G02 / R99: stronger all-volume polynomial parent stability
Same-agent candidate and strict dependency audit. Original G02 unchanged.

The complete theorem in work-03-uniform-relative-remainder.txt removes
the large finite-volume crossover penalty in the R87 gap proof. With
d0=min(1,beta_c-beta), beta_c=2log(1+sqrt2), and the same exact doubled
parent Q_g, it proves
 gap(Q_g)>=c_gamma d0^(4C_*)/2, for |g|<=c d0^p_new,
 p_new=4C_*(2a0+29)+2C_*+17,
where C_*>=1 is a fixed integer critical reference-gap exponent and a0 is
any fixed integer greater than C_*+14. Constants are independent of L,
beta and g; the theorem covers every square torus L>=3.

The physical Hamiltonian is the frozen one,
H(g)=-(1/4)sum_<i,j> ZiZj-g sum_i Xi.
The parent retains separate conjugated mismatch and heat-bath annihilators.
Its exact unique ground vector is the TFD of this Hamiltonian. The final
physical mixed Gibbs state is obtained by discarding the second copy.

What changed mathematically.
R87/R90 had imposed, in addition to the local stability restrictions,
 p>q+dB max(q,2), q=4C_*, B=3a0+38, d=2.
For C_*>=1 this is p>q(6a0+77), the dominant old restriction. It came
from bounding the global tail in absolute operator norm above a very
large delta-dependent size and using Weyl perturbation below that size.

The exact leftover R already annihilates the reference ground projection.
The same classical sources also supply a finite-torus reference gap
lambda0(L)>=a(1+L)^(-C_*), uniform up to criticality. Therefore
 |<psi,R psi>|<=a^-1(1+L)^C_* ||R|| <psi,Q0 psi>.
Its previously derived norm is
 ||R||<=C M0[L^(d-a0)+sqrt(C0)L^(3d/2)e^{-L/(C xi)}],
 M0=CJ gamma^-A, gamma=c d0^q, xi<=C d0^-2, A=2a0+29.
Taking the supremum in L now costs only a fixed power xi^(C_*+3d/2).
Together with the local relative bound it gives
 |<psi,(U_s^*H_sU_s-Q0-E_s I)psi>|
 <=C J gamma^-A a^-1[1+sqrt(C0)(1+xi)^(C_*+8)] <psi,Q0 psi>.
This closes the gap bootstrap uniformly, with no large-volume/small-volume
crossover. Fixed tori below size128 use their finite critical gap directly.
Only p>max{q(M+1),qA+2E} is required; the second term dominates, yielding
the displayed p_new. The old dominant threshold is strictly larger.

Detailed proof and audit map.
* work-01-norm-and-kernel-audit.txt: exact canonical local kernel including
  unused shell ancillas; common boundary-mixture channel for arbitrary
  doubled observables; support-weighted O(|g|) perturbation norm.
* work-02-classical-scope-audit.txt: correct beta/FK normalization, arbitrary
  shell boundary, rate-one conventions, and BOTH reference-gap bounds.
* work-04-flow-parameter-audit.txt: fixed filter moments, the J factor in
  the difference of filters, bounded spectral-flow LR exponent, and exact
  ground-state annihilation after scalar subtraction.
* work-03-uniform-relative-remainder.txt: full new relative-tail theorem,
  all volumes, bootstrap and improved field exponent.

The norm audit uses R97's concrete parent interaction match and retains
the small factor J for the perturbation. It does not replace local Hilbert
dimension by an unproved constant. The classical comparison uses the
proof's wired crossing UPPER bound, not monotonicity of a Glauber gap.
The LTQO bound is applied to A_r^2 and retains its support prefactor.
The flow uses the linear stability interpolation only for the spectral
argument; it does not assert this is the preparation path.

Relation to the original algorithm candidate.
The beta path at zero field stays farther from criticality than its
endpoint, so its reference gap remains >=c d0^q. Along the physical field
path h=s g the improved endpoint theorem applies at every s. These are
the same two gap hypotheses required by the R88 compiler. Its separate
fixed threshold |g|<=2^-48 can be included by shrinking c; it adds no
delta exponent. Thus, CONDITIONAL on that compiler's correctness, R90's
same resource exponent 28q+1 applies on the improved field window.
This round has audited the central gap dependency, not independently
certified every nested compiler operation. The original full algorithm
remains an unaccepted candidate. G02 is not newly marked solved.

For the original promise with delta instead of min(1,delta), decrease
c by max(1,beta_c-beta_min)^p_new; this pays the entire input interval.
The later finite-bit rounding result in R98 remains a separate standard
error-control input, not a replacement for this gap proof.

Attribution and restraint.
Classical inputs: Lubetzky--Sly, arXiv:1001.1613v4; Duminil-Copin--Garban--Pete,
arXiv:1111.0144v4. Spectral-flow/relative-bound strategy: Nachtergaele--Sims--Young,
arXiv:2010.15337v3, and earlier gap-stability work of Michalakis--Zwolak.
R83, R87, R90 and R97 inputs are inherited. The R99 quantitative repair
uses their finite-volume information more efficiently. It is not a claim
of a new general stability theorem or global priority. A publication would
need independent mathematical and source review of the whole algorithm.

Original gap change: the core existence/algorithm candidate already existed
in R90; it is not counted as solved again. The central parent theorem has
a strictly stronger explicit sufficient field window and a cleaner
all-volume proof. Canonical original-core stall accounting is not reset
merely because this quantitative intermediate proof improved.
