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Projected Commutator Cancellation for Pauli-Type Channels: A Conditional Parity Gate for BBGKY Bridge Diagnostics

Panasenko

Published Jun 19, 2026• 0 citations
Publisher

Abstract

This preprint develops a conditional projected parity-cancellation gate for commutator terms in Pauli-type projected channels within a BBGKY bridge diagnostic framework. It replaces a naive empty-window commutator obstruction by a verifiable conditional criterion involving resolution loss, parity leakage, and tail control. The result is a conditional gate theorem, not a proof of unconditional cutoff-free Coulomb BBGKY closure.

DOI
10.5281/zenodo.20760370

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This preprint develops a conditional projected parity-cancellation gate for commutator terms in Pauli-type projected channels within a BBGKY bridge diagnostic framework. It replaces a naive empty-window commutator obstruction by a verifiable conditional criterion involving resolution loss, parity leakage, and tail control. The result is a conditional gate theorem, not a proof of unconditional cutoff-free Coulomb BBGKY closure.
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This preprint develops an anisotropic Pauli-channel optimization layer for projected BBGKY bridge estimates under CPTP constraints. It refines the isotropic depolarizing-channel bridge by replacing a scalar attenuation parameter with a sectorwise Pauli-diagonal channel, formulating the admissible channel set through the Pauli CPTP tetrahedron, and deriving a weighted convex projection problem with KKT and active-set certification. The paper preserves the conditional nature of the underlying bridge estimates: it does not claim cutoff-free Coulomb BBGKY closure or automatic sectorwise commutator decay.
We formulate a conditional projected-channel renormalization theorem forcutoff-regularized Coulomb BBGKY residuals. The starting point is the spectralobstruction that a similarity transform Z−1W Z cannot attenuate a non-zeroHermitian matrix witness in full operator norm. The replacement mechanismstudied here is a non-similarity observation channel: in the minimal model, aPauli-diagonal completely positive trace-preserving channel acts on an M2(C)internal layer and damps declared contrast observables while preserving thetrace sector. The main result is a conditional dynamic gate theorem: if theresidual source, comparison defect, derivative budget, interaction-transportcommutator, and dual-orbit attenuation assumptions are supplied by a concretecutoff model or by an upstream higher-correlation gate, then the projectedwitness component is controlled with an explicit error budget. The theoremis deliberately conditional; it is not a cutoff-free Coulomb closure, not apropagation-of-chaos theorem in trace norm, and not a derivation of thesource budget from the full BBGKY hierarchy. Its contribution is to isolate theexact price of an E6-type operator-valued renormalization: contrast damping,observable-comparison loss, derivative compatibility, and source-budget controlmust all be paid separately.
This preprint is written in Ukrainian. It develops a toy-model theorem note for the E4/E6 escape gates in a post-obstruction BBGKY/Coulomb closure program. The manuscript proves a spectral obstruction to full witness attenuation by invertible similarity renormalization and formulates projected, quotient, and non-similarity channel alternatives with explicit comparison budgets. The work is a toy-gate impossibility and certification note; it does not claim cutoff-free Coulomb BBGKY closure. Препринт написано українською мовою. У роботі розроблено toy-model theorem note для шлюзів E4/E6 у пост-обструкційній програмі BBGKY/Coulomb closure. Основний результат — спектральна перешкода для повного поглинання witness через інвертовану similarity-ренормалізацію, а також projected/quotient/non-similarity channel альтернативи з явними comparison budgets. Робота не стверджує cutoff-free Coulomb BBGKY closure.
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We develop a decoder-agnostic logical-class diagnostic layer for abelian Pauli stabilizer codes, motivated by fault-tolerant quantum computing. An ancilla fermionic register defines a nilpotent supercharge whose clean degree-zero harmonic sector is the code space. Projected-return blocks derived from graded commutators organize the recovery test of whether a decoder-selected correction preserves every encoded input for every error in a declared same-syndrome family; the passing syndromes form the certified faithful window. Demoting the supercharge to an exact binary diagnostic formulation classifies residuals modulo stabilizers, separating a check violation, a harmless stabilizer residual, and a syndrome-silent nontrivial logical action, with a two-sector form for CSS codes. Because the unknown fault cancels, the relative logical class of two record-compatible candidates is computable online. We verify the diagnostic formalisms on the Steane code and specify a static--circuit bridge audit protocol for the Gross bivariate-bicycle quantum low-density parity-check (qLDPC) code. The method supplements existing decoders without additional physical qubits for software use.