R87/G02 — quantitative parent-gap interface for weak-field 2D Ising
Status: current-agent analytic candidate for a central intermediate theorem. G02 remains OPEN. This is not a complete preparation algorithm, a publication-ready novelty claim, or an independent review.

Statement.
Let H(g)=-(1/4)sum_<ij> Z_i Z_j-g sum_i X_i on every L by L square torus, L>=3. Let beta_c=2 log(1+sqrt(2)), 0<=beta<beta_c, d0=min(1,beta_c-beta). On a doubled lattice take the exact R82–R83 parent
 Q_g=sum_i[(K P_i K^-1)^*(K P_i K^-1)+(K h_i K^-1)^*(K h_i K^-1)],
 K=exp(-beta H(g)/2)exp(beta H(0)/2).
Here P_i is the mismatch projection and h_i is the matched-pair heat-bath projection from work-01. There exist positive constants c,c' and fixed finite exponents p,q, independent of beta,L, such that for |g|<=c d0^p,
 gap(Q_g)>=c' d0^q.
The unique ground vector is exactly the thermofield purification of exp(-beta H(g))/Tr exp(-beta H(g)). The state is not replaced by a diagonal Gibbs law. The constant c is conservative and potentially extremely small.

Proof dependencies and details.
1. Exact local kernels: work-01, equations(1)–(4). The canonical local restriction includes only stars wholly within the region; its unused shell ancillas are free labels. Every interior-mismatch sector has energy>=1. Each fully matched shell sector is the symmetric heat-bath generator with its physical shell boundary. For any inner doubled observable, its local-ground-space expectations are obtained by applying a single measure-and-prepare channel to a classical nearest-neighbor boundary marginal. Thus classical marginal L1 bounds imply bounds for all doubled quantum observables, without an exponential support-dimension loss.

2. Quantitative classical inputs: work-04, with the following geometry clarification. Only rectangles of a fixed bounded aspect ratio are needed in the subcritical recursion; starting from a square and splitting its longer side in the middle third preserves an aspect ratio bounded by a fixed constant. No assertion about fitting bounded-aspect crossing rectangles inside arbitrarily thin domains is needed. In a bounded-aspect domain, a grid of mesh between u/16 and u/8, aligned with the domain boundaries, reduces a physical path of diameter u to a crossing of one of O(R^2) contained squares or 2:1 rectangles. Domain Markov and wired boundary domination permit application of Duminil-Copin–Garban–Pete Lemma2.13 to these rectangles. Wires identifying boundary vertices are never counted as physical paths.
The proof of Lubetzky–Sly's critical upper mixing bound uses only the upper wired crossing bound, not a lower crossing bound, in its spatial coupling and gap recursion. Stochastic monotonicity in the FK parameter extends that bound to every beta<=beta_c. Thus all required local restricted gaps are at least a(1+r)^(-C_*), for universal a>0,C_*<infinity. This argument extends the proof, not a nonexistent monotonicity assertion for Glauber spectral gaps.
Duminil-Copin–Garban–Pete give a crossing length ell<=C d0^-1 log(e/d0)<=C d0^-2. The wired physical-cluster exploration yields joint boundary-marginal TV at most C|C|exp[-buffer/(C ell)]. The exact channel in step1 therefore gives support-weighted LTQO with a fixed prefactor C0 and xi<=C d0^-2.
For the global gap, overlap w(R)=ceil(K ell log(R+2)) yields a two-block gap>=1-R^-4 and an averaged site coverage<=1+C w(R)/R. The product of recursion losses is bounded above uniformly in the starting size, stopping at R0=C ell[1+log(ell+2)]^2<=C d0^-4. The local critical gap at the leaves gives a uniform box gap>=c d0^(4C_*). For the torus, one may use directly the two successive constant-overlap reductions in Lubetzky–Sly Section4.2: torus to cylinders to fixed-boundary rectangles. Their two-boundary coupling also only needs the critical crossing upper bound, so its constants are uniform for beta<=beta_c. These are two constant losses, not a loss at every length scale. Small tori use the critical periodic estimate. Set q=4C_* and gamma=c' d0^q<=1.

3. Finite-field locality is inherited, not rediscovered: R83/proof-v1.txt proves an exponentially summable anchored expansion of Q_g-Q_0 of strength J<=C|g|, with fixed spatial decay and constants uniform for beta<=beta_c, provided |g|<=g0 for a fixed positive g0. Keep P_i and h_i as separate annihilators in Q_g. Although P_i+h_i is a projection at g=0, conjugating that sum and squaring it at g!=0 would introduce cross terms and change the parent.

4. Polynomial locality of spectral transport: work-05 establishes the needed estimate on a bootstrap interval with gap>=gamma/2. In dimension d=2 choose an integer a0>C_*+2d+10, M=a0+3d+8 and A=2M+1. If J<=c gamma^(M+1), the transported and scalar-shifted Hamiltonian is
 U_s^*H_sU_s-H_0-E_s I=sum_x Z_x(s),
 Z_x P0=0, ||Z_x||<=CJ gamma^-A,
 ||Z_x-Pi_{x,r}Z_x||<=CJ gamma^-A(1+r)^-a0.
The fixed-moment filter bounds and the restriction on J keep the spectral-flow interaction norm bounded. In particular the LR exponential for flow time1 does not become exp(poly(gamma^-1)). This is the only quantitative modification required of the standard NSY spectral-flow strategy.

5. Relative form bound and all volumes.
Write M0=CJ gamma^-A. Let P_r be the local reference ground projector and A_r=Pi_{x,floor(r/2)}Z_x. Set W_x(r)=(I-P_r)A_r(I-P_r). Since Z_x Omega0=0, the global expectation of A_r^*A_r is <=C M0^2(1+r)^(-2a0). Apply the support-weighted LTQO from step2 to A_r^*A_r to obtain
 ||Z_x-W_x(r)||<=CM0[(1+r)^-a0+sqrt(C0)(1+r)^(d/2)exp(-r/(C xi))].
Each increment W_x(r)-W_x(r-1) is supported on radius r and annihilates P_r, since the P_r are nested. The local gap gives a form bound by a^-1(1+r)^C_* times the corresponding local reference Hamiltonian. On summing anchors, each reference term occurs at most C(1+r)^d times. Hence, stopping at r*=floor(L/8),
 |<psi,W psi>|<=b <psi,H_0 psi>,
 b<=CJ gamma^-A a^-1[1+sqrt(C0)(1+xi)^E], E=C_*+8.
The remainder R annihilates P0 and satisfies
 ||R||<=CJ gamma^-A[L^(d-a0)+sqrt(C0)L^(3d/2)exp(-L/(C xi))].
Here small radii are absorbed into fixed constants; this construction never needs a support-independent LTQO bound at fixed buffer.
To make the finite-size step explicit, put k=a0+2d+2, Y=2+gamma^-1+xi+a^-1+C0 and B=A+k+3. The inequality exp(-u)<=C_k u^-k shows that L0=C Y^B can be chosen so ||R||<=gamma/8 for all L>=L0 and J<=1. Indeed the polynomial remainder is bounded by C Y^(A+k+1)L^(-(a0-d)), and a0-d>1. Choose J small enough that b<=1/4. Orthogonal energies of the rotated Hamiltonian are then at least(3/4)gamma-gamma/8>gamma/2. Continuity improves and closes the bootstrap. For L<L0, impose additionally J<=c gamma L0^-d and use the operator-norm perturbation bound ||V||<=CJ L0^d; this keeps the gap above gamma/2 as well.
All restrictions are polynomial. With Q=max(q,2), it suffices to choose a fixed p strictly greater than
 max{q(M+1), qA+2E, q+dBQ}
and decrease c to cover the fixed R83 threshold and all prefactors. This proves the stated intermediate gap bound. No exponent is optimized.

What this does and does not close.
It supplies a concrete analytic candidate for the previously missing all-observable local-gap/LTQO and polynomial perturbative-gap interface. It does not supply the required N delta^-q polylog(N/epsilon) preparation cost. Efficient term access, derivatives along a preparation path, local simulation, input precision, and total global error must still be paid. Treating a dense matrix on a polylog-radius ball as a free oracle is not allowed. A known quasi-local state-conversion existence result also does not by itself pay those costs.

Primary attribution.
Lubetzky–Sly, arXiv:1001.1613v4, Theorem3.1, Section3.2 and Section4.2: critical fixed/free/periodic heat-bath gap and FK couplings.
Duminil-Copin–Garban–Pete, arXiv:1111.0144v4, Theorem1.2 and Lemma2.13: near-critical wired crossing scale and exponential decay beyond it.
Nachtergaele–Sims–Young, arXiv:2010.15337, support-weighted LTQO, spectral-flow decomposition(4.16)–(4.27), and LR framework: structural stability method. Michalakis–Zwolak, arXiv:1109.1588: earlier local-gap/LTQO stability framework.
New work in this campaign is the explicit application and parameter bookkeeping for the frozen parent. Worldwide novelty of the resulting quantitative intermediate statement has not been independently certified. Numerical files check only finite algebraic identities, not steps2,4,5.
