R99 G02: remove the small-volume crossover penalty
New quantitative repair within the staged all-volume parent-gap main.
Same-agent proof candidate, with inherited locality and classical inputs
explicitly identified. No independent or worldwide-priority certification.

Statement in the notation of R87.
Take d=2, a fixed critical-gap exponent C_*>=1, and integer
 a0>C_*+2d+10; M=a0+3d+8; A=2M+1; E=C_*+8.
Let the frustration-free reference parent H0=Q0 have unique ground P0 and:
 (i) uniform gap >=gamma, 0<gamma<=1;
 (ii) all square local gaps >=a(1+r)^(-C_*);
 (iii) full torus gap >=a(1+L)^(-C_*);
 (iv) support-weighted LTQO with prefactor C0 and length xi>=1.
The constants a<=1,C0>=1 are volume independent. Assume the actual
perturbation V=Q_g-Q0 has the exponential interaction norm J proved in
work-01. The standard fixed-moment flow construction, rechecked in work-04,
gives, on a gap>=gamma/2 bootstrap interval and for J<=c gamma^(M+1),
 U_s^*(H0+sV)U_s-H0-E_s I=sum_x Z_x,
 Z_x=Z_x^*, Z_x P0=0,
 ||Z_x||<=M0, ||Z_x-Pi_(x,r)Z_x||<=M0(1+r)^(-a0),
 M0=C J gamma^(-A).                                      (1)

Claim: at EVERY torus size L>=3,
 |<psi,(sum_x Z_x)psi>| <= b <psi,H0 psi>,
 b<=C M0 a^-1[1+sqrt(C0)(1+xi)^E].                       (2)
This has no volume crossover L0 and no additional inverse gamma factor.
It implies gap(Q_g)>=3gamma/4 after shrinking J so b<=1/4 and closing
the usual bootstrap. The numerical 3/4 may be weakened to 1/2.

Proof for L>=128.
Use square balls in the infinity metric; choose r*=floor(L/8). These
restrictions are genuine non-wrapping squares, to which (ii) and (iv)
apply. Let P_r be the canonical local reference ground projection, extended
by identity outside its square. It contains the global ground vector.
For r>=r_min=4, put
 A_r=Pi_(x,floor(r/2))Z_x, W_x(r)=(I-P_r)A_r(I-P_r).
For the finitely many smaller radii set the preceding W to zero and absorb
the resulting first increment into fixed constants, which may depend on
a0. Sizes L below 128 are handled by the finite-size paragraph below.
No estimate depends on a delta-dependent cutoff.

By (1), ||A_r||<=C M0 and ||A_r Omega0||<=C M0(1+r)^(-a0).
Apply all-observable LTQO to A_r^2, whose support remains the radius-r/2
square. It yields
 ||P_r A_r^2 P_r||
 <= C M0^2[(1+r)^(-2a0)+C0(1+r)^d e^{-r/(C xi)}].
Since ||A_r P_r||^2=||P_r A_r^2 P_r||, and
 A_r-(I-P_r)A_r(I-P_r)=P_r A_r+A_r P_r-P_r A_r P_r,
we have
 ||Z_x-W_x(r)||<=C M0 f(r),
 f(r)=(1+r)^(-a0)+sqrt(C0)(1+r)^(d/2)e^{-r/(C xi)}.       (3)
This is where the square root of the LTQO error and its support factor
are required. Replacing A_r^2 by A_r would not prove the needed norm bound.

Define B_x,r=W_x(r)-W_x(r-1), including the first increment. Frustration
freeness implies P_r<=P_(r-1), so both summands annihilate P_r. B_x,r is
self-adjoint and supported on its radius-r square. For any vector psi,
 |<psi,B_x,r psi>| <= ||B_x,r|| <psi,(I-P_r)psi>
 <= a^-1(1+r)^C_* ||B_x,r|| <psi,H0_(x,r)psi>.             (4)
For all but the first fixed radii, its norm is at most
C M0[f(r)+f(r-1)], by (3). The first radius obeys the same bound with a
larger fixed constant since xi>=1. Summing anchors counts every reference
interaction at most C(1+r)^d times. Therefore W=sum_x W_x(r*) obeys
 |<psi,W psi>| <= b_W <psi,H0 psi>,
 b_W<=C M0 a^-1 sum_(r>=1)(1+r)^(C_*+d)[f(r)+f(r-1)]
 <=C M0 a^-1[1+sqrt(C0)(1+xi)^(C_*+3d/2+1)].              (5)
The polynomial sum converges because a0>C_*+d+1. The exponential sum is
bounded by its elementary integral; all exponents and constants are fixed.
For d=2, E=C_*+8 is larger than the exponent C_*+4 in (5).

The exact global remainder R=sum_x[Z_x-W_x(r*)] satisfies
 R=R^*, R P0=0,
 ||R||<=C M0[L^(d-a0)+sqrt(C0)L^(3d/2)e^{-L/(C xi)}].     (6)
The identity R P0=0 is exact: each Z_x annihilates P0 by spectral filtering
and scalar subtraction, while each W_x annihilates the local kernel and
therefore the global kernel. It is not an approximate ground-state bound.

Now use hypothesis (iii), which the previous crossover argument left
unused at this point. Set Q=I-P0. Then
 |<psi,R psi>|=|<Qpsi,R Qpsi>|
 <=||R|| <psi,Q psi>
 <=a^-1(1+L)^C_* ||R|| <psi,H0 psi>.                     (7)
Neither commutation of R with H0 nor positivity of R is needed. Its
annihilation on the one-dimensional kernel and self-adjointness suffice.
Combining (6),(7), and absorbing 1+L versus L, gives
 b_R<=C M0 a^-1[L^(C_*+d-a0)
                  +sqrt(C0)L^(C_*+3d/2)e^{-L/(C xi)}]
 <=C M0 a^-1[1+sqrt(C0)(1+xi)^(C_*+3d/2)].               (8)
For the last inequality, a0>C_*+d and
 sup_(x>=0) x^v e^{-x/(C xi)}=(C xi v/e)^v
for fixed v=C_*+3d/2>0. Thus the bound is uniform in L and polynomial
in xi. Equations (5),(8) prove (2) for the non-wrapping regime.

Fixed small tori.
For every L below the chosen fixed cutoff, do not invoke a nonexistent
inner ball. Instead use (1), N=L^d, and the global annihilation directly:
 |<psi,(sum_x Z_x)psi>|
 <=C M0 L^d <psi,Q psi>
 <=C M0 a^-1 L^d(1+L)^C_* <psi,H0 psi>.
There are finitely many such L, so this is bounded by a fixed constant
times M0/a and is included in (2). The constant depends on C_* and the
fixed cutoff, never on beta, delta, g or L. This treats L=3 explicitly
at the level of assumptions. No delta-dependent Weyl regime is needed.

Close the bootstrap.
Choose J so that J<=c gamma^(M+1) and the right side of (2) is <=1/4.
The rotated Hamiltonian minus its ground energy has form lower bound
 H0+sum_x Z_x >=(3/4)H0.
Its reference ground vector has energy zero, and every orthogonal vector
has energy at least 3gamma/4. The spectrum is invariant under U_s.
Starting at s=0, continuity of finite-dimensional eigenvalues therefore
precludes a first point with gap gamma/2. The bootstrap holds through s=1
at each finite L with the SAME constants. At the endpoint Q_g's exact
positive-parent construction identifies its unique ground as the TFD.

Critical-distance substitution and actual improvement.
From work-02, gamma=c_gamma d0^q, q=4C_*, xi<=C_xi d0^-2,
with a,C0 fixed. Work-01 gives J<=C_J |g|. It now suffices to choose
 p>max{q(M+1),qA+2E}.                                   (9)
The second expression dominates the first. Choosing integer C_* once,
one explicit admissible exponent is
 p_new=qA+2E+1=4C_*(2a0+29)+2C_*+17.                     (10)
Shrink a fixed c so all constants, J<=1, and the R83 convergence threshold
are met. This proves the endpoint gap for |g|<=c d0^p_new.

R87/R90 additionally imposed p>q+dB max(q,2), where B=3a0+38 in d=2.
For C_*>=1 this discarded term was q(6a0+77), and was the largest one.
It exceeds the new threshold qA+2E by
 q(4a0+48)-2C_*-16>0.
Thus the repair removes a strictly larger critical-distance exponent
restriction. Even though fixed prefactors remain conservative, the
admissible field window is asymptotically larger as delta tends to zero.
No optimized exponent or practical threshold is claimed.

To state this in the original |g|<=c_orig delta^p form, let
delta_max=beta_c-beta_min and choose c_orig=c/max(1,delta_max)^p.
Then the original promise implies |g|<=c d0^p also when delta>1.

What is new and what remains.
The spectral-flow construction, the local relative-bound construction,
and the classical critical gaps are inherited/established. The R99 repair
is the use of the exact global annihilation together with the finite-torus
polynomial reference gap in (7), eliminating the artificial crossover
restriction throughout the all-size theorem. It strengthens the central
parent-gap candidate. It does not by itself independently verify the
R88 compiler, solve a new original, or establish worldwide novelty.
