R99 G02: classical source hypotheses and the two reference gaps
Same-agent source audit; inherited classical results remain attributed.

Normalization: coupling b=beta/4, FK p=1-e^{-beta/2}, beta_c=2log(1+sqrt2).
For 0<=beta<=beta_c, the derivative of p lies between two positive fixed
constants. Therefore p_c-p is comparable to beta_c-beta near criticality.
Use d0=min(1,beta_c-beta) everywhere to cover the rest of the interval.

Primary inputs checked at their theorem/proof locations:
Lubetzky--Sly, https://arxiv.org/html/1001.1613v4 , Thms3.1,3.2,
Claim3.6, Thm3.7/Cor3.8, and Thms4.2--4.4. The mixing upper proof uses
the wired crossing upper bound. Its probability is monotone in p; hence
the proof extends uniformly to p<=p_c. The lower-gap argument in Sec4.1
uses other inputs and is not being imported. Mixed-spin boundaries are
handled by the source's conditional FK exploration, including the event
of no plus/minus connection. No reciprocal of its possibly tiny probability
is introduced. Fixed, free, and periodic versions are distinguished.
Duminil-Copin--Garban--Pete, https://arxiv.org/html/1111.0144v4 , Lemma2.13,
supplies exponential wired physical-crossing decay beyond a scale
ell<=C d0^-1 log(e/d0)<=C d0^-2. Boundary identification wires are not
physical open paths. These are established source inputs, not new results.

1. The polynomial FINITE-VOLUME gap is retained, not discarded.
There are universal a>0 and fixed C_* (increase once and take C_*>=1)
such that both
 gap(Q0 restricted to a radius-r square)>=a(1+r)^(-C_*),      (C1)
 gap(Q0 on the full L-torus)>=a(1+L)^(-C_*).                 (C2)
The second statement also holds at criticality. The first allows every
shell configuration in work-01. All rates are one per site. Since an
interior mismatch costs one, the doubled parent inherits these bounds.
L<3 or a different Hamiltonian normalization is not part of the claim.

Uniformity below criticality follows at the exact source-proof interface:
the FK exploration couples outside components connected physically to the
changing boundary. Its domination under conditioning uses FKG and domain
Markov, both valid for all p. The only scale-independent critical input
is that a fully wired fixed-aspect rectangle has crossing probability
bounded strictly below one. This upper bound persists when p decreases.
The geometric recursive comparison and its one-site endpoint therefore
retain fixed constants. The two-side variant handles periodic directions.
This argument never asserts monotonicity of a Glauber spectral gap.

2. Strong mixing away from the critical point.
A point a distance u from the varying boundary is separated from it by
an interior square annulus. Covering that annulus by a fixed number of
overlapping bounded-aspect rectangles, and applying domain Markov with
wired domination, bounds its connection probability by C e^{-u/(C ell)}.
For a specified set C, the union bound costs |C|, not the system volume.
The same exploration yields joint marginal TV <=C|C|e^{-u/(C ell)}.
Through work-01 this is the support-weighted all-observable estimate
 ||P_r A P_r-<Omega0,A Omega0>P_r||
 <= C0 |supp A| ||A|| exp[-(r-r_A)/(C xi)], xi<=C d0^-2,    (C3)
for concentric square restrictions before wrapping. Inner supports near
a boundary require the stated buffer; no uniform-in-support shortcut is used.

3. The uniform thermodynamic-size gap is a DIFFERENT bound.
At box size R choose overlap w=ceil(K ell log(R+2)); stop when
R<=R0=C ell[1+log(ell+2)]^2<=C d0^-4. Above R0 the probability that
one block update fails to couple the other block's complement is <=R^-4.
The coupled random-scan chain has nonabsorbing bad/good transition bound
 [[eta,1-eta],[eta/2,(1-eta)/2]], eta=R^-4.
Its spectral radius is (1+eta)/2, so the continuous chain with one clock
per block has gap >=1-eta. Averaging middle-third cut positions gives
site coverage at most 1+Cw/R. The product of Poincare-comparison losses
is bounded, since sum_scales ell log(R+2)/R is bounded at this stop scale.
Use (C1) at the leaves. Two constant-overlap initial reductions remove
the torus's periodic directions, with (C2) for small tori. Thus
 gap(Q0,L)>=gamma=c d0^q, q=4C_*, uniformly in L.            (C4)

The two reference bounds coexist: lambda0(L)>=max{gamma,a(1+L)^(-C_*)}.
The R87 proof used (C4) for a large-size absolute remainder and a Weyl
argument at smaller sizes. R99 will additionally use (C2) to turn that
same remainder directly into a relative quadratic-form bound. Nothing
in (C4) negates the stronger finite-size information (C2).

Scope limits. This audit rechecks the finite-volume, boundary, rate and
critical-distance hypotheses. The source constants are fixed universal
ones; no useful decimal exponent or practical field threshold is claimed.
There is no assertion of a new classical mixing theorem or independent
certification. The complete algorithm still depends on its separate
explicit compiler and global-error analysis.
