[ToikY]

Change and mathematical scope

PR1662 leaves the two-target optimum between 14 and 16 with arbitrary dirty workspace. This contribution proves that extra workspace cannot lower that unit call count: C_r=C_2 for every finite r>=2, while the exact value remains 14,15 or16.

The new uniform lemma applies to every full formal UT4(F2) central-target word with r>=w>=2 and length L<=7w+1. Summing the six scalar span inequalities permits at most one extra span dimension. All factors therefore share a common upper/lower block orientation; active-corner compression preserves multiplication over the full noncommutative group algebra. Compressed nonzero rank-one factors remain legal, and zero identity factors are dropped. Every 14/15-call two-target word consequently reduces to exactlytwo roles. The direct 16-call two-role upper word then proves equality of the minima.

Only research/ut4-one-slack-workspace/ is added (12 files). No 14/15-call construction or nonexistence census is claimed. This does not compress arbitrary 16-call words or establish weighted-tariff workspace independence. No physical supplier, rational-frame identity, kappa or selected-frontier change.

Task and overlap

Base main a18b4b0. Revision3 task bit-new-register-model, nonce e5da093483cfe0c1dfea3262d0f0935b, scope 7f5f4426ef796683c485. Scope explicitly changed from a shortest-identity construction attempt to a useful missing-workspace reduction for that same construction problem; TASK.json retains the original card and reason.

This complements Aaron Wang/navinor's active two-role14-call search, which leaves extra dirty roles open. We claim only that missing implication, not an overlapping census or an unpublished search result. Complementary claim.

Entry overlap covered537 open PR bodies including drafts; pre-submit numbered pagination found461 distinct PRs and only predecessor1662 in the UT4 scope. The helper also searched local metadata/ledger; live1163/1166 and relevantDiscussion claims were read. Concurrent merges prevent an atomic snapshot. Full Discussion pagination, all code diffs and unpublished work were not covered; no novelty guarantee. Six guidance/catalog/selected/model blobs and predecessorhead were rechecked unchanged at418e40eb2572d930b58c1cf9b82b41c0fc7ee08d and again at d78fe4b before publication.

Validation

Run python3.12 -B scripts/research_package.py --package research/ut4-one-slack-workspace. Registered replay passed in 11.855s on Python3.12.14/macOS26.5.2 arm64, standard library only; six fault-test groups and 20 workflow tests passed. Optimized isolated verification, source pins and staged whitespace checks pass.

Exact corroboration:3,969 ordered triangular pairs per orientation (7,938 recorded cases,7,209 distinct;729 overlap),1,524,096 full regular input columns,2,268 actually dropped factors,105 rank-slack profiles,10 known upper witnesses/2,560 basis columns. Separate literal4x4/regular-operator checker has four compression fixtures and actual mixed-orientation feedback controls. The inherited fixed17-term quotient replays4,096 products/4,913 basis triples. Real source/vendor/receipt corruption, illegal inputs, occupied outputs and optimized read-only relocation reject or pass as appropriate. Finitefixtures corroborate the written proof; they are not14/15-call witnesses. VALIDATION.json lists unrun obligations, including selected cold replay and kernel/physical/analytic transfer.

Credit and next question

Credit @ToikY/PR1662 at 01cc02b for the six UT4 span containments and middle quotient. Its quotient.py is copied byte-for-byte with Gitblob/SHA256 in VENDOR.json, so replay needs no unmerged dependency checkout. This retained implementation is not claimed independently new.

Credit steph4n-gh/#1573 at 0b8c5a6, Arham Tahir/#1589 at 010ec8c, Hanson Sun/@bbMisk/#1607 at 9d61179 and their retained lineage. Aaron Wang/navinor is credited for the complementary ongoing search, without endorsing its outcome. Apache-2.0; prepared for @ToikY with substantial OpenAI Codex assistance and two bounded subagents. No independent human/kernel review claimed.

Next: independently audit navinor's eventual complete 14-call certificate; a rigorous r2 exclusion now transfers to every finite workspace. If14 is excluded, decide 15 on the same two-role 27-factor alphabet. Exact minima and complete search certificates remain open.

[JosieElliston]

Independent read-through note from #1816, which uses this lemma at L = 15 = 7w+1. I re-derived each step of PROOF.md and found no gap:

• the six span containments from abc, cba, bac (BAC+BCA) and acb (ACB+CAB);
• n_g >= u_g+v_g from X_g Y_g = 0;
• n_b >= w+u_b+v_b via ker X / im Y from XY = UY = XV = 0 and UV = P;
• the at-most-one-excess argument;
• the common block-triangular orientation;
• multiplicativity of the active-corner projection over the noncommutative ring;
• dropping zero projected factors.

This is a reader's check, not a formalization or human review. With #1662 (≤ 13), #1690 (14) and #1816 (15, exhaustive at r = 2), it gives exactly 16 calls for two targets at every r ≥ 2.

[Mister-Meeseeks]

Batch review at a281330e84e64ad357e35b338796c96144a345ce.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: UT4 one-slack workspace compression: six target-span containments force a common block orientation at length<=7w+1, allowing active-corner projection in the full group algebra. Two-target unit optimum is workspace-independent in14..16; weighted/larger-slack claims excluded.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.