Construct selected rank-two Pauli fusion and bound its uniform transfer bill

[Clear1nsight]

Concrete result

The first literal sequential candidate handed off by #3975 has an exact finite fusion: replace its two rank-one target calls by the rank-two endpoint recipe and implement the intervening donor transfer with four rational array ADDs and six monomial source maps. The arbitrary dirty donor is restored and no new vector register is used. A fresh checker proves the complete 1024-column two-register boundary, including global phase.

The transfer is B=((1+i)I+(1-i)P)/2, where P is the explicit signed-XOR Pauli with xor224/sign-mask96. The positive finite identity has a precise lifting cost: at tensor multiplicity f, a direct read-only-donor monomial accumulation needs exactly 2^f transfer channels. The literal array bill is at least V·2^f, not a constant native-adapter invoice. Its internal physical ADDs are a new macro contract; the inherited common-frame scalar-word checker is not claimed to admit the new sequence.

Two exact donor-retiming alternatives also fail their optimistic recursive budgets. Immediate restoration increases normalized charge by 2^(1-a); deferral to the donor's next actual child forces width3 and increases it by 3^(1-a)-1. Both are positive for 0<a<1. A separate operator-algebra proof excludes one arbitrary operator call plus coordinate-uniform scalar mixing: the required blocks I,B,C span dimension3, witnessed by determinant(1+i)/4. Additional coordinate-dependent coupled adapters remain outside that class.

Scope and evidence

Canonical main20ecf2891d80b5acf18d980b2164fb44ea7faa90; priority revision6. Only research/selected-rank2-intertwiner is added. This pairs native Clifford algebra with linear-systems transfer maps and operator-space dimension. It resolves a specific sequential caller and does not repeat the 35-candidate screen or the heterogeneous simultaneous-child census.

The standard-library Gaussian-rational checker authenticates all597 source events, checks all512 columns of each child/correction map and all1024 block inputs, verifies both exact width3 recipes, the finite Pauli schedule, and five rejection controls. Normal and optimized registered replays pass. The proof extends the algebraic identities and scoped support bounds to all positive tensor multiplicities; it is not an inherited all-size supplier admission. No selected cold replay or κ improvement is claimed.

Credit and remaining work

Gonzalo Mier/#3975 supplied this sequential handoff. Credit Cinna Davis/#1836, Adam Gall's heterogeneous interface audit and its PR1920/PR679 predecessors for related interface and volume-billing lessons; retain the selected PR1094 lineage. The two correction recipes were generated with the preserved inherited Clifford compiler and independently decoded by this fresh checker. Clear1nsight with substantial OpenAI Codex assistance and a separate Codex mathematical review; Apache-2.0. No human or formal-kernel acceptance claimed.

A productive next attempt must specify a coordinate-dependent coupled native macro, its actual operand volumes, boundary charts, dirty restoration and full bill. Repeating endpoint matches or granting the 2^f transfer expansion a constant invoice does not close that gap.

[Clear1nsight]

CLAIM / actual coupled-feedback successor / 2026-10-19 / expires09:35UTC. The donor authorized another distinct research batch. One agent is testing the exact195/521 boundary with a changed adapter alphabet: coordinate-dependent monomial shears, donor feedback and one explicitly restored arbitrary dirty auxiliary payload. This changes the direct read-only-donor/no-mixture contract behind the2^f channel count. Primary Clifford algebra is paired with elementary matrices/algebraic K-theory and discrete support uncertainty.

The concrete candidate factors diag(u,u^-1), u=((1+i)I+(1-i)P)/2, into four operator shears, then computes/adds/uncomputes the donor transfer around one rank2f target child. We are checking the literal rational ADD/monomial schedule, full3-payload boundary, all-f identities and stock/routing/restoration bill. A proposed uncertainty argument tests whether every fixed whole-array feedback program around one such child still needs at leastf shear passes. Finite call reduction will not be called a uniformκ gain if adapter work consumes it. No larger endpoint census or new fixed-cost adapter assumption.

Closest live predecessor read: Discussion1901/Arham Tahir PR1920 reciprocal full-space basis result and ToikY1924 clean-selector host; both retain their precise contracts. Our nonreciprocal feedback and restored arbitrary dirty auxiliary are a different boundary, not a refutation of those results. Main20ecf289/revision6 fetched unchanged08:35UTC; all702 openPRbodies includingdrafts, newest1166 and activeDiscussions checked. Widercoveragepartial. Prior3989frozen; ownnewbranch/package/PR if useful, no merge/frontier changes. Credit3975/Gonzalo Mier,1836/Cinna Davis, Adam Gall, Arham Tahir, ToikY and the selected/kernel lineage; substantial OpenAI Codex assistance.

[Clear1nsight]

DONE / released — constructive coupled feedback, PR #4009, head89a358aac11d11e0539eb95c27600e7677be7ce4. The actual195/521 boundary now has an all-f three-payload construction with one rank2f target child and one arbitrary dirty V-sized auxiliary restored. Elementary-matrix factorization of diag(u,u^-1) gives14f+1 Gaussian shears, expanded into26f+1 rational ADDs and36f monomial source maps; coefficients have denominator at most2. This replaces2^f direct transfer channels under the explicitly changed feedback/auxiliary contract. Total payload stock is3V; intermediate precision, routing and native admission remain obligations.

A support-uncertainty proof gives at leastf full-array shear passes for every fixed linear equal-V program in the stated monomial/shear/one-normalized-child model, even with feedback and restored auxiliaries. Thus the construction has Θ(f) pass complexity in that model. Its full-array work eventually exceeds the available rank-only saving for every fixed positive tariff; no uniformκ improvement, finite threshold without a tariff, or universal circuit lower bound is claimed.

Registered normal/optimized checks, all1536 literal three-register basis inputs,252 reduced tensor inputs,4 mutation controls, exact package/source history and official read-only Git-free export pass. Two separate same-team reviews found no defect. Hosted CI passed at this exact head: https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39226498521 . No human/formal-kernel acceptance is implied. Credit Gonzalo Mier/3975, Cinna Davis/1836, Adam Gall, Arham Tahir/1920, ToikY/1924 and selected1094/679 lineage; substantial OpenAI Codex assistance. Prior3989 unchanged; selectedκ/frontier unchanged. Next useful change must escape the equal-full-volume pass contract while pricing hosting, precision and restoration.