[petalas]

Summary

Scoped all-field algebraic proof: a normalized rank-one word with arbitrary nonidentity permutation labels and full endpoint Q_A direct-sum I_dirty needs at most

w + floor((sum_i inv(g_i) - L) / min_i inv(g_i)) total roles.

Here w is the original active width and L is the ordered-pair endpoint crossing rank. One spectator-only scalar basis change fixes the active coordinates and preserves every original paid factor, label, order and complete dirty endpoint. For constant inversion count d the bound is w+m-ceil(L/d). A canceling-pair example proves coefficient sharpness for preserving these factors; no endpoint-optimal workspace claim is made.

This answers the non-simple-label Open next question in Brian/JoyboyBrian's #4783 at c46cf15df53f3dad2ce59faec9cdc54184e643e5. The new step is the inversion-dimensional crossing decomposition and its weighted local charge. The predecessor's charge framework, nested splits and sharpness method retain credit. Non-involutive labels are peeled with their literal inverse. Only research/permutation-slack-workspace/ is added.

This contribution is produced by Nick Petalas. Credit Brian/JoyboyBrian #4783/#4765, Ross Richardson #1989/#1966, Adam Gall #1941, Sagar Thakrar #3888, Arham Tahir #2998, Nick Petalas #2583/#3408 and the retained register-model lineage. The proof is self-contained; no predecessor executable or finite-exclusion receipt was replayed. Independent human and formal proof-kernel review are unclaimed.

Advance CLAIM and investigation.

Evidence

Before, #4783's slack theorem covered simple adjacent S4 labels. After, the written proof covers arbitrary symmetric groups, permutation labels and scalar fields under the fixed-word boundary above.

All four registered local checks pass, with warnings treated as errors in normal and optimized Python. Primary checks cover 5,625 exact peels, 576 literal crossing decompositions, four full formal words, 300 regular-action dirty-input basis columns, two constructive mixed-label compressions and seven boundary controls. The local active maps are deterministic samples, not an exhaustive GL census. A separate set-based oracle imports no primary code and checks all 840 S3 cases by enumerating actual 512-element subspaces and their intersections.

The four registered commands took 0.867, 0.837, 0.234 and 0.234 seconds locally, below their 120-second limits. These are execution-duration records, not a performance-improvement claim. A generated hash-bound receipt passes optimized comparison; changed scientific receipt data, changed proof bytes and an occupied output path reject.

Reproduce from the repository root using CPython 3.12 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/permutation-slack-workspace
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/permutation-slack-workspace/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/permutation-slack-workspace/oracle.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/permutation-slack-workspace --check-source-history

Substitute a local CPython 3.12 path on another machine. Committed-byte/source-history preflight passes for all seven package files at 93a05b2221fb68c9e134d45b70e289bd7b7bc289. The replay plan selects only this package. Official committed export and integrity checks pass; both optimized direct scripts also pass from the read-only Git-free export with an empty PATH, leaving all eight export files unchanged. Generated receipts and replay outputs remain outside the checkout.

Limitations

This is a scoped proof with finite corroboration, not a complete multiplication theorem or arbitrary-word search exclusion. It preserves a supplied factor list. Physical basis-conversion cost, native realization, restricted gauges, access, ownership, hosting, precision, tape, full supplier and outer admission remain open. No new kappa or selected-frontier change is claimed. No independent human or proof-kernel acceptance is claimed, and no Lean, Docker or historical heavy replay was run.

The selected source-binding positive control passes. The historical card command research_baseline.py raises KeyError: 'bit_lower' against the current selected certificate schema and is unchanged. That is orientation metadata, not a CI failure or evidence against a construction.

Canonical main was fetched before submission and remains 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. Entry overlap covered 380 open PRs and 3,860 Discussion records with complete pagination. The sparse local scan covered 69 of 6,319 notes; separate distinctive main Git-object README searches, merged searches, current claims and live relevant PR bodies were inspected. Arbitrary diffs, unretained closed work and unpublished investigations remain outside coverage. No novelty guarantee is claimed.

Related open work

• Bound S4 spectator workspace by inversion slack #4783 proves the simple adjacent S4 spectator-slack bound. This result answers its explicit non-simple-label question using inversion-sized crossing spaces.
• Prove exact fixed-factor dirty workspace and optimal split counts (no κ) #4847 gives the exact scalar-coupling minimum and optimal-splitting counts for a supplied factor list under the same basis restriction. It does not prove the changed-alphabet geometric slack inequality supplied here; no scalar classification or count is extended.
• Prove shortest S3 word rigidity over every field (no new κ) #4851 classifies shortest formal S3 reversal words over every field. This result bounds workspace for an arbitrary supplied permutation word and makes no shortest-word classification or competing numerical claim.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22T22:17:30Z were refreshed. The nine other submissions concern different frame, chart, Clifford, echo, layout or QQ16 quantities. None supplies a competing permutation-label spectator-slack result or fixture value. Existing same-object Related open work remains listed above; no additional comparison computations were run.

Open next questions

• Specify a changed-label target and paid budget for which the finite role bound reduces a necessary search, including its complete physical allowance.
• Derive a crossing-space decomposition and spectator split charge for one higher-rank scalar idempotent. The rank-one proof does not cover it.

CI notes

Both hosted checks passed on exact contribution head 93a05b2221fb68c9e134d45b70e289bd7b7bc289 in run 39864912110: planning in 4m28s and registered package replay in 1m12s. Run metadata confirms that head, completed status and success. The registered replay executes all four warning-strict normal and optimized commands. Local committed-byte/source-history and read-only Git-free export checks also pass. These automated checks certify the bounded replay; independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way.

Blast Radius: isolated research package. Shared code, frozen predecessor evidence and the selected frontier are unchanged.