M07 original-scope audit, following the R05 candidate.
The physical filter is exactly source Eq17, fhat(u)=exp(-(beta u+1)^2/8+1/8),
and the coherent multiplier is exactly source Eq20,(i/2)tanh(beta u/4).
The unrescaled sum is retained. A new definition-level control builds the
interacting three-mode jumps and Gibbs density directly in the energy basis:
the full generator fixes that density to <1e-14; removing G gives residuals
up to.0023. At lambda0 the total G vanishes, as required. This checks the
normalization and sign without extending a source theorem beyond its scope.

Parity twist: J(X)=PX is an isometry and J^2=I. Hence e^(t J L J)=J e^(tL)J
is contractive on ALL operators even though it is not CP. Moving the full
perturbed twisted generator to the homogeneous side is essential: retaining
only the free twisted generator would leave an extensive perturbation norm.
Outside a local shell, CAR supplies precisely this twist in the derivative
commutation rule. Inside, the tracial local expectation eliminates a local
unital-annihilating perturbation on the complement and gives the factor3.

The matrix kernel uses the same physical-site derivatives on both sides.
Exponential weighted row bounds plus spectral positivity give an integrable
polynomial-times-exponential unweighted row/column bound, independently of N.
The Volterra estimate uses its integral A_beta, not a presumed instantaneous
unit-rate contraction. This avoids the nonlocal-eigenmode interaction gap.

Coherent locality: the positive logarithmic p_beta kernel is integrable at0,
and its Fourier identity was independently checked numerically at two beta/u
pairs. Before integrating it, i[H,B_i] is uniformly local by a shell expansion
and bounded interaction overlap. Its difference tends to zero in a slightly
weaker exponential local norm. The LR time split then supplies a further
positive spatial exponent, however small at large fixed beta. All polynomial
shell factors remain summable. No bound on extensive ||V|| is used.

Use half the available perturbative threshold when selecting lambda_* to
make the closed |lambda|<=lambda_* endpoint explicit. N=1 even states form
a two-state birth/death process with initial total variation<=1 and uniformly
positive rate, so t<=r^-1 log(1/epsilon) handles that case without a prefactor.
No new gap identified. Validity is internally supported; the uniform-locality
and oscillator estimates still warrant outside expert review. Novelty and
publication are separate, and no release has been authorized.
