M07-v1 — independent strict review, 22 October 2026

Judgments
Validity: supported internally; no proof gap or refutation found.
Original coverage: complete original for each fixed beta>0 and fixed bounded
finite-range Hermitian hopping class in D>=2, sufficiently small even local
interactions, the prescribed unrescaled Gaussian KMS generator including its
coherent correction, and even initial states. No uniformity as beta->infinity
or claim for arbitrary strong interactions is made.
Source comparison: difference supported in a bounded current primary-source
comparison; no identified unresolved covering theorem. Bibliography should
reflect the current one-dimensional paper version, as noted below.
External status: no external expert acceptance established; no publication or
submission performed by this review.

Target and difficult proof bridges
Original campaign/originals/M07.tex:6–23; all candidate files and manuscript
hashes are in frozen-inputs.json. The main theorem is manuscript main.tex:94–107.
1. proof.txt:8–30: complex hopping needs a transpose for creation coefficients.
The CAR identity uses R and R^T respectively and the total free coherent term
vanishes because total loss commutes with H0. It does not falsely set every
site-labelled coherent term to zero. Left parity multiplication Jp is an
isometry; exp(t Jp L Jp)=Jp exp(tL)Jp is contractive even though it is not CP.
This is enough for the homogeneous derivative equation.
2. A spectral lower bound for R alone would not give a uniform absolute row
sum. Lines 32–46 use a weighted spatial matrix norm for the far tail and the
spectral norm for entries in a ball of radius vt. Polynomial volume growth
gives C(1+t)^D exp(-r0 t/2), both row and column sums, hence finite A_beta and
B_beta. No exponentially small Gibbs eigenvalue appears in trace conversion.
3. Lines 48–96 retain the full coherent correction. The positive normalized
p_beta kernel has a logarithmic integrable singularity, exponential tails,
and Fourier transform tanh(beta u/4)/(beta u/4). Multiplication by i beta u/8
is precisely the required tanh multiplier. Commuting the local shell expansion
with H costs a support polynomial, still summable. Duhamel at fixed time and
a separately controlled tail give uniform local continuity. This replaces,
rather than misapplies, the source's non-hopping perturbation lemma.
4. Lines 98–128 close physical-site gradients. Outside the local shell the
graded commutation identity has exactly the parity twist; inside, unital
annihilation and telescoping bound forcing by 3||E_X|| times local gradients.
The FULL interacting twisted generator is on the homogeneous side, so no
extensive error is left there. The spatial kernel commutes with the operator
evolution. kappa A_beta<=1/2 yields exponential gradient decay and trace mixing.
5. Lines 130–141 identify the exact Gibbs stationary density using KMS
fixation. The self-adjoint source coupling convention is respected because
(c,c†) is a unitary label rotation of the two Majoranas. N=1 is treated directly
as a uniformly positive birth–death process. Only even-state uniqueness is
needed by the original.

Primary sources and current equivalents
• Smid et al., 2510.04954v2, §II.1, §IV.3, Appendices A/B: exact Gaussian
sampler, coherent completion, and locality. Its perturbative rapid-mixing
reference has no hopping. The reviewed proof adds physical-site gradients
and their spatial kernel; it does not import the non-hopping theorem wholesale.
https://arxiv.org/html/2510.04954v2
• 2610.05848v2 (21 October 2026), Theorems 1/2: a one-dimensional spin
sampler with single-site Pauli couplings and O(log^4 N+log(1/epsilon)) bound.
It does not cover the D>=2 fermionic logarithmic claim. The manuscript currently
cites v1 at line 434; update/acknowledge v2 in a subsequent release edit.
https://arxiv.org/html/2610.05848v2
• 2610.03578v1, Lemma 3.1 and the smoothed heat-bath/QBP block-reset
construction: its positive Fourier kernel is inherited and credited. The
strict quantum Dobrushin–Shlosman result is for its own sampler and assumptions;
it does not establish the prescribed physical-jump generator here.
https://arxiv.org/html/2610.03578v1
• Han–Zhao–Liu, 2609.35159v1 (28 September), Theorems A.1/A.2: sufficiently
weak local fermionic Gibbs states are Gaussian mixtures, with a polynomial-time
classical algorithm sampling Gaussian leaves. I read the formal hypotheses
and output definition. The theorem does not assert logarithmic trace mixing
for this Lindbladian. It is relevant new background, not a covering theorem.
https://arxiv.org/html/2609.35159v1
• Kim, 2610.01190v1 (1 October), Eq. (3), Theorem 1.1: logarithmic mixing
for constant-block Petz recovery updates on one-dimensional finite-dimensional
spin chains. The geometry and generator both differ from this result.
https://arxiv.org/html/2610.01190v1

Controls and manuscript correspondence
Both clean copies pass check-car.py and check-kms.py. They exercise the
complex-hopping CAR identity and exact KMS completion; their finite sizes do
not certify the uniform theorem. The complete analytical closure above is
needed. Manuscript sections on free gradients (109), spatial kernel (160),
full sampler locality (198), coherent correction (225), forcing lemma (337),
and Volterra inequality (388) agree with the frozen proof. No unsupported
scope expansion found. Retain; make the bibliographic version update without
claiming exhaustive priority clearance.
