R99 G02 strict challenge of the all-volume relative repair
Reviewer /root; independent=false. This is a same-agent mathematical
review, not an independent certificate or an original-completion verdict.

Result: the new relative-remainder proof survives the checks below. The
old R87/R90 gap statement is not refuted; its sufficient field exponent
can be improved by removing its unnecessary crossover restriction.

1. Is the key annihilation only approximate?
No. The exact gap filter has zero ground/excited matrix elements, and the
exact spectral transport sends the perturbed ground projection to P0.
Each Y_x commutes with P0, and rank-one scalar subtraction gives Z_x P0=0
exactly. Canonical local projectors contain the global frustration-free
ground vector, so W_x(r)P0=0 exactly as well. Their difference and its
sum R therefore annihilate P0. The proof must use the exact analytic
flow, not a numerically truncated implementation, at this step.

2. Can a global nonlocal remainder be relatively bounded this way?
Yes. For self-adjoint R with RP0=0, also P0R=0, and
 |<psi,Rpsi>|<=||R|| ||(I-P0)psi||^2.
The global reference gap a(1+L)^(-C_*) then gives (7) of work-03. This
does not assume [R,H0]=0, R>=0, locality of R, or a small extensive norm
before division by the reference energy. A bound by ||psi||^2 would
not suffice; exact ground annihilation is the essential additional fact.

3. Does the L supremum hide exponential critical-distance dependence?
No. After using the finite-size gap, the two factors are
 L^(C_*+d-a0) and L^(C_*+3d/2)e^{-L/(C xi)}.
The first has negative exponent. The second has maximum
(C xi v/e)^v for fixed v=C_*+3d/2. This is a fixed power of xi,
not exp(xi). All powers are independent of beta, delta, L and g.
The local increments cost a slightly larger power xi^(C_*+3d/2+1),
already bounded by the inherited E=C_*+8 in d=2. Thus the new remainder
requires no stricter field window than the local relative estimate.

4. Are the local increments actually allowed to use P_r?
Yes. P_r<=P_(r-1) for canonical frustration-free restrictions. Hence
W_x(r-1) also annihilates P_r, even though it was defined with P_(r-1).
The difference is supported in the larger square and self-adjoint.
Its form is bounded using that square's local gap. The sum over anchors
costs its volume O(r^d); the sum over radii costs one additional power
in the exponential term. Both are explicitly present. LTQO is applied
to A_r^2, with support prefactor retained before taking a square root.

5. Are very small and near-crossover tori omitted?
No. For L>=128 the cutoff r*=floor(L/8) leaves genuine square balls.
The initial increment begins at r=4; all its constants are fixed. For
3<=L<128 use exact global annihilation of sum Z_x and the same finite
critical gap directly. The finite maximum L^d(1+L)^C_* is a fixed
constant, not a delta-dependent crossover. Intermediate L comparable
to xi or gamma^-1 are included in the elementary supremum in item3.

6. Is the finite critical gap available in the correct model?
The parent on its matched sector is the symmetric heat-bath transform,
with one update clock per site. Mismatch sectors have energy >=1.
Lubetzky--Sly's fixed and periodic proofs were checked separately. Their
mixing upper proof uses only the upper wired crossing bound; its
extension to subcritical beta follows at that proof step. It is not a
claim of monotonicity of the generator's spectral gap. Taking one larger
integer C_* and one smaller a covers local and full-torus statements.
The FK parameter 1-e^{-beta/2} has the correct factor for H0=-sum ZZ/4.

7. Could the spectral-flow constant itself still be exponential?
Work-01 verifies the support-weighted perturbation norm O(|g|), with
the anchor and support-cardinality factors both paid. Work-04 retains
the small factor J when differentiating the real-time filter and bounds
the parameter-time LR norm by kappa=CJ gamma^(-M-1). Imposing kappa<=1
before exponentiation keeps the prefactor linear in kappa. Fixed filter
moments supply only fixed powers of gamma^-1. No comparison of full
Gibbs densities or condition number e^{beta||H||} enters the proof.

8. Exponent arithmetic.
In d=2: M=a0+14, A=2a0+29, E=C_*+8, B=3a0+38, q=4C_*.
For C_*>=1, max(q,2)=q. The old third threshold is q(6a0+77).
The new second threshold is q(2a0+29)+2C_*+16 and dominates q(M+1).
Their difference is q(4a0+48)-2C_*-16>0. Choosing one integer above
the new threshold gives p_new in proof-v1. Shrinking the fixed prefactor
also pays delta>1, the fixed convergence cutoff and the compiler's
separate |g|<=2^-48 cutoff. A useful numerical exponent is not claimed.

9. Bootstrap and downstream scope.
The whole rotated perturbation is relatively bounded by bH0, b<=1/4.
It therefore leaves an orthogonal energy >=3gamma/4 throughout any
assumed gap>=gamma/2 interval. Continuity rules out its first failure.
The linear interpolation need not be frustration free. At the endpoint
the exact parent kernel identifies the physical TFD; the physical
preparation path can use the endpoint theorem at every intermediate
field. The R88 compiler has not received an independent review here.
Its conditional use of the improved gap window is valid at the level
of those gap hypotheses, not an independent verification of its circuit.

10. Research-integrity assessment.
The initial audit scout found that R90 already contains much of the
proposed source/parameter audit. That repeat is disclosed in the program
refinement; it is not credited as a new original advance. The main
deliverable is the complete all-size relative repair and strictly weaker
sufficient delta-power restriction. Its proof is analytic, not a small
test. Further work solely to consume the nominal token range would not
improve this closed target. A next main must address a genuinely new
compiler or algorithmic interface, or obtain independent review.

Original complete=false; new accepted original=false; core-gap reset=false.
Central quantitative intermediary strengthened. Decision: route the
versioned gap theorem to verification; park automatic renewal of this
same parent-stability audit without new mathematical input.
