M07 complete-original candidate, version1, 20 October2026
Physical-site graded derivatives give stable logarithmic mixing for bounded
finite-range Hermitian hopping and sufficiently weak bounded even interactions,
with the original Gaussian KMS sampler, physical jumps and coherent completion.
All constants below depend only on beta and the fixed local/geometry bounds.
No diagonalization of interacting local terms into nonlocal modes is used.

1. FIXED FREE GENERATOR AND GRADIENT IDENTITY
Use the source Gaussian fhat(u)=exp(-(beta u+1)^2/8+1/8), and its unrescaled
sum of jumps. For complex M, the annihilation coefficients are fhat(-M), and
creation coefficients are fhat(M)^T. Put
 R=fhat(-M)^2+fhat(M)^2=2 exp(-beta^2 M^2/4) cosh(beta M/2).
If ||M||<=J, R>=r0 I with r0=2exp(-beta^2 J^2/4)>0. The total free loss
commutes with H0, so its prescribed coherent correction is zero. Individual
site-labelled coherent terms need not vanish; only their total is used here.

Let P be total parity, Jp(X)=PX, and L^tw=Jp L Jp. For any Lindbladian L,
exp(t L^tw)=Jp exp(t L) Jp is an operator-norm contraction, since left
multiplication by P is an isometry and the Heisenberg semigroup is unital CP.
No positivity of the twisted semigroup is asserted.
On even observables define partial_i O=[c_i,O] and barpartial_i O=[c_i^dagger,O].
For a linear odd jump l, direct CAR algebra gives
 partial_i D(l)O=D(l)^tw partial_i O
   -( {c_i,l^dagger}[l,O]+{c_i,l}[l^dagger,O] )/2.
The anticommutators in the final expression are scalars. Thus for the free
generator, in column-vector notation,
 partial L0 O=L0^tw partial O-(R/2)partial O,
 barpartial L0 O=L0^tw barpartial O-(R^T/2)barpartial O.
The finite complex-hopping controls in R04 test this exact identity; the
displayed CAR calculation is its proof, not an assumption about eigenmodes.

2. A VOLUME-INDEPENDENT INTEGRABLE MATRIX KERNEL
For a matrix A define ||A||_mu=max_i sum_j exp(mu dist(i,j))|A_ij|. The metric
triangle inequality makes this norm submultiplicative. Finite range and
bounded local strength give ||M||_mu<=J_mu uniformly in the volume. The entire
series for R gives a finite bound B_mu on ||R||_mu, depending only on beta,J_mu.
Consequently for S_t=exp(-tR/2), ||S_t||_mu<=exp(B_mu t/2), while its spectral
norm is <=exp(-r0 t/2). Split the absolute row sum into dist<=v t and its
complement, choosing mu v>=B_mu/2+r0/2. Polynomial ball growth then gives
 K(t):=max_i sum_j |(S_t)_ij| <=C_beta(1+t)^D exp(-r0 t/2).
The tail is bounded by exp(-mu v t)||S_t||_mu; every near entry is bounded
by the spectral norm. Hermiticity gives the same column bound and hence the
same bound for R^T. For gamma=r0/4, both
 A_beta=integral_0^infinity exp(gamma t)K(t)dt,
 B_beta=sup_t exp(gamma t)K(t)
are finite uniformly in N and the bounded instance.

3. FULL SAMPLER PERTURBATION, INCLUDING THE COHERENT TERM
Write the two filtered jumps at site i as K_i,sigma(lambda), and group their
loss into B_i(lambda)=sum_sigma K_i,sigma^dagger K_i,sigma. Fermionic LR
for the fixed finite-range H_lambda, |lambda|<=1, and the Gaussian filter imply
uniform exponential local tails and uniform bounded norms for K_i,sigma.
Odd localization can use the even-operator-times-Majorana construction in
M03, or the tracial CAR expectation. Local Duhamel gives for fixed T
 ||K_i,sigma(lambda)-K_i,sigma(0)||
 <=C|lambda|T exp(vT)+2 integral_{|t|>T}|f(t)|dt.
Let T grow slowly as lambda->0. Thus these differences tend to zero uniformly
in i,N; their spatial tails are uniform as well. By splitting shell sums at
a fixed radius, they tend to zero in every weaker exponential local norm.
Products show the same properties for B_i(lambda).

The apparent singular kernel for the coherent correction need not be treated
as an absolutely integrable 1/t kernel. Define the normalized positive kernel
 p_beta(t)=4/(pi beta) log coth(pi |t|/beta),
whose Fourier transform is tanh(beta u/4)/(beta u/4). It has an integrable
logarithmic singularity at zero and exponential tails. The original source
coherent term with multiplier (i/2)tanh(beta ad_H/4) is EXACTLY
 G_i(lambda)=(beta/8) integral p_beta(t) tau_t^{H_lambda}
                    ( i[H_lambda,B_i(lambda)] )dt.
This identity is spectral calculus: multiply the kernel transform by i beta u/8.
It is valid on every finite-volume Bohr frequency, independent of the optional
frequency cutoff used to represent the same source correction. It preserves
the source normalization and does not remove the coherent term.

Commutation with a bounded finite-range Hamiltonian maps an exponentially
local observable about i to an exponentially local one, at a slightly weaker
exponent and with polynomial support factors: expand it in local shells,
and only terms meeting a shell support contribute. The resulting polynomial
factor is summable. Thus C_i(lambda)=i[H_lambda,B_i(lambda)] has uniform
exponential tails and C_i(lambda)-C_i(0)->0 in a weaker local norm. Indeed
the difference is i[H0,B_i(lambda)-B_i(0)]+i lambda[V,B_i(lambda)].
The second commutator is uniformly bounded by the same local shell estimate.
Apply LR to the p_beta integral, splitting time at a constant times the
localization radius. Its exponential time tail and the LR bound give uniform
exponential spatial tails for G_i(lambda). Split time at fixed T and use local
Duhamel, then take T large, to prove ||G_i(lambda)-G_i(0)||->0 uniformly.
All these are LOCAL per-site estimates; no global ||V||=O(N) bound is used.

Let E_i=L_i(lambda)-L_i(0), including both dissipators and i[G_i, .]. Localize
jumps and G_i to B(i,r), use their differences at consecutive radii, and subtract
the corresponding free approximants. This gives E_i=sum_{r>=0}E_i,r with
support X_i,r=B(i,r), total even parity, and E_i,r(I)=0. Uniform bounds give
 ||E_i,0||<=C eta(lambda),
 ||E_i,r||<=epsilon_r(lambda):=C min{eta(lambda),exp(-c r)} for r>=1,
where eta(lambda)->0. These are full induced operator norms. It follows that
 kappa(lambda):=6 C_ball^2 sum_r(1+r)^(2D)epsilon_r(lambda)->0.

4. PERTURBED DERIVATIVE CLOSURE
Set S(O)=sum_i(||partial_i O||+||barpartial_i O||). On even observables,
fermionic tracial expectations T_i give
 ||O-tr(O)I||<=S(O), S(O)<=4N||O||.
For example ||O-T_iO||<=||[c_i,O]||+||[c_i^dagger,O]|| follows by writing
T_i as the average of conjugation by I and the three local CAR Pauli unitaries.
T_i commute and are contractions on the even algebra, so telescoping proves
the first estimate.

For any local even superoperator E_X annihilating identity,
 partial_i E_XO-E_X^tw partial_i O=0 outside X.
Inside X its norm is <=3||E_X|| sum_{j in X}(||partial_jO||+||barpartial_jO||).
To see this, E_X T_X O=0 by locality and unital annihilation; hence
||E_XO||<=||E_X|| sum_X derivatives. The first commutator term costs a factor2,
and the twisted second term costs a factor1. The outside identity follows
termwise from CAR, with the twist giving the sign for odd left/right factors.
The same assertions hold for creation derivatives.

For O(t)=exp(tL_lambda)O, move the FULL L_lambda^tw onto the homogeneous side:
 d partial O(t)/dt = L_lambda^tw partial O(t)-(R/2)partial O(t)+F(t),
and analogously for creation derivatives, where the summed forcing norms are
at most kappa(lambda)S(O(t)). Matrix-index evolution commutes with the operator
superoperator. Contractivity of exp(tL_lambda^tw), the K(t) bound, and Duhamel
give
 S(O(t))<=K(t)S(O)+kappa integral_0^t K(t-u)S(O(u))du.
Choose lambda_*>0 smaller than half a valid perturbative threshold so that
kappa(lambda)A_beta<=1/2 for all |lambda|<=lambda_*.
Multiplying by exp(gamma t) and taking the supremum up to time t yields
 S(O(t))<=2B_beta exp(-gamma t) S(O).
This is a contraction estimate in the PHYSICAL lattice, not a gap-to-trace
conversion involving an exponentially small stationary eigenvalue.

5. EXACT TARGET AND MIXING
The source detailed-balance construction fixes the Gibbs density of H_lambda.
It is even. Trace duality can be restricted to even observables for differences
of even states. Consequently
 (1/2)||exp(tL_lambda^*)rho-rho_beta,lambda||_1
       <=8B_beta N exp(-gamma t), for every even rho.
For N>=2 this implies C_beta log(N/epsilon) mixing, by absorbing the fixed
prefactor into C_beta. N=1 is a single-mode even Hamiltonian, hence a diagonal
birth/death process with uniformly positive total rate on the bounded input
class; it obeys the same form after increasing C_beta if needed. The even
stationary density is unique by contraction. The claim in M07 concerns even
initial states, so no stronger odd-sector uniqueness assertion is required.

SOURCE AND REVIEW BOUNDARY
Inherited: exact Gaussian sampler and KMS fixation, arXiv:2510.04954v2 II.1;
fermionic LR and localization, 1705.08553v3 and2510.04954v2 App.B; the positive
kernel Fourier identity also appears in2610.03578v1 Lemma3.1. The source App.A
perturbation lemma assumes a non-hopping reference; it is NOT invoked outside
its scope. Section3 above supplies the needed hopping-reference estimates.
New interface here: physical-site commutator gradients, parity-twisted
contractivity, and an integrable spatial matrix kernel for the Volterra bound.
Candidate validity and novelty require separate audit. Numerical controls
confirm finite CAR identities but are not a proof of the uniform estimates.
