R99 G02: quantitative spectral-flow interface used by work-03
Reconstruction of the inherited R87 work-05 estimates, with norm (A) from
R99 work-01. This is an audit of a known strategy, not a new general flow.

Fix gamma from the REFERENCE gap; it is constant in s. On a bootstrap
interval H_s=H0+sV has a unique eigenvalue separated by gamma/2. Let
w be a real even Schwartz function, integral one, with Fourier support
inside (-1/8,1/8). Set w_gamma(t)=gamma w(gamma t). An odd integrated
kernel F_gamma has the inverse-frequency multiplier outside that support
and bounds
 int |t|^k|w_gamma(t)|dt<=C_k gamma^-k,
 int |t|^k|F_gamma(t)|dt<=C_k gamma^(-k-1).
All moment orders below are fixed independently of delta and volume.

T_s(A)=int w_gamma(t)e^{itH_s} A e^{-itH_s}dt removes every ground/excited
matrix element. D_s=int F_gamma(t)e^{itH_s}V e^{-itH_s}dt, with the
spectral-flow sign convention, generates exact projector transport.
This is the standard NSY construction, arXiv:2010.15337v3, Sec4.2.
Positivity of w is not required, but norm estimates use its absolute
moments. No inverse minimum eigenvalue of the Gibbs density is involved.

Let b=a0+2d+4 and M=b+d+4. The exponential interaction bound (A) gives
a real-time light cone with velocity and decay constants uniform in s,
J<=1, delta and L. For fixed-support A, splitting time at c r yields
 ||T_s(A)-Pi_(x,r)T_s(A)||<=C||A|| gamma^-M (1+r)^-M.
The analogous anchored generator estimate is
 ||D_s,x-Pi_(x,r)D_s,x||<=CJ gamma^(-M-1)(1+r)^-M.
The anchor V_x has an exponential shell decomposition, so summing its
support/radius factors changes only fixed constants in this estimate.

Decompose D_s,x into consecutive-radius shells. For a radius-r ball
containing two specified sites there are at most C(1+r)^d possible
anchors. With F_b(distance)=(1+distance)^-b, this gives
 ||D_s||_(F_b)<=CJ gamma^(-M-1) sum_r(1+r)^(b+d-M)
 <=CJ gamma^(-M-1)=kappa.
The sum has exponent -4. Since b>d, F_b is summable and its convolution
constant is volume independent, including finite tori. Enforce kappa<=1
by J<=c gamma^(M+1) before using the parameter-time LR theorem.

For alpha_s(A)=U_s^* A U_s, a fixed local support X then obeys
 ||(alpha_s-id)A||<=C kappa |X| ||A||,
 ||(id-Pi_X(r))(alpha_s-id)A||
 <=C kappa |X| ||A|| (1+r)^(d-b).
The first bound integrates i alpha_u([D_u,A]) and sums interactions
intersecting X. The second follows from the usual F_b commutator bound
and normalized partial trace outside X(r). The LR exponential is
e^{C kappa}-1<=C' kappa, not e^{poly(gamma^-1)}. Normalized partial trace
is contractive and does not introduce a local-dimension factor.

The difference T_s-T0 must retain a factor J. Duhamel gives
 partial_s tau_t^s(A)=i int_0^t tau_(t-u)^s([V,tau_u^s(A)])du.
Inside the real-time cone, the number of anchors is O((1+|u|)^d);
outside, the summed exponential LR tail is bounded by the same order.
Consequently for fixed-support A,
 ||partial_s tau_t^s(A)||<=CJ||A|| |t|(1+|t|)^d.
For |t|<=c r, split V anchors at distance r/4, truncate their exponential
shells at r/8, and truncate the inner evolved A at r/2. Near commutators
are supported within r/2; localizing the final real-time evolution to r
costs CJ times a fixed polynomial in r times e^{v'|t|-mu' r}. Far anchors
are bounded directly by their summed LR tails. Choosing c small makes
this short-time contribution exponentially small in r, retaining J.
For |t|>c r use the norm bound and the M-th filter moment. It gives
 ||(T_s-T0)A-Pi_(x,r)(T_s-T0)A||
 <=CJ||A|| gamma^-M (1+r)^(-(M-d-1)),
with the same coarser bound for the norm at r=0. Subtracting two separate
locality bounds would lose J and would not prove this statement.

Finally define
 Y_x=(alpha_s-id)T_s(h_x)+(T_s-T0)(h_x)+s alpha_s T_s(V_x)
     =alpha_s T_s(h_x+sV_x)-T0(h_x).
T_s(h_x+sV_x) commutes with the perturbed ground projection; alpha_s
transports it to P0. T0(h_x) commutes with P0 as well. Thus [Y_x,P0]=0.
Its scalar c_x=<Omega0,Y_x Omega0> is real, and Z_x=Y_x-c_x I has
Z_x P0=0. Summing uses T_s(H_s)=H_s, T0(H0)=H0, and
sum c_x=E_s. It does not assume T0(h_x)=h_x term by term.

For the first term of Y_x, decompose T_s(h_x) into radius-R shells.
Each norm is <=C gamma^-M(1+R)^-M, each support has O((1+R)^d) sites.
For R<=r/2 use the flow tail and for R>r/2 its norm bound. This gives
 C kappa gamma^-M[(1+r)^(d-b)+(1+r)^(d-M+1)]
 <=CJ gamma^(-2M-1)(1+r)^-a0.
The differentiated-filter estimate gives a stronger radius exponent for
the second term. Filtering and transporting the exponentially decaying
V_x shells gives CJ gamma^-M times the required radius tail for the third.
Their norms at r=0 obey the same coarser prefactor. Subtracting c_x I
preserves every localization tail and at most doubles the norm.
Hence (1) in work-03 holds with A=2M+1 and M0=CJ gamma^-A.

The audit therefore finds no unmatched exponential or support-dimension
factor in this interface. Its quantitative strength is conservative; a
better flow estimate could further improve the field exponent. R99's
proved repair does not require such an improvement.
