays in a read-only Git-free export with PATH=/nonexistent agree with the checkout receipts. Receipts are generated and bind the package files by SHA256.

From the repository root, CPython 3.12 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/fixed-line-crossing-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/fixed-line-crossing-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/fixed-line-crossing-floor
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/fixed-line-crossing-floor --check-source-history --check-modes

Use the corresponding Python 3.12 path elsewhere. Scientific marker: PASS_FIXED_LINE_CROSSING_FLOOR. Registered marker: PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations and unrun obligations

All public frames stay fixed during the exchange; scalar gates are local to equal frames; no input copies are preloaded elsewhere; owned MOVEs join strictly nested frames. The result does not exclude a different public schedule, an exchange interleaved with the Y bounce, Gamma coupling or another Whitehead native. Only owned MOVEs are optimized. Scalar operations, allocation, peak storage, physical phases and delivery, the complete signed invoice and uniform framework assumptions are separate obligations. No kappa improvement or supplier admission is claimed.

No predecessor scientific suite, selected cold replay, physical native, Lean or Docker run was performed. Selected source-binding orientation passed. Independent human and formal proof-kernel review are unclaimed. The finite matrix tests corroborate the construction; they do not replace the universal proof.

Open next questions

• Can a public-frame change or an exchange interleaved with the Y bounce reduce the complete Whitehead bill? Start with one literal schedule, check all endpoint coefficients, and charge every public and owned frame change. A legal complete schedule and favorable full bill are missing.
• What is the minimum peak owned storage at the minimum 2s MOVE count? Start with a single permutation cycle and interleaved transport; the one-carrier-per-moved-source construction here is not a storage optimum claim.

Related open work

The closeout check read this account's nine other open PRs created since the CLAIM: #7203, #7204, #7205, #7206, #7207, #7210, #7211, #7212 and #7213. None computes this fixed-line Whitehead permutation-exchange minimum. No additional scientific computation was run to compare them.

• #7207 concerns fixed common bases for idle coalescing in the connected-root family. This package counts entrances and exits in an owned permutation exchange.
• #7211 reduces sufficient component width for a once-rounded complete MOVE. This package proves an owned-MOVE invocation floor under fixed public frames; no shared numerical quantity is compared.
• #7213 proves scalar-projector-rank cost for diagonal targets in a formal group-gate alphabet. The locality and nested-frame MOVE contract here is different, and no numerical comparison is claimed.

No other open PR studying this exact fixed-line optimum was found in the paginated open-PR body snapshot and fresh precommit claim/PR check. #6939 is the merged predecessor; its broader changed-frame and Gamma routes remain open. These overlap searches are snapshots, not a novelty guarantee.

CI notes

CI did not certify this package at unchanged head 291a8768b4121a5a0688f71e9486fbd25d6f76e3. I read the completed checks once and inspected the plan job 130497997453 log in run 43013294091. The "Onboarding helper tests" step reports exactly one failure among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

Its AssertionError says that 'make frontier7000-verify' was not found in the CI verification document. On canonical base e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. I inspected those base bytes and confirmed that both files and the failing test are unchanged by this PR.

I also read the comparison logs and verified the same sole failing test, missing-command assertion, 760-test total and 24 skips in all six runs:

• 42905094314
• 42903313790
• #7203 run 43012497455
• #7204 run 43012524632
• #7210 run 43013336222
• #7211 run 43013361819

The dependent package replay job was skipped because plan failed. All mathematical replay evidence remains local as recorded above. No repair, retrigger commit, push, shared-file edit or job rerun was made. The repository-level navigation repair is left to maintainers.

Merge Danger

Two-way. The change adds one isolated research package with two bounded opt-in checks. It changes no selected result, predecessor package or shared implementation.