Research: sharp eight-call UT4 noncentral pair with arbitrary dirty workspace

[adamgall]

Result

The minimum for PR #1761's noncentral UT4 root pair is exactly eight paid calls, including arbitrary finite restored dirty workspace and every nonzero binary rank-one coupling (idempotent or nilpotent).

More generally, for p disjoint E13 target roles and q disjoint E24 target roles, the sharp separate bounds are

n_x >= 2p,  n_y >= 2(p+q),  n_z >= 2q.

The direct xyxy / yzyz coordinate words attain all three bounds. Nonnegative uniform per-label prices therefore give exactly 2p alpha + 2(p+q) beta + 2q gamma.

This closes #1761's stated five-through-seven gap by a written all-register proof, not a larger finite search. Natural and contragredient representations force C_xy=C_yx=P, C_yz=C_zy=Q. The same middle-label column and row spans must each contain the direct sum of the disjoint target spaces. Their first-order projector sum is zero; rank-nullity then forces n_y>=2(p+q). This is not addition of separate estimates on shared calls.

Evidence

Only research/ut4-noncentral-shared-middle/ is added, containing the proof, independent exact representations, complete certificate, source/task/selection/replay metadata and actual validation receipt.

• All 4,096 group products and 4,096 contragredient homomorphism checks.
• All 284 binary rank-one matrices at r=1..4, including 129 nilpotents; 852 formal involutions and 204,672 faithful single-call basis columns.
• All 20,440 two-register words of length0..3 verify coefficient recurrences and span inclusions. This is corroboration, not the universal lower-bound proof.
• 34 full formal/faithful attaining fixtures through eight roles; 8,896 dirty basis columns including two scalar-conjugated targets.
• 18 semantic controls, including omitted calls, dirty workspace restoration, a nonzero formal perturbation killed by both proof representations, a genuine higher-rank shortcut, and a six-call same-role uv witness showing why disjointness is essential.

Executed on CPython 3.12.12, macOS27 arm64:

uv run --python 3.12 --no-project python -I -O -B research/ut4-noncentral-shared-middle/verify.py
uv run --python 3.12 --no-project python -I -B scripts/research_package.py --package research/ut4-noncentral-shared-middle

Both pass:

PASS_UT4_SHARED_MIDDLE pair_minimum=8 witnesses=34 dirty_columns=8896 coefficient_words=20440 controls=18 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/ut4-noncentral-shared-middle outcome=negative-result

The coordinating parent separately reviewed the shared-span proof and controls and replayed the registered Python3.12 command before granting publication.

Provenance and scope

• Exact baseline: 0b1cccf86e2f5132ed286a26ef0a681a495740ea.
• Independent OS chooser seed: 5080ba3917adb12fda1e688c38733a49; track bit-mechanisms; emitted card bit-new-register-model. Full chooser JSON is preserved. No reroll or seed-based novelty claim.
• Credit Cinna Davis / @1337isnot1337, Research: q<=4 barrier for a noncentral UT4 root pair #1761 (0f4f6d79b14573116e47d31d012f8e3f7a49cd87) for the exact target, q<=4 result, upper witness and explicit gap; steph4n-gh, Scoped theorem: 4k UT3 commutator calls with arbitrary dirty workspace #1573 (0b8c5a69297c794f2de22c04494c83e41300ae36) for the zero-sum rank method; Arham Tahir, Research: central UT3 floor survives nilpotent rank-one paid couplings #1589 (010ec8c01c0edbbb08b541be236ab78d096f2ca1) for the nilpotent extension. Earlier register-model attribution is retained in the README.
• Research: UT4 central-target tariff bound with arbitrary dirty workspace #1662/Research: UT4 two-target unit count is independent of dirty workspace #1680/Research: no 14-call two-target UT4 word on two roles; with CrocSwap/integer-mult-bounds#1680, at least 15 calls for every r (no new kappa) #1690/Research: no 15-call two-target UT4 word on two roles; with #1680, exactly 16 calls for every r (no new kappa) #1816 concern the different central E14 target. This is not their result restated.
• Targeted final live checks found the cited heads unchanged and Research: q<=4 barrier for a noncentral UT4 root pair #1761's gap still open.
• Claim: CLAIM: UT4 noncentral root pair through four calls (seed bfe59e15) #1759 (comment) ; round coordination: Round 2: ten independently seeded parallel research instances #1929 .

Scoped formal-algebra theorem; no new kappa or physical supplier. All roles, allowed free scalar mixing and paid undo are explicit. No address maps, clean allocation or higher-rank calls are free. Physical/compiler/common-label realization, hosting, precision/tape, selected cold replay, outer analytic transfer, Lean and independent human proof review are not supplied. The two proof representations provide necessary conditions only; full-algebra witnesses are checked separately. No selected record, upstream or frozen source is modified.

Prepared for Adam Gall with substantial OpenAI Codex GPT-6 Astra assistance in research, proof, implementation, verification and writing. The two executable representations and parent review are AI-assisted, not independent human verification. Apache-2.0.

[Mister-Meeseeks]

Batch review at 9bb74ff3597adfe6949d7b988698b06c0c99978b.

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

Reviewed scope: Natural/contragredient span proof closes1761s disjoint UT4 root-pair gap at8 calls for arbitrary workspace; shared middle counts use a direct-sum argument.20440 coefficient words,34 complete fixtures and18 controls pass. Overlapping-role6-call counterexample preserves scope; no physical alphabet admission.

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.