R87/G02 — detailed classical reference bounds
Current-agent derivation, pending final challenge review. This is an application and quantitative assembly of classical sources, not a claim of novelty for their ingredients.

Notation. The dimensionless Ising coupling is b=beta/4; p=1-exp(-beta/2). Put d0=min(1,beta_c-beta). The constants in this note are independent of beta<=beta_c and of the box/torus size unless explicitly indicated. Let C_* denote a universal exponent large enough for all fixed, free and periodic bounded-aspect critical-gap bounds in Lubetzky–Sly, arXiv:1001.1613v4. Increasing that exponent and changing a fixed prefactor is harmless.

Lemma A: the critical local-gap upper bound extends uniformly to every beta<=beta_c.
This assertion is about a proof, not monotonicity of spectral gaps. In Section3.2 of the source, the mixed-boundary Ising coupling is constructed for a general FK parameter: its domination step uses the domain Markov property and FKG conditioned on the decreasing event forbidding a plus/minus boundary connection. The only critical input to the upper-gap proof is the fully-wired crossing UPPER bound c_+<1 in Theorem3.7 and Corollary3.8. That upper bound remains true for all p<=p_c by stochastic monotonicity in p. All subsequent two-block comparisons and recursive coverage bounds in Section3.1 are geometric, and the terminal one-site gap is uniformly positive. Section4.2 treats free/periodic boundaries by the same exploration with one or two changing sides; it again needs the same crossing upper bound. Therefore the same proof supplies
 lambda_HB(R,tau)>=a(1+r)^(-C_*)                            (A1)
for every rectangular region with fixed arbitrary spins, where r may conservatively be taken as its maximal side. The corresponding small periodic boxes also satisfy(A1). No high-temperature comparison of Gibbs densities, which could cost exp(delta r^2), is used.

Lemma B: a quantitative physical-cluster bound.
Duminil-Copin–Garban–Pete, arXiv:1111.0144, Lemma2.13 provides a scale ell<=C d0^{-1}log(e/d0) and exponentially small crossing probabilities on dyadic larger rectangles under fully wired boundary. Increasing ell gives ell<=C d0^{-2}. Dyadic rounding changes only fixed constants. A path crossing an annulus of thickness u contains a crossing of one of a fixed number of bounded-aspect rectangles with short side comparable to u. By domain Markov and boundary domination, its probability is <=C exp[-u/(C ell)]. Covering a finite bounded-aspect rectangle by possible such starting boxes gives
 P_wired(exists a physical FK component of diameter>=u)
       <= C R^2 exp[-u/(C ell)]                            (B1)
for u above a fixed multiple of ell. For components touching a boundary, use inward-shifted covering rectangles contained in the domain; the possible rectangle shapes form a fixed finite list. Boundary identification wires are NOT physical open paths. The same local bound applies on cylinders/tori when u is below a fixed fraction of the circumference, by cutting the local patch and dominating its induced boundary by wired boundary.
For a specified interior point at distance u from the boundary, no union over the entire domain is needed: a single interior annulus gives
 P_wired(point connected physically to the outer boundary)
       <=C exp[-u/(C ell)].                                (B2)
The mixed-spin-boundary exploration from Lemma A gives a coupling agreeing outside the wired boundary-connected components. Thus for any set C a distance u inside the box,
 TV(pi^tau|_C,pi^{tau'}|_C)<=C |C| exp[-u/(C ell)].           (B3)
This is the bound needed for the all-observable quantum reduction, rather than an unsupported diagonal-to-quantum inference.

Lemma C: near-critical global gap with polynomial dependence.
Take a rectangle whose larger side is R and fixed arbitrary exterior spins. Let w(R)=ceil(K ell log(R+2)). When R>=R0=C ell[1+log(ell+2)]^2, choose C so w<=R/12. For every integer cut center s in the middle third, take blocks B1 ending at s+w and B2 starting at s-w. Updating B1 leaves only one variable boundary adjacent to its complement. Under any two starting configurations, the FK exploration couples the spins in B1\B2 except with probability
 eta_R<=C R^2 exp[-w/(C ell)]<=R^{-4},
with K fixed sufficiently large. The same holds with B1,B2 exchanged.
The rate-one-per-block two-block heat-bath dynamics has gap at least1-eta_R. One way to check the near-one constant is the repeated random-scan coupling from the source: after an update, with probability at least1-eta_R the complement of the OTHER block agrees; a following update of that other block coalesces. The survival matrix on bad/good flags is bounded by
 [[eta_R,1-eta_R],[eta_R/2,(1-eta_R)/2]],
whose spectral radius is(1+eta_R)/2. The discrete random-scan chain consequently has nontrivial eigenvalue at most that number, and multiplying its generator by2 gives the stated gap. A two-step constant Doeblin bound would be too weak for the recursion and is not substituted here.
For a test function f, the block Dirichlet form is the sum of the two conditional variances. Conditional Poincare on each block bounds it by lambda_*^{-1} times the sum of site conditional variances in that block, with lambda_* the infimum of the block gaps over their induced boundary spins. Average this inequality over the cut centers. Each site is covered once plus an average overlap <=Cw/R. The averaged block gap is still at least1-eta_R, by averaging its Poincare inequalities. Hence
 lambda(R)^(-1)<= (1+Cw(R)/R)/(1-eta_R) * max_child lambda(child)^(-1). (C1)
A split of the longer side yields children whose split-side length is <=3R/4. If the unsplit side is larger, split it next; after at most two steps the maximal side has shrunk by3/4. Thus the logarithm of the accumulated factor in(C1) is bounded by a constant times
 sum_{R_j>=R0} [ell log(R_j+2)/R_j + R_j^{-4}],
where at most two R_j occur at each geometric scale. This sum is bounded independently of the initial R. At the leaves use(A1), giving
 lambda_box >= c R0^{-C_*}.                                (C2)
For tori of side L>=C R0, first cover the torus by two overlapping cylinders, with overlaps a fixed positive fraction of L. Bound(B1) gives a two-block gap bounded below by a constant. Comparing with site Dirichlet forms costs at most the coverage2. Cut the remaining periodic direction in each cylinder once more, obtaining ordinary fixed-boundary rectangles. This introduces only another constant factor, not a factor at every system-size scale. For L<C R0 use the periodic form of(A1). Therefore(C2) holds uniformly over every torus L>=3.
Since R0<=C d0^{-4} after a conservative absorption of logarithms,
 lambda_HB(torus)>=c d0^{4C_*}.                            (C3)
The per-site clock rate is1 throughout; no hidden factor N is gained by rescaling time.

Consequences for the exact R82 parent.
Work-01 gives all restricted local gaps >=a(1+r)^(-C_*) (up to the constant mismatch gap1). The torus parent gap is >=c d0^{4C_*}. Its support-weighted all-observable LTQO has a decay length xi<=C d0^{-2}, via(B3) and the exact boundary-mixture channel. The bound compares to the global torus ground expectation as well: that global classical interior law is a mixture of the same fixed-shell conditional laws, so the maximum shell-to-shell estimate also bounds shell-to-global variation.

Audit limits. The finite algebra controls only check the kernel and mixture identity. The global estimates rely on the explicitly cited FK crossing results and their stated boundary conditions. Constants C_* and a are inherited universal constants from those sources; no numerical optimization or unsupported explicit decimal exponent is claimed. Full finite-g stability still requires the polynomial spectral-flow estimate in work-03, and actual preparation costs remain a separate obligation.
