R87/G02 — extracting polynomial parameter dependence from the parent-stability route
WORKING PROOF, not yet a final reviewed claim. This is the central attack; the finite controls in check-local-parent.json do not establish it.

A. Quantitative hypotheses to be paid
Consider a two-dimensional frustration-free reference H0=sum_x Q_x, with uniformly bounded finite range and fixed local dimension. Its torus ground state is unique. Suppose:
(i) global gap >=gamma in (0,1];
(ii) restricted rectangular gaps >=a(1+r)^(-C_*), with a>0,C_* independent of volume and of the parameter delta;
(iii) for an observable A supported in radius k, compression to a local ground space with buffer ell differs from its global ground expectation by at most C0 |ball(k)| ||A|| exp[-ell/xi], for all buffers before torus wrapping;
(iv) V=sum_x V_x has a volume-uniform exponentially decaying anchored interaction expansion of strength J, with decay rate and normalized prefactors fixed.
Here C0 may be replaced by any polynomially bounded parameter. The desired conclusion is a stability threshold that is an inverse polynomial in gamma^{-1},xi,C0,a^{-1}, not merely a positive function of them. C_* is fixed. R83 supplies(iv) for the actual finite-g parent, with J<=C|g| and fixed constants; work-01 and the classical program address(i)-(iii).

B. Why a black-box existence theorem is insufficient
A stability theorem may hide spectral-flow constants growing in gamma^{-1}. Simply citing it would reproduce the previous fixed-beta gap. The proposed extraction below uses fixed-order spatial moments throughout; the required smallness of J controls the velocity of the spectral flow itself. It also explicitly keeps the support-volume factor in LTQO. This is an adaptation/quantitative unpacking of the standard BHM/Michalakis–Zwolak/Nachtergaele–Sims–Young proof strategy, not a new claim of inventing gap stability.

C. Polynomial locality estimate to prove and audit
Fix a sufficiently large exponent a0>C_*+12. There should be constants A,C, depending only on a0,C_*,dimension,interaction range and the normalized decay in(iv), such that on any parameter interval where H_s=H0+sV has a gap >=gamma/2, one can write
 U_s^* H_s U_s-H0-E_s I = sum_x Z_x(s),
where U_s transports the unique reference ground vector, Z_x(s) is Hermitian, Z_x(s)P0=0, and it has local approximants Z_x^{<=r} satisfying
 ||Z_x-Z_x^{<=r}|| <= C J gamma^{-A}(1+r)^(-a0),
 ||Z_x|| <= C J gamma^{-A}.                                  (C1)
These bounds are required uniformly in volume. A safe choice of A must be exhibited by the locality argument, rather than silently included in C.

Route to(C1): use an even normalized energy filter w_gamma(t)=gamma w_1(gamma t), whose Fourier transform is supported strictly inside the gap and whose every fixed moment is finite. Its k-th absolute moment scales exactly gamma^{-k}. The odd integrated filter F_gamma has k-th moment O(gamma^{-(k+1)}) and Fourier transform proportional to(1-hat(w_gamma)(omega))/omega. It generates exact ground-space transport on this gapped interval.
Real-time locality of H_s has a bounded velocity when J<=1. To estimate a spatial tail r, split time at c r: short times give an exponentially small Lieb–Robinson error, while the long-time tail is bounded by a fixed moment of the filter. This loses only a fixed power of gamma. For the derivative with respect to s, use Duhamel and sum commutators with local terms of V over the light cone; its volume grows polynomially in time. The derivative is linear in J and loses only additional fixed moments.
The resulting spectral-flow generator has a sufficiently high polynomial weighted interaction norm <=C J gamma^{-A1}. Impose J<=c gamma^{A1}; its parameter-time Lieb–Robinson exponential is then uniformly bounded on s in[0,1], instead of retaining exp(C gamma^{-A1}). Applying the flow to the filtered local energy terms, and subtracting their unperturbed counterparts, yields(C1) with A a fixed sum of these moment losses. Precise weighted-norm/convolution exponents still need to be written down before this lemma is accepted.

D. Local block diagonalization once(C1) is available
Put M=CJ gamma^{-A}. Let P_r be the local ground projector in a radius-r square centered at x. Let A_r=Z_x^{<=floor(r/2)} and define
 W_x^{(r)}=(I-P_r) A_r (I-P_r).
Because Z_x P0=0, ||A_r P0||<=M(1+r)^(-a0). Apply support-weighted LTQO to A_r^* A_r. For sufficiently small r compared with the torus side,
 ||A_r P_r|| <= C M[(1+r)^(-a0)+(1+r)^{d/2} sqrt(C0) exp(-r/(C xi))].
The global-ground expectation in this application is ||A_r Omega||^2 and is small by the tail bound. Hence
 ||Z_x-W_x^{(r)}|| <= C M[(1+r)^(-a0)+(1+r)^{d/2} sqrt(C0) exp(-r/(C xi))]. (D1)
This is the point at which a diagonal-only correlation statement would have been insufficient; work-01 supplies all observables including A_r^*A_r.

Take increments w_x(r)=W_x^{(r)}-W_x^{(r-1)}. The local ground projectors are nested, P_r<=P_{r-1}, so each increment annihilates P_r and has support in the radius-r square. Its norm is bounded by the sum of the two errors in(D1), with finitely many small radii treated directly. For a normalized arbitrary vector psi,
 |<psi,w_x(r)psi>| <= ||w_x(r)|| <psi,(I-P_r)psi>
                   <= ||w_x(r)|| a^{-1}(1+r)^{C_*}<psi,H_{ball(r)}psi>.
Sum over x. Every reference local term lies in at most C(1+r)^d such squares. Thus the full locally block-diagonal part has relative form bound
 |<psi,W psi>| <= b <psi,H0 psi>,
 b <= C J gamma^{-A} a^{-1}[1+sqrt(C0) xi^{C_*+3d/2+1}].       (D2)
The polynomial tail sum converges if a0>C_*+d+1. In d=2 a conservative exponent C_*+8 on(1+xi) absorbs all rounding and small-radius constants. This calculation does not assume a volume-independent LTQO bound at fixed buffer for arbitrarily large supports.

E. Finite-size remainder and the all-size quantifier
Stop the local increments at r*=floor(L/8) to avoid wrapping. The remaining sum R_s has norm bounded by
 C J gamma^{-A}[L^{d-a0}+sqrt(C0)L^{3d/2}exp(-L/(C xi))].       (E1)
It annihilates P0 because both Z_x and each W_x^{(r)} do. Choose a polynomial L0 in gamma^{-1},xi,C0,a^{-1} large enough that(E1)<=gamma/8 for all L>=L0 and J<=1. Such a polynomial cutoff exists: the exponential term can be bounded using any fixed polynomial tail, with an additional fixed power of xi. Impose the polynomial smallness from(D2) so b<=1/4. Then the rotated Hamiltonian has its unique zero vector and all orthogonal energies >=(3/4)gamma-gamma/8>gamma/2. Continuity closes the usual gap bootstrap.
For L<L0 use ||V||<=C J L0^d and impose J<=c gamma L0^{-d}; Weyl then preserves at least half the reference gap. This extra restriction is polynomial in the parameters and covers every small torus, rather than leaving an uncontrolled system-size threshold.

F. Conditional consequence and outstanding work
If(C1) with fixed polynomial constants, and the quantitative classical inputs in work-02, are fully established, then there exist fixed exponents p,q such that the R83 parent of the actual weak-field TFIM has gap >=c delta^q for |g|<=c' delta^p, uniformly in N. This closes a central spectral-stability interface but is still not the full G02 preparation algorithm.
Outstanding: give a complete weighted-norm derivation for(C1), finalize the classical recursion including periodic geometry, and separately charge an efficient implementation of the quasi-local parent and its spectral transport. A dense matrix for a polylog-radius ball is not a polynomial-cost oracle. These are active proof obligations, not implicit assumptions hidden in a PASS label.
