bibkey: codex2026paulientropy
authors: Codex CLI (canonical exposition of supplied ordinary manuscripts)
year: 2026
doi: null
url: https://github.com/the-omega-institute/trureturing/blob/dev/docs/develop/theory/PAULI_CHANNEL_ENTROPY_GEOMETRY.md
title: "Pauli-channel entropy geometry: support rotation and support-aware covariance funding"
claim: "Local full-complex-fiber rank-three optimization with strict diagonal restriction cost, all-domain BKM covariance plus entropy-PSD lower bounds, and two obstructions to universal auxiliary funding; the original deficit sign remains unresolved."
strata_touched: []
license: citation-only
triage: anchor

Pauli-channel entropy geometry

The canonical mathematical exposition is Pauli-channel entropy geometry. It is repo-derived ordinary mathematics, reconstructed from the supplied unpublished manuscripts below. The first two author lines are “Ordinary mathematical implementation”; the third is “Ordinary mathematical implementation; source-only continuation”, and the fourth is “Ordinary mathematical continuation”. Codex CLI is responsible for this canonical conversion. The manuscripts remain substantive proof inputs, not substitutes for the mathematical derivations retained in the volume.

The relative locators identify the supplied private research collection quantum-zhang-yeung-20261030; they are descriptive provenance locators, not public repository links. The repository URL identifies only the canonical exposition and does not publish the private TeX manuscripts. No attribution to any other named author, transfer of manuscript copyright, or independent priority is asserted.

The common domain is every actual complex $4\times 4$ PSD trace-one source, with $K=X\otimes X$, $L=Z\otimes Z$, and independently variable $a,b\in [0,1]$. Natural-log entropy and the original deficit are retained. The nine real Pauli correlation coefficients and six marginal coordinates parametrize the full Hermitian space, including matrices with complex entries. No global real, rank, separability, equal-bias, or axis restriction is imposed.

Theorem 2.1 is a local theorem about the minimum on full complex fixed-margin fibers near $r=9/10$, $\beta =1/20$, and $a=b=1/2$. The analytic rank-three optimizer is unique against every physical competitor in its fiber, including singular-output competitors. Its unique saturated rank-one slack and its source fail to commute with either marginal. The optimal marginal-diagonal source has the strict cost $\gamma {\epsilon }^2+O({\epsilon }^4)$, where $\gamma =D^2/(B+10m)>0$ with the exact constants in the volume. An explicit rank-three comparison source realizes the same quadratic improvement. The open continuation varies all six marginal coordinates and both channel biases, including unequal biases. This local optimizer result asserts neither a negative physical deficit nor its universal sign.

Theorems 3.3, 4.2, and 5.1 have the entire original source and bias domain. They retain supported BKM paths, all six local covariances, the full normalized cross-operator norm ${\kappa }_t$, the augmented Schur residual, the exact weighted entropy identities, and the entropy-compatible PSD witness bound. The affine Hermitian reference ${\rho }_0$ has physical source margins but need not be PSD. Supported Gibbs references and full Hilbert–Schmidt remainders retain singular cases. Zero Schur complements use the range of the actual augmented tangent expectation vector; all $t=1$ BKM uses are improper integral or support limits. Strictness of the entropy remainder uses the support trace-average and the attained Gibbs optimizer. Propositions 6.1–8.1 give the full transverse product-Z calculation, a strict entire-fiber entropy calibration, and the optional finite algebraic covariance form.

Proposition 10.1 corrects the sufficient funding part of Assertion 9.1: a nonreal rank-two state with unequal biases makes every allowed PSD-witness and augmented-score certificate negative, even with all joint score observables. The original deficit there is at least $\frac{3}{2}\log 2>0$. Proposition 10.3 proves the actual-correlation endpoint lower bound, including singular states. Proposition 10.4 gives a separate full-rank state where even full BKM curvature cannot fund that smaller visible budget: $U<-0.024$, while the original deficit exceeds $0.307$. The volume contains exact input states, spectral polynomials and rational interval bounds with their tail estimates. These are ordinary mathematical arguments, not Lean/kernel certificates. The old lower bounds and Proposition 9.2’s exact pinching identity remain valid. Open assertion 10.5 retains only the original unrestricted sign question; the two refuted sufficient funding assertions are not future proof targets.

The standard intermediate ingredients are literature-attested at their usual scope: finite-dimensional relative-entropy joint convexity and CPTP data processing, Gibbs variational optimization and Pinsker (Watrous, The Theory of Quantum Information, 2018, DOI 10.1017/9781316848142); logarithmic-mean quantum covariance and Fisher-information background (Petz, Covariance and Fisher information in quantum mechanics, arXiv:quant-ph/0106125); finite-dimensional matrix functional calculus (Bhatia, Matrix Analysis, 1997, DOI 10.1007/978-1-4612-0653-8); and the real analytic implicit function theorem (Krantz and Parks, The Implicit Function Theorem: History, Theory, and Applications, 2002). These references support intermediate methods, not the particular Pauli optimizer or universal funding. No finer theorem/page locator or new online literature verification is claimed. The volume uses the standard facts inside the substantive proofs and does not present them as new stand-alone theory results.

The supplied ordinary proofs are the attribution basis for this repo-derived exposition. There is no Lean or kernel verification claim, no formal module registration, no official challenge acceptance, and no unique-solution, priority, or novelty claim. strata_touched: [] records that this is an ordinary-only source note. No exhaustive novelty search is asserted.