Audit 5: replacement and functional universality

Gaussian and Rademacher couplings match their first three moments. The fourth moments differ. In a fourth-order replacement calculation, normalization contributes $n^{-2}$ while the number of edges is of order $n^2$. These powers alone give only order one, not an error tending to zero.

The source therefore uses clock-history cutoffs, projected scores, derivative estimates including cutoff derivatives, and a functional tightness argument OAI227History,OAI227Scores,OAI227Replacement. The general replacement methodology has a separate literature Chatterjee.

A full audit must verify both removal of the cutoffs and control between time-grid points. Comparison of finitely many evaluations alone does not prove convergence in $C_{\mathrm{loc}}$.

Universality in the frozen statement concerns the two named disorders. It does not assert universality for every mean-zero disorder distribution.