M03: quasilocal stability of fermionic amplitude damping
Version 2, 20 October 2026. Complete-original candidate; review pending.
Sources: original M03.tex; R01 locality interface; arXiv:2503.15827v3,
Appendix I; arXiv:1705.08553v3, Theorem 3.1 and Lemmas 4.2--4.3.
This is a derivation for internal review, not an external verification claim.

ASSUMPTIONS AND REDUCTION
Use the fixed geometry, range, local strength, one-particle gap and filter
assumptions in the original. Constants below depend on these fixed parameters,
not the volume. Balls satisfy |B(i,r)| <= Cb(1+r)^D. The filter has integrable
stretched-exponential time tails. Write its negative-band plateau as 1.
The R01 real particle-hole transform gives CAR modes d_i, a free Hamiltonian
d^dagger |M| d plus a scalar, and an exponentially summable interaction lambda V.
The free Hamiltonian also has an exponentially summable interaction. Label
unitary invariance of the full 2N jump list identifies the ORIGINAL generator
with jumps K_i^- = F_H(d_i), K_i^+ = F_H(d_i^dagger). At lambda=0 these are
exactly d_i and 0, since the spectrum of |M| lies in the filter plateau.

UNIFORM JUMP ESTIMATES
Fermionic Lieb--Robinson bounds for exponentially summable even interactions
(1705.08553, Theorem 3.1, with F(r)=exp(-a r)/(1+r)^(D+1)) give uniform local
approximants to K_i^sigma with tail g(r) <= C exp(-c r^b), 0<b<=1.
Here is a parity-safe construction. Choose a Majorana gamma_i at i and let
E_B be the even conditional expectation of 1705.08553, Sec.4. Set
 K_i^{sigma,r} = E_{B(i,r)}(K_i^sigma gamma_i) gamma_i.
The argument of E is even; its output is even and local, so the approximant
is odd and local. This map is contractive, preserves d_i and 0, and its tail
is bounded by Lemma 4.3. For even distant B, [d_i(t) gamma_i,B] is bounded by
the LR commutator; for odd distant B it equals minus the LR anticommutator
times gamma_i. Summing the LR spatial bound over the exterior of the ball
is volume independent. Split the filter integral at time proportional to r:
the interior part is exponentially small in r and the exterior part is
bounded by the stretched-exponential filter tail. Polynomial factors can be
absorbed by decreasing c. This proves the stated g uniformly for |lambda|<=1.

Local Duhamel and the same LR bound give, for either coupling A_i,
 ||tau_t^{H_lambda}(A_i)-tau_t^{H_0}(A_i)||
       <= C |lambda| |t| exp(v|t|), uniformly in N.
For any T>=1 this implies
 eta(lambda):=sup_i,sigma ||K_i^sigma(lambda)-K_i^sigma(0)||
 <= C |lambda| T exp(vT) + 2 integral_{|t|>T}|f(t)| dt.
Taking T growing sufficiently slowly as lambda tends to zero proves
eta(lambda)->0. For example T=log(1/|lambda|)/(2v) works when v>0.

LOCAL SUPEROPERATOR DECOMPOSITION
All dissipators in this note are Heisenberg dissipators
D(K)(O)=K^dagger O K-{K^dagger K,O}/2.
Put A_i^r=D(K_i^{-,r})+D(K_i^{+,r}),
 E_i,0=A_i^0-D(d_i), and E_i,r=A_i^r-A_i^{r-1} for r>=1.
Then L_i(lambda)-D(d_i)=sum_r E_i,r in operator norm, and each E_i,r
is parity preserving, supported on X_i,r=B(i,r), and annihilates identity.
The elementary bound ||D(K)-D(L)|| <= 2(||K||+||L||)||K-L|| gives
 ||E_i,r|| <= epsilon_r(lambda),
 epsilon_0=C eta(lambda),
 epsilon_r <= C min{eta(lambda),g(r-1)} for r>=1.
All norms here are operator-norm induced superoperator norms. No complete
bounded norm is needed. The majorant sum_r (1+r)^(2D) g(r-1) is finite.

OSCILLATOR SEMINORM AND LOCAL ESTIMATES
Work only on the even observable algebra. For each site let T_i be the
normalized fermionic partial trace/expectation onto its complement, let
delta_i=I-T_i, and P_i(O)=(O+Z_i O Z_i)/2, Q_i=I-P_i, Z_i=1-2d_i^dagger d_i.
All T_i,P_i,Q_i are contractions on this algebra; ||delta_i||<=2.
Write Delta_i,+ = delta_i P_i, Delta_i,-=delta_i Q_i,
 s_i(O)=||Delta_i,+ O||+||Delta_i,- O||, S(O)=sum_i s_i(O).
Then S(O)<=4N||O|| and, by telescoping the commuting T_i,
 ||O-Tr(O)I/2^N|| <= S(O).

For any even local superoperator E_X annihilating identity, of the form a sum
of products p O q with p,q supported on X and total even parity, the CAR and
the normalized trace imply:
 (i) E_X T_X O=0;
 (ii) [Delta_i,sigma,E_X]O=0 for i outside X;
 (iii) ||E_X O|| <= ||E_X|| sum_{k in X}s_k(O).
For (iii), telescope O-T_XO and use contractivity of the other T_k. Assertion
(ii) is also Lemma 14 of 2503.15827 plus commutation with local parity outside X.
The annihilation assertion uses both locality and E_X(I)=0; it is NOT true for
an arbitrary local superoperator without the latter condition.
Consequently for i in X,
 ||Delta_i,sigma E_X O|| <=2||E_X|| sum_X s_k(O),
 ||[Delta_i,sigma,E_X]O|| <=3||E_X|| sum_X s_k(O).

DECAY
For the free local dissipator the exact identities are
 Delta_i,+ D(d_i) O=-Delta_i,+ O,
 Delta_i,- D(d_i) O=-(1/2)Delta_i,- O.
Let O(t)=exp(t L_lambda)O. For X_i,sigma(t)=Delta_i,sigma O(t), use the
contractive unital CP semigroup generated by sum_{j != i}L_j(lambda), together
with the damping above. The remaining forcing is
 Delta_i,sigma E_i O(t) + sum_{j != i}[Delta_i,sigma,E_j]O(t).
Duhamel and the local estimates, summed over i and sigma, yield
 S(O(t)) <= e^{-t/2} S(O) + kappa(lambda) integral_0^t e^{-(t-u)/2}S(O(u))du,
where a sufficient bound is
 kappa(lambda)=6 Cb^2 sum_{r>=0}(1+r)^(2D) epsilon_r(lambda).
Indeed, summing over sites affected by E_j,r costs |X_j,r|, and counting its
centres j whose ball contains a fixed k costs at most Cb(1+r)^D again.
Dominated convergence gives kappa(lambda)->0 uniformly in volume.
Choose lambda_*>0 smaller than half the threshold above, so that
kappa(lambda)<=1/4 for every |lambda|<=lambda_*.
This choice is quantitative without volume dependence: first choose r_* with
weighted g-tail <1/8, then choose eta small enough that the finite initial
sum is <1/8. Gronwall gives S(O(t))<=exp(-t/4)S(O).

GROUND STATE, UNIQUENESS, AND MIXING
The filter vanishes at every nonnegative frequency, including zero. Therefore
every ground vector is annihilated by every jump. Each ground-state projector
is stationary. Since H is even, it has a parity-eigenstate ground vector; its
projector is even. For any two even states rho,sigma, trace duality and the
seminorm estimate give
 (1/2)||exp(t L_lambda^*)(rho-sigma)||_1 <= 4N exp(-t/4).
Hence there is exactly one even stationary state. Distinct ground vectors
would give distinct even stationary projectors after choosing a parity basis,
so the ground space is one dimensional. This identifies the even stationary
state with the pure ground-state density rho_gs. Any stationary state can be
parity twirled; the result must be rho_gs. Extremality of a pure state then
forces the original stationary state itself to be rho_gs.
For even initial rho, total-variation/trace distance <=4N exp(-t/4), hence
t>=4 log(4N/eta) suffices. Both spectral bands are nonempty, so N>=2. For 0<eta<1,
4 log(4N/eta)<=12 log(N/eta), exactly the required form.

REVIEW ITEMS
Independently check the local CAR/expectation identities, the original filter
tail assumption, quantitative uniformity of the LR norm after particle-hole
transformation, and exact trace-distance convention. No numerical gap is used
as evidence for the thermodynamic limit. Novelty compared with the source's
positive-spectrum theorem is separate from validity of this extension.
