# Verification of the spatial mollification theorem

The analytic proof handles the coalesced Dyson entrance-law bridge itself.
A weighted drift identity bounds the integral of inverse particle gaps
by Gaussian norms. Bounded half-bridge absolute continuity and reversal
give exponential moments of every order. The covariance kernel obeys a
uniform inverse-gap bound, so the omitted mutual-path counterterm tends
to zero with a single exponentially integrable dominator.

Spatial convolution contracts each known occupation-correlation kernel.
Chaos orthogonality gives the full-Wick L2 limit. Replica identities and
the O'Connell-Warren all-exponential intersection-local-time estimates
give arbitrarily high uniform moments. Holder's inequality then controls
the separate-Wick correction, without assuming independence of overlapping
replica pairs. The GUE gap density and the two endpoint layers yield an
O(epsilon) first moment; exponential domination upgrades this to the
stated convention-comparison rate.

The standard-library checker performs 2,000 exact rational controls:
1,200 weighted drift cases, five tensor/replica cases, 360 counterterm
cases, 99 rate-exponent cases and 336 Gaussian antiderivative cases.
Normal and optimized Python runs agree. The complete local runs also
check the exact frozen/public AIM statement agreement and source number.
Portable archive runs use --algebra-only and explicitly omit that local
source check. These controls are not formal verification of the analytic
argument or numerical testing of the stochastic limit.

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The accepted theorem has passed originating-model self-audit. It is not
independently peer reviewed. The existing chaos object and key estimates
are credited to their authors. The bounded literature search is not a
certificate of absolute priority.