R83 G02 — a volume-uniform finite-field quasi-local parent continuation

This extends the exact parent Q_g of R82, not the R79 diagonal-polymer route. Put A_0=-H_0, V=sum_j X_j, t=beta/2, and tau_s(O)=exp(s A_0)O exp(-s A_0). Then K_g=exp(t(A_0+gV))exp(-tA_0)=tau_t(U_I(t)), where U_I'(s)=g tau_{-s}(V) U_I(s), U_I(0)=I. Thus
 K_g A_i K_g^{-1}=tau_t[ U_I(t) tau_{-t}(A_i) U_I(t)^{-1} ].
Here A_i is either of the R82 local projectors P_i,h_i, of norm1 and physical support at most the five-site star (plus the one ancilla).

All terms in A_0 commute. For O supported on physical S, only interaction edges touching S affect tau_s(O). There are at most4|S| such terms, of norm1/4, so ||tau_s(O)||<=exp(2|s||S|)||O|| and its support lies in S plus its immediate neighbors. In particular ||tau_{-s}(X_j)||<=exp(2t), with star support. The initial tau_{-t}(A_i) has norm at most exp(10t) and at most13 physical sites. Ancillas do not enter this count because A_0,V act only on physical sites.

Expand the conjugation by U_I in nested commutators, with time-ordered m-fold integrals. A nonzero next star must intersect the current physical support. A set of s physical sites intersects at most5s stars; each added intersecting star enlarges its support by at most4. Consequently there are at most product_{r=0}^{m-1}5(13+4r) nonzero ordered star choices. Each commutator contributes norm at most2 exp(2t); the time-simplex volume is t^m/m!. Final tau_t adds norm at most exp(2t(13+4m)). Therefore the norm sum of all degree-m terms is bounded by
 exp(36t) q^m (13/4)_m/m!,  q=40 t exp(10t)|g|.                 (1)
This bound deliberately overcounts. It is uniform in the torus size, including wraparound; smaller supports only improve it. The m-th term is supported within graph radius3+2m of i. The m=0 term is exactly A_i, since tau_t tau_{-t}=id.

For q<1 the series converges absolutely by sum_m (a)_m q^m/m!=(1-q)^(-a). In finite dimension it agrees with the analytic ODE solution, hence with the exact conjugated annihilator; convergence bounds are volume uniform. If q<=1/4, the tail beyond order M is at most exp(36t) 2^(-M)(1-2q)^(-13/4). The norm of R_i(g)-A_i is at most exp(36t)[(1-q)^(-13/4)-1]=O(|g|), uniformly for beta<=beta_c. Products R_i^*R_i inherit an exponentially decaying local expansion, by convolution of these summable tails. Counting centers in a radius-r ball costs only O(r^2), so the interaction norm per site of Q_g-Q_0 is O(|g|) at a fixed sufficiently small spatial exponential weight. Q_g remains the exact positive parent with unique |TFD_g> from R82.

One explicit conservative constant is |g|<=1/[160 t_c exp(10t_c)] approximately1.0543e-6, t_c=log(1+sqrt2). This is not a useful optimized physical threshold, but is strictly positive and independent of N and delta. It closes the FINITE-g locality/remainder interface left open in R82; it does not supply a polynomial-delta spectral gap or an efficient ground-state preparation circuit. General gap-stability theorems need local-gap/ground-space-indistinguishability hypotheses and controlled constants, not merely a global classical gap. These remain to be tested. Mark this as a scoped core-gap change of the parent route, with original G02 OPEN. No claim of worldwide novelty or independent-agent verification is made. The proof is the convergent commutator majorant, not the numerical value of g0.
