G02 — combined near-critical weak-field preparation candidate v1
25 October 2026. Current-agent reviewed; no independent certification.

Statement.
Fix beta_min>0 below beta_c=2 log(1+sqrt(2)) and a finite universal gate set.
There exist positive fixed c,p,C,q,a such that, on every square L by L
torus L>=3, for beta_min<=beta<beta_c, delta=beta_c-beta,
|g|<=c delta^p and 0<epsilon<1/4, the actual Gibbs state of
H(g)=-1/4 sum_<i,j> Z_i Z_j-g sum_i X_i
can be prepared to trace distance epsilon using at most
C N max(1,delta^-q)[1+log(N/epsilon)]^a poly(b)
finite-set gates, classical bit operations and ancillary space.
N=L^2, the input parameters have finite b-bit descriptions, and arbitrary
gate connectivity is permitted. No quantum oracle, qRAM, Gibbs copies,
postselection or dense-patch operation is part of the input model.

Parameter choice and proof map.
Choose a fixed critical reference-gap exponent C_* from the classical
sources. Work in dimension d=2 and take an integer a0>C_*+2d+10. Put
M=a0+3d+8, A=2M+1, E=C_*+8, k=a0+2d+2, B=A+k+3,
q0=4C_* and Q=max(q0,2). It suffices to choose
p>max{q0(M+1),q0 A+2E,q0+d B Q}, q=28q0+1.
The fixed constant c is decreased to meet the finitely many gap and LCU
thresholds, including |g|<=2^-48 when d0=min(1,delta) is used. The fixed
logarithmic exponent a includes reversible arithmetic and gate synthesis.
No numerical optimization or practical threshold is claimed.

The proof consists of the complete dependency files pinned in inputs.json:
* R87/work-01: exact canonical local kernels and the common boundary-mixture
  channel controlling all doubled quantum observables.
* R87/work-04: polynomial local/global classical gaps and support-weighted
  boundary decay, with the normalization and geometries checked in
  R90/work-01 sections 1–2.
* R83/proof-v1: the volume-uniform finite-field connected expansion.
* R87/work-05 and proof-v1: polynomial spectral transport, relative form
  bound and all-volume bootstrap, checked in R90/work-01 sections 3–4.
* R88/work-01..03 and consolidated proof-v1: explicit Pauli-path access,
  Gaussian approximate transport, spatial decomposition and local circuit
  compilation, checked in R90/work-01 section 5.

The gap theorem supplies BOTH preparation-path hypotheses that were
conditional in R88. Bell pairs initialize the beta path; the field path
begins at its endpoint. The last operation discards the purifying copy, so
the output is the actual mixed Gibbs state in the frozen target. Parent
truncation and all circuit errors are globally budgeted before this trace.

Coverage and contribution.
The candidate matches the positive formal target of G02, including all
volumes, fixed exponents, critical-distance factors, classical compilation,
finite precision and mixed output. The literature distinction is a
quantitative refinement of established preparation and stability methods;
source-check.txt records the closest results and the limits of the search.

Verdict.
The combined proof has passed this current-agent dependency audit. It is
an original-level mathematical candidate, not an independently certified
solution or an authorized publication. A later review should target the
quantitative classical-to-parent stability and the nested compiler access
model. This file does not close R90 or change the original catalog status.
