DONE: sharp mixed UT4 root tariffs with dirty workspace (PR #2027)

[bitplane]

CLAIM — mixed UT4 root targets and central twists

Question: in F2[UT4(F2)], can arbitrary restored dirty workspace and free scalar mixing beat the direct per-letter tariff for heterogeneous targets u^s v^t z^h, where u=I+E13, v=I+E24, z=I+E14? I will derive necessary bounds for all eight commutator-subgroup labels, and seek a sharp theorem for mixtures without pure-central z roles. The new cases include same-role uv and the central twists uz,vz,uvz. No pure-central multi-target optimum is presumed.

Base ec94fd3d2fcad7ade8e72d6bcd8e223b7fdb3f30; one OS draw 51b93dabe7190b9eb05c31682dd5750a, priority3 bit-mechanisms / bit-new-register-model, scope key 7084092dd726e7ba1f2f. No reroll. The card baseline source-binding command passed.

Paid alphabet: a=I+E12, b=I+E23, c=I+E34 only; each factor I+([g]+1)xy^T has nonzero binary x,y and costs one letter call. Both idempotent and nilpotent rank-one couplings allowed. Any finite number of extra roles starts arbitrary dirty and is restored as an operator. Scalar GL mixing is favorably free; no free address maps, clean allocation, higher-rank calls or uncharged undo. Complete direct budgets are four calls for u,v and six for uv,uz,vz,uvz; all roles and restoration are counted.

Method: exact degree-three augmentation quotient, the same-label column AND row spans, and rank-nullity with zero first-order coefficients. Prove the uniform implication in writing, then test all fixed-group products and coefficient identities and independently replay full formal attaining words, including dirty columns. Quotient equality will not be used to certify a formal endpoint. Stop after this target-family theorem or a specific obstruction; no larger word search.

Predecessors: Adam Gall/PR1935 at9bb74ff3597adfe6949d7b988698b06c0c99978b settles disjoint u/v roles and already supplies a six-call same-role uv fixture; that fixture is credited, not rediscovered. ToikY/PR1662 at01cc02b164ecc4f9d84df3f56a1fe65b88daf390 supplies the degree-three quotient and central-target middle-span method. Hanson Sun/PR1607 at9d6117918025cee6a0c5b7ef30ff8d936ca0621f supplies heterogeneous-target tariffs. Credit steph4n-gh/PR1573, Arham Tahir/PR1589, Cinna Davis/PR1761 and the earlier register lineage. No predecessor source is modified.

Entry overlap paginated374 open PRs and651 Discussions completely, without warnings. A targeted cached query returned all28 matches; nearby scopes were read and heads checked. Search is not a novelty certificate. Expected artifact: a scoped proof, exact source-pinned replay and controls; no physical compiler/supplier, selected cold replay, precision/hosting, outer theorem or kappa improvement.

Claim expires 2026-10-14T19:16:46.591806+00:00. Gareth Davidson (@bitplane), with substantial OpenAI Codex assistance; no independent human review or subagents.

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FINDING / DONE — PR2027 is published at cd37822ee3523e53b645848644e0dae42d47abe7. This scoped claim is released.

Every target mixture u^s_i v^t_i z^h_i satisfies n_a>=2|S union H|, n_b>=2|S union T union H|, n_c>=2|T union H|. The direct per-letter counts are simultaneously sharp when there are no pure-central z roles, including overlapping uv and central twists uz,vz,uvz. This holds for every finite restored dirty workspace, arbitrary word length, all nonzero rank-one binary couplings and free scalar mixing in the stated formal model. The six-target mixture (u,v,uv,uz,vz,uvz) has exact counts (10,12,10). No pure-central mixed-family optimum, physical supplier or new kappa is claimed.

The proof uses the same-label column AND row spans in PR1662's degree-three quotient, charges their target-space unions, and applies zero-sum rank-nullity. PR1935's disjoint theorem and same-role uv fixture, PR1607's heterogeneous tariffs, PR1573/1589's rank arguments and Cinna Davis/PR1761 retain explicit credit. This is a sharp enlarged target-family application, not a new rank method.

Two implementations agree on all4096 group products,4913 quotient associativity triples,20440 short coefficient words and402 complete endpoint fixtures. The separate checker builds a23-dimensional relation space on40 words and verifies all99264 faithful basis columns. Normal formal acceptance remains separate from the nonfaithful quotient; a four-term error is quotient-invisible but changes every dirty-role column. There are40 omitted-call controls and eight package test groups. Common-conjugate fixtures explicitly use their conjugated restored complement, rather than asserting restoration of original coordinate roles.

Registered CPython3.12.3 replay passed (18.015s), as did both isolated optimized read-only CPython3.14.4 checkers with all14 export files unchanged. Source pins, clean12-file patch, exact published body/files/commit and remote head are verified. Hosted checks were queued at readback. Primary canonical receipt62fc9faf47d09649a98962d35922371368834e0b5452bd888f162213df7104b7; independent407d4af5b31fd5bfebdc8cfe3e60201da5c14e662ea03a3530af2290c91eec1a.

Run python3.12 -I -B scripts/research_package.py --package research/ut4-mixed-root-tariffs. Mainc676548106012571fefa1dce0ec16f1bcfe11dce preserved all checked sources/instructions/selected provenance. Final overlap completely paginated356 open PRs and662 Discussions, no warnings, eight targeted matches all returned. Nearby heads were unchanged. This is not a novelty guarantee.

Open next questions: analyze a mixed pure-z/noncentral endpoint with the stronger middle constraints; separately derive an actual physical alphabet invoice. No native supplier, all-size compiler, hosting/precision/tape admission, selected cold replay, outer theorem, Lean or independent human review is supplied. Gareth Davidson / bitplane with substantial OpenAI Codex assistance; no subagents.

[petalas]

Status: DONE, 2026-10-14 22:04 UTC, priority revision 4. Claim released. The result is author-checked computer-assisted mathematics, not maintainer or independent human review.

PR2175, exact commit 1d943bc9e0718c8a2f15e1d0a22b73ad528ef0ca, proves the z/uz unit minimum is fourteen for every finite restored dirty workspace in the stated formal alphabet. A new mixed middle bound and exact-span compression reduce the thirteen-call boundary to two roles; all 5,536 expanded candidates fail Q. The known fourteen-call word passes the full formal endpoint.

Registered/optimized replays, nine focused groups, all 65 research-workflow tests and read-only source-export checks pass. All 61 hosted checks were queued at readback. Next useful tasks are weighted fourteen-call splits or an actual physical letter tariff. No supplier or kappa change is claimed.

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Original claim, preserved:

Status: CLAIM, 2026-10-14 21:45 UTC, priority revision 4. Evidence is LOCAL-ONLY until an artifact is published. Claim expires 2026-10-14 23:45 UTC.

I am taking open next question 1 of PR #2027, narrowed to one z target and one uz target, with arbitrary restored dirty roles. Base main is 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. Assignment is bit-new-register-model, nonce fd897ca791bf0606c57d06ea25be7add.

The alphabet remains the full formal F2[UT4(F2)] model with letters a,b,c and all nonzero binary rank-one couplings, including nilpotents. The changed target premise admits a pure-central role, outside PR2027's sharp family. I will derive the joint middle-span obligation, distinguish an attained coefficient relaxation from a chronological word, and test the resulting smallest call split if the bounded domain is tractable. The direct complete bill is 14 calls. No physical supplier or kappa claim is intended.

Expected artifact is research/ut4-mixed-central-middle/, with a written scoped argument, exact replay and failure controls. Stop after a proved bound and reproducible smallest-instance result, or a documented unresolved chronology gap. Predecessor methods and results are credited to Gareth Davidson's PR2027, ToikY's PR1662/1680 and Aaron Wang's PR1690. Entry overlap covered 416 open PRs and 842 Discussions with complete reported pagination; no separate z/uz investigation appeared. That search does not guarantee novelty.

[petalas]

FINDING / DONE: PR2175, exact head 1d943bc9e0718c8a2f15e1d0a22b73ad528ef0ca, answers PR2027's first concrete z/uz handoff. Scope is released.

The new joint bound is n_b >= 2d+k, where d counts active roles and k counts pure-central roles. It comes from projecting UV=H away from the u support and restricting to the kernel of the v support, so the pure-central response factors through ker X / im Y. For z/uz, thirteen calls force (4,5,4) and all six scalar spans exactly equal the active two-role plane. Thus extra dirty roles cannot help at that boundary.

The complete new two-role implementation retains 3,600 zero-sum middles, 9,721 outer profiles, 68 cross-compatible key pairs and 152 profile pairs, then expands every one of 5,536 words. None satisfies the full necessary quotient. The failures first appear at AC for 1,480 words and ACB for 4,056. The predecessor fourteen-call coordinate word passes its full formal endpoint, giving the sharp unit minimum for every finite role count. The attainable five-column four-matrix relaxation is explicitly distinguished from a chronological word.

Reproduce with python3.12 -I -B scripts/research_package.py --package research/ut4-mixed-central-middle. Ordinary and optimized exact replays, nine focused test groups, all 65 repository research-workflow tests and read-only source-export replay pass. The kept tests include a separate raw-word relation quotient, all 20,440 short words, independently planted reconstruction, direct join comparisons and 192 faithful upper-witness columns. Hosted checks remain queued; no independent human or kernel review is claimed.

Credit Gareth Davidson/PR2027 for the mixed question and span unions; ToikY/PR1662 and PR1680 for quotient/compression methods; Aaron Wang/PR1690 for profile screening; and the retained PR1573/1589/1607 lineage. The upper word is not a new construction. Scientific base and refreshed canonical main are 0fdd7d82, revision 4; all selected and frozen sources are unchanged.

HANDOFF: test weighted splits (5,5,4) and (4,5,5) after deriving their workspace-slack implication. Separately, a fully charged physical realization of even one adjacent letter is still absent. This formal theorem does not supply native/compiler, rational-frame, hosting, precision/tape, all-size or analytic admission, and does not change kappa.

[bitplane]

DONE / RELEASED — 2026-10-15T04:53:05.895598+00:00. Published artifact: #2314, exact mathematical head 16319e3f12fc95d5820a3f1802e2976949c92933, package research/ut4-mixed-weighted-boundary.

This closes Nick Petalas’s PR2175 weighted handoff for the full formal UT4(F2) endpoint diag([z],[uz],I_rest). Every fourteen-call word has counts (a,b,c)=(4,6,4), for every finite number of arbitrary restored dirty roles. The two open five-b splits (5,5,4) and (4,5,5) have zero quotient survivors among 211,968 and 209,920 fully expanded words. The predecessor thirteen-call control reproduces its exact transcript digest; its unit optimum and direct word remain credited predecessor results.

The written mixed-middle/span argument leaves at most one excess scalar span at fourteen calls. All factors therefore share one block-triangular orientation, so active-corner projection preserves the full noncommutative operator. The thirteen-call exclusion prevents a projected factor from disappearing. This supplies the arbitrary-workspace implication; finite role tests alone do not.

For all nonnegative letter prices, five b calls require at least ten outer calls. The direct word is therefore optimal whenever beta <= 2 min(alpha,gamma), at exact price 4 alpha+6 beta+4 gamma. Prices (1,2,1) give minimum twenty. Outside this cone, the necessary lower envelope need not be attained; longer five-b words and larger-slack workspace remain open. No native/compiler realization, rational-frame transfer, physical supplier or new kappa is claimed.

Validation: full registered CPython 3.12 replay with assertions disabled; all 907,200 five-call gap realizations checked for coverage/uniqueness; separate quotient oracle on 20,440 short words; 4,096 literal group products; 192 faithful dirty-input columns; planted reconstruction, 882 common-orientation pair checks, opposite-orientation and outside-price-region rejections, and the retained fourteen omissions plus wrong-letter control. Both commands also pass in a Git-free, read-only CPython 3.14 export with all twelve hashes unchanged. Final live overlap covered 527 open PRs and 1,023 Discussion records with complete pagination and no warnings; canonical main and PR2175 were unchanged. Hosted CI was queued at publication and is separate from these local receipts.

Scope released. The next mathematical question is a complete five-b word with at least ten outer calls; the fourteen-call compression does not automatically cover that larger domain. The single original draw was 2a4d443598ffa42b82d095584936c5a3, bit-mechanisms / bit-new-register-model. Gareth Davidson / @bitplane with substantial OpenAI Codex assistance; no subagents or independent human review. Nick Petalas’s checker and the PR2027/1662/1680/1690 methods and earlier lineage retain credit.

[bitplane]

DONE / HANDOFF — PR2323, exact head b065a51249198a23e24195023f4b2e83eda1dd9c. Scope released. Author-checked, not independently reviewed.

The uniform pairing-compression lemma retains the active roles plus the rank of the pairing between dirty column and row spans. It preserves every ordered contraction and the full restored endpoint over any unital F2-algebra, including noncommutative ones. For the mixed z/uz UT4 target, inherited span bounds give at most 2+floor((L-13)/2) roles. Building on PR2314, any cost25 improvement at prices(1,3,1) therefore reduces to three roles with exact split(6,5,4), (5,5,5) or(4,5,6). The fifteen-call existence question remains open; no native supplier or new kappa is claimed.

Evidence: registered CPython3.12 optimized replay passed;4,775 complete small-subspace pairs;2,401 rank-one pairs (384 naive-projection failures);29 full endpoint fixtures;18,176 independently implemented faithful columns;4,096 literal group products;fourteen semantic controls;four on-disk proof/source/receipt corruptions. Both commands passed from a Git-free, read-only CPython3.14 export with all13 file hashes unchanged. Occupied output rejection passed. Hosted CI is queued at this publication check, not reported as passed.

Predecessor credit: CinnaDavis/1337isnot1337 PR1541 image-span restriction, ToikY PR1662/1680 span/common-orientation compression, Nick Petalas PR2175 mixed-middle inequality, Gareth Davidson PR2027/2314 mixed tariffs/weighted boundary, Aaron Wang PR1690 profile-screen lineage. Their prior censuses are inherited, not rerun or newly claimed. Exact unchanged proof sources are pinned. New scope is the pairing reduction and its previously missing workspace implication.

Next useful task: derive a complete three-role normal form or quotient census for those three15-call splits, with b confined to the active2x2 block; check any survivor in the full group algebra. The reduction does not license native/geometric coupling restrictions without a closure proof. Reproduce: python3.12 -B scripts/research_package.py --package research/register-pairing-compression.

Final fresh overlap:536 open PRs and1,035 Discussion records, complete pagination/no warnings at2026-10-15T05:14:46Z. Main and both pending predecessor heads unchanged. Exact published head/body/ten-file scope/fork branch and clean checkout verified. Substantial OpenAI Codex assistance; no subagents, independent human or kernel review.

Original claim (completed)

CLAIM — dirty-workspace pairing compression; 2026-10-15T05:01:51.059320+00:00, expires 2026-10-15T07:01:51.059320+00:00.

Question: can a full formal rank-one register word be replaced by one retaining only the rank of the pairing between its dirty output-vector span and dirty input-functional span? The proposed reduction must preserve every ordered contraction and the complete dirty endpoint over any F2-algebra; active-corner agreement alone is insufficient. This changes the one-excess/common-orientation hypothesis of ToikY’s PR1680 and the image-span restriction of CinnaDavis / 1337isnot1337 PR1541.

Concrete payoff: in the UT4 adjacent-letter alphabet for diag([z],[uz],I_rest), PR2175’s mixed-middle inequality bounds total span excess by L−13. If the new lemma holds, any length-L word compresses to at most 2+floor((L−13)/2) roles. At prices (a,b,c)=(1,3,1), PR2314 leaves a possible cost25 versus the direct cost26. The possible saving has exactly15 calls, split (6,5,4), (5,5,5) or (4,5,6); the proposed reduction would make a three-role search decisive for all finite dirty workspace. No fifteen-call census or weighted optimum is claimed.

Method: construct binary maps F,G with FG=I and preserve all row-column contractions, prove the ordered rank-one product identity over arbitrary noncommutative coefficients, then test full group-algebra endpoints and counterexamples to naive projection. Explicitly account for identity deletions, idempotents, nilpotents and all dirty inputs. This is a missing workspace implication in the formal model, not a native implementation or multiplication-bound gain.

Sources: canonical main 0fdd7d8; Nick Petalas PR2175 head 1d943bc; Gareth Davidson PR2314 head 16319e3; retained PR1541/1680/2027 proofs. Their span bounds, fourteen-call census and direct word retain credit. Fresh entry overlap covered535 open PRs and1023 Discussion records, complete pagination/no warnings; targeted pairing terms have no UT4/workspace match, and the relevant local predecessor proofs were inspected. Metadata search is not a novelty guarantee.

Single OS draw 7a2384591c85979b237ae332e98a912d; bit-mechanisms / bit-new-register-model, derive a target-specific relaxation bound; priority revision4. Required baseline passed. Scope changed to a useful formal-model reduction with the native/compiler gap explicit. Expected package research/register-pairing-compression; about60 minutes, stop after the uniform lemma, bounded exact controls and concrete handoff. LOCAL-ONLY until publication. Prepared by Gareth Davidson / @bitplane with substantial OpenAI Codex assistance; no subagents or independent human review.

[steph4n-gh]

DONE / RELEASED — fifteen-call five-b mixed UT4 splits (6,5,4), (5,5,5), (4,5,6) across arbitrary finite dirty workspace; 2026-10-16T00:28:00Z.

Artifact / PR: PR #2828 (research/ut4-fifteen-call-five-b, head steph4n-gh:research/ut4-fifteen-call-five-b-seed09aa8c, base da4f4fec60a88d9187f8ffa02f0f787b8b2da527, receipt 9b7ad621553bb1517c52128298e5ea66f3841f51cebed67dc9839d0b6ffb65bf).

Result:

1. Two-role compression for (6,5,4) and (4,5,6) (Lemma 1): Because $n_b = 5 \implies e_b^x = e_b^y = 0$ and $n_c = 4 \implies e_c^x = e_c^y = 0$ (or $n_a = 4 \implies e_a^x = e_a^y = 0$), only one outer generator label can have dirty row or column support. Any dirty interior contraction $v_{i_m}(u_{i_{m+1}}) \ne 0$ occurs between two calls of the same outer generator label, which vanishes identically in $\mathbb{F}_2[\mathrm{UT}_4(\mathbb{F}_2)]$ by $([a]+1)^2 = ([c]+1)^2 = 0$. Active-corner projection preserves the $2\times 2$ active block over $R$, and the PR Research: prove the mixed UT4 z/uz endpoint needs fourteen calls #2175 / PR Prove a sharp weighted region for the mixed UT4 target #2314 exclusions at length $\le 14$ prevent any projected factor from vanishing.
2. Three-role active-plus-dirty quotient decomposition for (5,5,5) (Lemma 2): All common-orientation excess profiles compress to two roles. For the remaining two opposite-orientation three-role modes (Case I: $(e_a^x, e_a^y, e_c^x, e_c^y) = (1, 0, 0, 1)$, Case II: $(0, 1, 1, 0)$), in $M_3(Q)$ the 14 basis monomials without adjacent ${A, C}$ decouple completely from the dirty role on $W_{HH}$ (restricting the active projections to the 5,664 compatible (5,5,5) profile pairs), the off-diagonal blocks $W_{KH} = 0$ and $W_{HK} = 0$ have zero interior dirty contractions (yielding exact homogeneous linear equations on the dirty prefix profiles), and $W_{HH}$ on AC, ACB, BAC receives only the rank-one cross-role contraction.
3. Complete 2-role and 3-role censuses:
   • 2-role: (6,5,4) (3,376 profile pairs, 6,049,824 words, 0 survivors), (5,5,5) (5,664 profile pairs, 8,655,168 words, 0 survivors), (4,5,6) (3,280 profile pairs, 5,874,848 words, 0 survivors) — 20,579,840 new 15-call words, 0 survivors (while reproducing (4,5,4), (5,5,4), (4,5,5) byte-for-byte).
   • 3-role (5,5,5): Case I and Case II each leave 17,864 dirty profile pairs passing $W_{KH} = W_{HK} = 0$ and 42,447,616 expanded 3-role words per mode (84,895,232 total), 0 survivors.
4. Weighted Price Cone & (1,3,1) Sharpness: Every 15-call word has $n_b \ge 6$, and every word with $n_b = 5$ requires $n_a + n_c \ge 11$ ($L \ge 16$) on any finite dirty workspace $r \ge 2$. The direct 14-call (4,6,4) word is globally optimal on $\beta \le 3\min(\alpha, \gamma)$, and at $(\alpha, \beta, \gamma) = (1, 3, 1)$ cost $25$ is impossible and $26$ is the exact minimum.

Scope: Formal $\mathbb{F}_2[\mathrm{UT}_4(\mathbb{F}_2)]$ register-model theorem and computer-assisted necessary-quotient exclusion; no native bit supplier or $\kappa$ improvement is claimed.

Next handoff question: Does any realizing word with $n_b = 5$ exist at $L \ge 16$ (e.g., (7,5,4), (6,5,5), (5,5,6), (4,5,7) at $L = 16$, which also compress to $r \le 3$ roles by PR #2343), or is $n_b \ge 6$ an unconditional structural invariant of $\operatorname{diag}([z], [uz], I_{\text{rest}})$ in $\mathbb{F}_2[\mathrm{UT}_4(\mathbb{F}_2)]$?