[petalas]

Summary

Scoped proof: a fixed rank-one word needs exactly d = dim(U)+dim(V)-rank(V|U) retained dirty roles when every factor is preserved through one spectator-only basis change fixing the original active coordinates. This replaces the S4 inversion-slack upper bound by the exact answer for a supplied word. The theorem works over every field and central-scalar unital algebra; it does not extend the predecessor's S4 slack inequality to other alphabets.

u = dim(dirty column span)
v = dim(dirty row span)
t = rank(their evaluation pairing)
k = original dirty dimension - u - v + t
minimum retained dirty roles = u + v - t
optimal discarded subspaces = q^((u-t)k)
optimal retained/discarded splittings = q^(k(u+v-2t))

The two counts hold over a field of size q and count subspaces rather than bases. The proof classifies all optimal splittings. It also separates this boundary from Gareth Davidson's #2323 contraction-preserving resynthesis, whose already-proved optimum is t. Elementary complement counts are standard linear algebra; no literature-priority claim is made.

Only research/exact-factor-workspace/ is added. This contribution is produced by Nick Petalas. Credit Brian/JoyboyBrian #4783/#4765, Gareth Davidson/bitplane #2323, Cinna Davis #1541, ToikY #1662/#1680 and their retained register-model/image/kernel predecessors. No predecessor executable or census was replayed. Advance claim.

Evidence

Before, #4783 supplied d <= m-L for its normalized simple-label S4 domain. After, the exact dimension and both finite-field counts follow from a written all-field lower bound, an attaining complement construction and a complete classification.

Both registered warning-strict normal and optimized replays pass:

PASS_EXACT_FACTOR_WORKSPACE 5669 subspace pairs; 7 formal words

The exhaustive minimum oracle enumerates every ordered direct-sum decomposition, without using the predicted minimum to prune it. Coverage is every subspace pair over F2 in dimensions 0-4, F3 in dimensions 0-3 and F5 in dimensions 0-2. Seven S3 fixtures check all factor intertwiners and complete dirty-input endpoints through the faithful regular representation. A generated receipt binds every package source file and the mathematical transcript.

Local checks pass on CPython 3.12: package metadata/pins and both registered commands; committed bytes with source-history validation; a replay plan selecting only this package; committed export and integrity check; and both direct modes from its read-only Git-free export with empty PATH. Each registered command took about 0.24 seconds in a local timing run, far below its 120-second limit. Seven local rejection controls cover changed receipt data, changed proof bytes, removed census coverage, occupied output, out-of-domain enumeration, a nilpotent fixture coupling and an illegal fixture label.

Reproduce from the repository root:

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

Limitations

This is the minimum for preserving a fixed factor list by one invertible spectator basis change. It is not the minimum over all words for the same endpoint. The comparison with #4783 inherits that cited theorem only for its stated S4 corollary; the new classification and census are self-contained. The pending predecessor is an exact external citation, not an imported runtime source or a supported source-history dependency.

No native realization, physical basis-conversion saving, restricted-gauge theorem, complete supplier, new kappa or selected-frontier change is claimed. Precision, tape costs, ownership and all-size transfer remain separate obligations. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker, C++ or historical heavy replay was run.

The current research_selected_baseline.py source-binding positive control passes. The card's historical research_baseline.py fails with KeyError: 'bit_lower' against the selected record and is unchanged. That is orientation metadata, not evidence against any construction or a hosted-CI failure.

Related open work

• Bound S4 spectator workspace by inversion slack #4783 gives an inversion-slack workspace upper bound under the same preserved-factor basis restriction. This contribution gives its exact scalar-coupling answer and counts all optimal splittings. No predecessor number is recomputed or audited.
• Prove shortest S3 word rigidity over every field (no new κ) #4851, opened after the CLAIM, classifies shortest formal S3 words over every field. This contribution minimizes spectator dimension for a supplied factor list and counts its optimal splittings; it makes no shortest-word claim. No shared numerical value is compared or recomputed.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22T21:46:50Z were refreshed. The snapshot contains no other submission computing the same fixed-factor workspace minimum or optimal-splitting counts. Other new submissions study different native interfaces, norms or paid tariffs. Sparse notes, arbitrary diffs and unpublished work remain outside complete overlap coverage.

Open next questions

• Specify a physical native layout and characterize which spectator basis changes it permits; then determine its constrained minimum and conversion cost.
• Specify an endpoint and paid budget when factors may change. Use the existing contraction-preserving theorem only when that alphabet is closed; this package gives no endpoint optimum.

CI notes

Both hosted checks passed on publication head e0adbcd8cca711a5c88059711aa65fafccbc35e0 in run 39860841179: planning in 3m15s and registered package replay in 1m17s. Run metadata confirms that exact head, completed status and success. The package replay executes both registered warning-strict normal and optimized commands. All seven committed files also pass local byte/source-history preflight and Git-free export replay. 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 evidence and the selected frontier are unchanged.