HISTORICAL DEPENDENCY NOTE: the partial status below records R01. The current
complete argument is proof.txt; second-review.txt audits that closure.

M03 / R01: the real particle-hole transformation and its locality interface

Exact original: campaign/originals/M03.tex. Imported baseline:
Zhan et al., arXiv:2503.15827v3, Theorem 11 and Appendix I.
The theorem assumes M > Delta. Its text remarks that quasilocal perturbations
can be accommodated, but the filled-band transformation also changes the free
quadratic matrix. That additional interface must be checked explicitly.

Algebraic lemma. For real symmetric M with spectrum in [-J,-Delta] union
[Delta,J], let P_+,P_- be its real orthogonal spectral projectors and set
d=P_+c+P_-c^dagger. Then
 {d_i,d_j^dagger}=delta_ij and {d_i,d_j}=0,
by P_+^2+P_-^2=I, P_+P_-=0 and symmetry. The transformation is involutive;
its 2N-by-2N Nambu matrix [[P_+,P_-],[P_-,P_+]] is real orthogonal.
The quadratic Hamiltonian becomes
 c^dagger M c = Tr(MP_-) + d^dagger |M| d.
This follows by splitting the positive and negative subspaces and applying
the CAR to the negative-band term. No localized Wannier basis is required.

Uniform locality lemma. Assume the lattice has polynomial ball growth, M
has fixed range R, and sup_i sum_j |M_ij| <= J0, uniformly in volume.
Choose a contour surrounding one spectral band at distance delta>0 from the
spectrum, with contour length depending only on J,Delta. Conjugating M by
diag(exp(mu dist(i,x))) changes its norm by at most
J0(exp(mu R)-1), using both row and column bounds and the Schur norm test.
Choose mu so that this is <=delta/2. The resolvent identity then bounds the
conjugated resolvent by 2/delta. Contour integration gives
 |(P_+)_ij|+|(P_-)_ij| <= C exp(-mu dist(i,j)),
with constants independent of volume. Since M has finite range, |M| has the
same type of exponential bound, possibly with changed constants.

For completeness, the norm transfer for local interactions can be established
without assuming their transformed support remains finite. Expand a local
even v_X in Majorana monomials. If |X|<=s0, at most 4^s0 monomials occur, each
of degree <=2s0 and coefficient magnitude <=||v_X||. Substitute the linear
particle-hole transform into each Majorana. Truncating the resulting linear
combinations to the r-neighborhood X_r costs at most C exp(-nu r), for any
fixed 0<nu<mu, by polynomial lattice growth. Telescoping products gives local
approximants v_X^(r) with
 ||tilde(v_X)-v_X^(r)|| <= C_s0 ||v_X|| exp(-nu r).
The shell decomposition v_X^(0)+sum_(r>=1)(v_X^(r)-v_X^(r-1)) has supports X_r.
Bounded overlap of the original X and polynomial ball growth imply, for
0<a<nu/2,
 sup_y sum_(X,r:y in X_r) |X_r| exp(a diam(X_r))
              ||v_X^(r)-v_X^(r-1)|| < C ||v||_local.
The convergent majorant is a fixed polynomial in r times exp(-(nu-2a)r).
This is a volume-independent weighted quasilocal interaction bound.

Jump-list interface. Filtering is linear in the coupling operator. A unitary
mixing of all 2N jump labels preserves sum_a D(K_a). The Nambu matrix above
is orthogonal, so after transforming the physical {c_i,c_i^dagger} list, the
same total dissipator can be represented by filtered {d_i,d_i^dagger}.
This is an identity of generators, not permission to change physical jumps.

Remaining original gap. A proof must extend the Appendix I oscillator-norm
stability estimates to exponentially decaying |M| and transformed interactions,
with a uniform contraction margin. The locality bound alone is not that
contraction estimate. Ground-state fixation, uniqueness, normalization of the
filter, and uniform O(log(N/eta)) mixing must then be traced through.

Status: internally supported interface lemmas; original M03 remains open in
this campaign. Next test: prove a weighted perturbation bound for the fermionic
oscillator seminorm using summable jump tails, and check the source's exact
conditional-expectation estimates.
