R87/G02 working theorem 1 — exact local kernels and all-observable boundary reduction
Status: derived argument under current-agent verification; not an original completion. No novelty claim for coherent Gibbs-state constructions. Parent and normalization are exactly those frozen in G02 and R82–83.

Let beta in [0,beta_c], H0=-(1/4)sum_<ij> ZiZj, and double every site by an ancilla. Write P_i=(I-Z_i Z'_i)/2. With the physical nearest-neighbor spins fixed, p_i(s) is the classical heat-bath conditional probability. Let h_i project onto sqrt(p_i(-))|00>-sqrt(p_i(+))|11>, and annihilate the mismatched two-dimensional subspace. Since P_i h_i=0, Q_i=P_i+h_i is a projection of norm1. It has star support and commutes with every mismatch flag P_j. The torus parent is Q0=sum_i Q_i.

1. Restriction convention must be specified.
For a finite square Lambda, let I be the centers whose whole star is contained in Lambda and S=Lambda\I its shell. Use the canonical restricted parent Q_Lambda=sum_{i in I} Q_i. In this convention shell mismatch projectors are NOT imposed. One may alternatively include P_i for all shell sites; this only removes free shell ancilla labels. Both versions have the same interior conclusions below. We do not silently use the more constrained version in a theorem about the canonical restriction.

2. Exact kernel.
Fix a physical shell configuration tau and a shell ancilla configuration eta. Both are conserved by Q_Lambda. All interior mismatches have energy at least1. In the fully matched interior identify |x,x> with x. The sum of the h_i is the symmetric ground-state transform of the rate-one-per-site classical heat-bath generator on I with shell boundary tau. At finite beta every flip has positive rate, so this generator is irreducible. Its normalized zero vector is
 |psi_tau>_I=sum_x sqrt(pi_I^tau(x)) |x,x>.
Consequently the local ground-space projector is exactly
 P_Lambda=sum_{tau,eta} |tau,eta><tau,eta|_S tensor |psi_tau><psi_tau|_I.     (1)
With shell mismatch penalties, restrict the sum to eta=tau. The nonzero local gap is at least min(1,min_tau lambda_HB(I,tau)); this is an equality up to any additional mismatch-sector eigenvalues, which are all >=1. In particular it is NOT inferred from the torus gap alone.

3. Reduction of arbitrary local quantum observables to a classical marginal.
Let A be inside I and let C=partial^+ A be its nearest-neighbor exterior. For a classical boundary assignment b on C define
 |phi_b>_A=sum_{x_A} sqrt(pi_A(x_A|b)) |x_A,x_A>.
The conditional distribution depends only on b, by the nearest-neighbor Markov property. Trace the doubled complement of A in |psi_tau><psi_tau|. Matching the two copies outside A forces identical outside configurations in bra and ket. Group these configurations by b to obtain the exact identity
 rho_A^tau=sum_b pi_I^tau(C=b) |phi_b><phi_b|.                (2)
No diagonalization of the observable is involved. The map b -> |phi_b><phi_b| is a fixed measure-and-prepare channel, independent of tau. Hence
 ||rho_A^tau-rho_A^tau'||_1 <= ||pi_I^tau|_C-pi_I^tau'|_C||_1. (3)
This bound has no factor exponential in |A|. An observable supported on doubled A has no off-diagonal matrix element in (1) between distinct (tau,eta), since it acts trivially on S. Thus the compression of any Hermitian O_A is diagonal in those shell labels, with eigenvalues Tr(rho_A^tau O_A). Its distance from any scalar lying between these expectations is at most
 ||O_A|| max_{tau,tau'} ||pi_I^tau|_C-pi_I^tau'|_C||_1.       (4)
The same bound holds for non-Hermitian O by the trace-norm duality or by comparing each diagonal entry directly. This establishes the all-observable part that was missing at R84, subject only to a classical boundary-marginal estimate.

4. Scope of the local-topological-order bound.
If the classical coupling bound is TV(pi_I^tau|_C,pi_I^tau'|_C)<=A0 |C| exp[-dist(C,S)/xi], then (4) gives a support-weighted LTQO bound of that form, with the L1 factor2. For square A of radius r and an outer buffer ell, |C|=O(r+1) and dist(C,S)>=ell-2. This is the volume/support-prefactor form used in Nachtergaele–Sims–Young, arXiv:2010.15337, Definition2.1, and must NOT be mislabeled as a bound uniform in the support radius r with a decay function of ell alone. The early Michalakis–Zwolak formulation needs its hypotheses checked or its proof adapted. In the balanced regime r<=ell, the polynomial factor can be absorbed into a longer exponential length, but that does not by itself establish the unbalanced version.

5. Remaining work.
Pay a polynomial-delta classical boundary-decay length and local/global gap, then audit the stability theorem's constants. Finally an actual efficient preparation circuit must still be built; a spectral gap or exact ground state alone is not a gate count. Equation(2), although a useful bridge, is not enough to close this main attack.
