R99 G02: exact algebra and support-norm audit
Same-agent audit. These inherited ingredients are not new theorem claims.
Frozen original and R87/R90 proof files are retained unchanged.

A. Which parent and which restriction?
Let A0=-H(0), t=beta/2, K_g=e^{t(A0+g sum X)}e^{-tA0}.
There are TWO annihilators per anchor: mismatch P_i and heat-bath h_i.
Q_g=sum_i[(K_g P_i K_g^-1)^*(K_g P_i K_g^-1)
          +(K_g h_i K_g^-1)^*(K_g h_i K_g^-1)].
The conjugates of P_i+h_i must not be squared as one term. At g=0 their
cross terms vanish, but no such claim holds after nonunitary conjugation.
K_g is invertible, so the common kernel is exactly K_g times the original
one-dimensional kernel. Acting on vectorized e^{-beta H0/2} gives vectorized
e^{-beta Hg/2}. Normalizing and tracing the ancilla gives the actual Gibbs
state. This statement is algebraic and independent of any gap argument.

Each doubled site is dimension4. The canonical restriction to a square
includes only anchors with the ENTIRE star inside that square. Outer
physical and ancilla labels are conserved. Interior mismatches cost at
least one; every summand is positive and commutes with each mismatch flag.
In a matched interior sector, the h_i sum is the symmetric transform of
the per-site-rate-one classical heat-bath generator, with fixed physical
shell spins. Its kernel is sum_x sqrt(pi^tau(x))|x,x>.
The full local kernel is a direct sum over BOTH physical shell tau and
free ancilla shell eta; shell mismatch penalties are not added silently.
The local nonzero gap is bounded below by the minimum of one and the
worst fixed-boundary heat-bath gap. This uses arbitrary boundary spins,
not merely a torus gap or a plus-boundary theorem.

For an inner doubled set A, trace out its doubled complement in that
coherent conditional Gibbs vector. The outside configuration is identical
in bra and ket, leaving the exact mixture
 rho_A^tau=sum_b pi^tau(partial^+ A=b)|phi_b><phi_b|,
 |phi_b>=sum_xA sqrt(pi(xA|b))|xA,xA>.
The map b -> |phi_b><phi_b| is independent of tau. It is a quantum channel
from a classical register, so its trace-norm contraction has no factor
4^|A|. Since inner operators do not change the shell labels, compression
to the entire local ground space is diagonal in those labels. Joint
classical marginal TV bounds therefore control ALL quantum observables.
The comparison with the global torus expectation follows by conditioning
the global classical law on the same shell and using the same channel.
The factor proportional to the support or its boundary is retained.

B. The actual perturbation has the norm needed by real-time LR.
R83 gives each annihilator R_i=sum_(m>=0)R_i,m, R_i,0=A_i, with radius
3+2m and summed norm bounded by
 e^{36t} q^m (13/4)_m/m!, q=40t e^{10t}|g|<=1/4.
This is a convergent connected-commutator majorant, not an expansion in
the extensive operator norm of g sum X. The support and norm constants
are uniform for t<=t_c=log(1+sqrt2), also on wrapped tori.

Group R_i^*R_i by k=m+l. Each group is Hermitian because it contains the
ordered pair and its adjoint, has radius <=3+2k, and its norm is at most
 e^{72t} q^k (13/2)_k/k!.
Only k>=1 occurs in V=Q_g-Q0. For a fixed doubled site x, at most
C(1+k)^2 anchors have their radius-(3+2k) support containing x. Each
support itself has <=C(1+k)^2 sites and diameter <=6+4k.
Thus with mu=(log2)/8,
 sup_x sum_(X contains x) |X| e^{mu diam X} ||V(X)||
 <= C sum_(k>=1)(1+k)^4 2^{k/2} q^k (13/2)_k/k! <= C' |g|.   (A)
To verify the last step uniformly down to g=0, take one factor q out:
q^k<=q(1/4)^(k-1). The remaining series is a fixed convergent
polynomial-times-geometric series. q<=40t_c e^{10t_c}|g|.
Torus identifications reduce, rather than increase, these cardinalities.
The reference finite-range term has the same norm bounded by a constant.
Hence H_s=Q0+sV has a uniform finite real-time LR velocity for J=C'|g|<=1.
No local Hilbert-space dimension is exponentiated. This is the exact
support-weighted norm also matched in R97/G05; here the essential extra
point is the O(|g|) bound for the DIFFERENCE interaction.

C. No dependence on the wrong interpolation.
For a fixed endpoint field g the stability proof uses H_s=Q0+s(Q_g-Q0).
Its intermediate terms need not be positive or frustration free. On a
bootstrap interval only a unique ground eigenvalue separated by gamma/2
is assumed. The spectral filter removes transitions out of that state,
and exact transport moves its projector back to P0. Its energy E_s must be
subtracted. The endpoint parent Q_g itself remains positive and has its
exact TFD kernel, independently of this interpolation.
The preparation algorithm instead follows the physical beta and field
paths. A uniform endpoint gap theorem can be applied at every point of
those paths, but the two paths must not be identified term by term.

Audit finding: no local-kernel or norm mismatch was found. R90 already
checked much of this structure, so repeating it is not an original core
advance. The new R99 target is the global relative remainder in work-03.
