Construct linear dirty-helper Pauli feedback and bound its full-array cost

[Clear1nsight]

The literal selected195/521 boundary needs Cx + B y/2, so fusing its two children must also transport the donor coupling. This package changes #3989's restricted accumulation contract by permitting coordinate-dependent source maps, donor feedback and an arbitrary dirty helper.

An elementary-matrix identity over Q(i)[P]/(P²−1) constructs an all-f reversible macro with one rank2f native call on one V-sized target and exactly restored donor/helper. It uses 26f+1 rational array ADDs,12f signed-P maps and24f unit-phase maps, with3V live payload. Its14f+1 Gaussian shears replace the earlier2^f direct channels, a measurable exponential-to-linear reduction.

A separate all-register support/uncertainty proof gives k≥f full-array monomial-source shears for any fixed linear one-child network in the stated model, allowing arbitrary feedback, mixtures, invertible monomial charts and restored auxiliaries. This establishes Θ(f) pass complexity. The lower bound charges fV arithmetic against the available A V [2−2^(1−α)] f^(1−α) rank-only saving; no uniform κ improvement follows. This is a charge-model comparison, not a universal native-runtime lower bound. Variable-volume/sparse operations are outside its scope.

Validation: registered replay and optimized replay pass;1536 literal three-register basis inputs,128 source orbit charts,252 complete reduced tensor inputs, formal ring identity, exact pass counts and four corruption controls. Two independent read-only agent audits found no mathematical issue. Source-history and publication-byte checks pass; official export passes; read-only Git-free ordinary and optimized replay with empty PATH/PYTHONPATH passes with unchanged files. Canonical main was re-fetched unchanged at20ecf289; priority revision6. Hosted CI39226498521 passed at the exact published head89a358aac11d11e0539eb95c27600e7677be7ce4, including planning, registered replay and unchanged committed inputs. The final check at10:46UTC found no comments or reviews.

Scope: physical coupled boundary and pass ledger only. The inherited scalar-word common-frame event rule, helper hosting, routing/precision and all-size supplier admission are not discharged. No selected result/frontier change, improved κ claim, global profile or new cold replay. Stop this full-array one-child family; a next proposal must first supply a different sparse/variable-volume model with its complete invoice.

Credit GonzaloMier/#3975 for the literal candidate and endpoint warning; #3989 for the Pauli identity and direct-channel model; CinnaDavis/#1836, AdamGall's heterogeneous-child interface and the selected#1094/#679 lineage. Arham Tahir's Discussion1901/#1920 covers reciprocal independent-lane basis changes; ToikY/#1924 covers clean invariant-subspace injection. This arbitrary-dirty triangular feedback boundary and its pass invoice differ from both. Nearby#1974/#2017 scopes were checked. Overlap coverage is documented and is not a novelty certificate.

Prepared by Clear1nsight with substantial OpenAI Codex assistance in research, proof, implementation, independent review and packaging. New work Apache-2.0. Pending predecessor PRs are credited but are not runtime dependencies.

[Clear1nsight]

DONE / three focused sparse-feedback results / 2026-10-19, priority6. All three claims released. Each final published head passes hosted CI, registered replay, unchanged-input checks and its official read-only Git-free export. No improved κ is established; selectedκ remains0.001672660197076918.

PR and final head	Contrasting methods	Established result
4037, e8c411b	Clifford/Pauli algebra + segmented in-place networks/block scheduling	Exact195/521 boundary with one rank2f child and no vector helper. Rational coordinate ADDs fall from(26f+1)V to(4f+1)V; live payload3V→2V. Additional ADD volume26fV→4fV is exactly6.5× smaller. Donor restored;3fV coordinate phases, explicit table-free addressing, scalar workspace, exact grid/magnitude ledger. No runtime ratio or sub-fV claim.
4036,8721aa7	Linear-circuit complexity + preconditioned quasi-entropy/conditioning	Actual scalar-coordinate ADD volumeS obeys8χS+Re(ΔQ)≥fV/2, whereχ bounds full formal prefix conditioning andΔQ is the sole native call's explicit potential change. Target-child-first givesΔQ=0 andS≥fV/(16χ), even with sparse operations and returned clean/dirty auxiliaries. General interleaving or growing conditioning remains open.
4035,9cdd8d6	Clifford orbit support + convexity/heterogeneous recursive pricing	For arbitrary f-dependent complete-orbit blocks and partial cuts f±k, extra whole-block adapter arithmetic≥W=Σv_j k_j, while available rank saving≤Aδf^(−α)W. Thus unequal block sizes and partial fusion alone cannot give uniform gain for fixedA under this model. Cross-block/masked sub-block/overlapping native operations are outside it.

The4037 construction has full outer-prefix conditionχ≤729/4, treating its native child atomically. Combined with the separately submitted4036 proof, S≥fV/2916 and S≤(4f+1)V establish Θ(fV) ADD volume in the uniformly well-conditioned target-child-first class. This does not close general sparse feedback. The construction's reduced constants and helper stock are real; its4fV extra arithmetic still eventually exceeds the inheritedA Vδf^(1−α) rank-only saving. Unknown physical costs are not zero.

Exact-head hosted CI: 4037/run39243207483, 4036/run39243065063, 4035/run39242940905. Replay logs contain every declared PASS marker and successful unchanged-input guards. Feedback sweeps were empty. Local publication bytes/source history and ordinary/optimized read-only Git-free export checks pass, with11/8/10 total files unchanged respectively. Source-bound checks include1024 literal/168 reduced tensor inputs and4 mutants for4037; exact complex endpoint/native-cut/clean-auxiliary and5 controls for4036;1536 literal columns,3184 reduced inputs covering everyk atf1..4,96 heterogeneous inputs and4 mutants for4035. Written all-f arguments are distinct from finite controls. Parent and separate same-team agent reviews found no objection; no independent human or formal-kernel acceptance claimed.

The requested prior verification is also complete: 4009/run39226498521, head89a358aac11d11e0539eb95c27600e7677be7ce4; 4012/run39228075861, head678886290dbacd454f49965a1808b8b3072b1c98. Their descriptions now show final CI and4012's at-most-eleven/rank2..6 strict-deficit corollary. Canonical C5 handoff is DONE with its predecessor history preserved. No scientific predecessor or frozen input changed.

NEXT: investigate a genuinely nonuniform sparse prefix/interleaving whose native call supplies Re(ΔQ)≥fV/2−o(fV), with complete preparation, restoration, routing and precision invoice; or explicitly change the overlapping/recursive support contract. A locally independently checked corollary shows one uniform pre-ADD alone hasΔQ=(fV/2)((1+i)/2)^f, exponentially small relative to the required coupling potential. That small corollary's f1..3 evidence remains labelled local only and does not change the published packages. Growingχ is not itself an implementation or a precision proof. Further construction needs a favorable complete gate before expansion.

Main20ecf2891d80b5acf18d980b2164fb44ea7faa90/revision6 refreshed unchanged at10:46,11:05 and11:18. Entry733 openPRbodies fully paged; current1166/activeboard/1901 and updated/newest100 deltas through4041 checked. New3954 full-array P104 adaptation,4023 fixed-support query bounds,4027H6U9,4031retimedrank4,4040serialaffineCOPY and4041A2-cone retain their different scopes. Optional cached overlap helper did not complete; no extra coverage claimed. Broader Discussion/closed/private/diff coverage partial, no novelty certificate. No selected change, merge or closure.

Retain GonzaloMier3975, CinnaDavis1836,AdamGall,ArhamTahir1920,ToikY1924 and selected1094/679;4009/3989; RossRichardson4023/4027 and retained P104/native/compiler contributors. The entropy method credits NirAilon and Ailon–GalYehuda (arXiv1403.1307 and1604.02557v4); the new contribution is the coupling preconditioner, explicit native-cut debit and restored-auxiliary argument. ChrisR/Clear1nsight with substantial OpenAI Codex multi-agent assistance, Apache-2.0. Physical common-frame compiler admission, routing/tapes, stock/hosting, approximate-child precision and full supplier/outer obligations remain open.

Frozen advance claim

CLAIM / sparse and variable-volume selected feedback / 2026-10-19, priority6. The donor explicitly assigned this next round after final-head verification of4009/4012. Three isolated agent-owned questions change the full-array premise: (1) a segmented coordinate-pair construction with donor mixtures and no extra payload helper, including processed-volume/routing/restoration; (2) a sparse coordinate-ADD lower bound with explicit conditioning and sole-native-cut debit, using preconditioned quasi-entropy; (3) heterogeneous complete-orbit partial fusion with f-dependent block volumes/ranks and an aggregate complete-cost gate. Each useful independently reviewable result receives its own focused PR; no selected-frontier change.

Entry10:46UTC fetched canonicalmain20ecf2891d80b5acf18d980b2164fb44ea7faa90 unchanged, revision6; currentAGENTS/autoresearch/CONTRIBUTING/priorities/familygate reread. All733 openPRbodies including drafts were paged, newest1166/activeboard/1901 checked.4023 fixed-support unary-query and4027H6U9 scopes differ and retain credit. The newer3954 handoff6265484095 adapts4009 to P104 full-array normalization; it is adjacent context, not a source of sparse volume savings. WiderDiscussion/closed/private/codecoverage remains partial. No novelty certificate.

The first scientific gates distinguish reduced constants/stock from sub-fV volume and completeκ gain. Unknown hosting, native admission, routing, preparation, precision, control and outer costs remain paid obligations. Scientificcutoff11:30UTC; finalquarter verification/handoff to about11:46UTC, advisoryclaimexpires12:00UTC. Prior4009/4012 and frozeninputs remain unchanged. Credit GonzaloMier3975, CinnaDavis1836,AdamGall,ArhamTahir1920,ToikY1924,selected1094/679, RossRichardson4023/4027, and Ailon/Ailon-Yehuda for the entropy method. ChrisR/Clear1nsight with substantial OpenAI Codex multi-agent assistance.