Finite-n Chernoff Errors under Laplace Transformation: Stationary Singularities, Scaling, and Chebyshev Control.

Shift-family Chernoff approximations of the heat semigroup produce a movinghigh-frequency "comb" artifact on non-smooth data at any finite number ofcompositions n. This note shows what happens to the comb under Laplacetransformation (the resolvent route): it decomposes **exactly** into astationary singular atom `w0(n) f / lambda` -- inherited from the centralweight of the shift lattice -- plus a C^2-smoothing mixture of Rayleighkernels. Consequences: protected (exactly computable) coefficients of allcomponents of the datum below the C^2 threshold, with the first componentabove it carrying a different, explicitly computable coefficient(verified on the regularity ladder |x|^{1/2}, |x|, |x|^{3/2} and by aparameter-free second-difference test against an analytic secondderivative); an empirical two-parameter scaling law`E* ~ K (w0/lambda) sigma_1^xi` collapsing 18 configurations per datum;a smooth-vs-kink ranking inversion between the first- and second-ordershift families in all 27 tested raw kink configurations (1.36--2.35x); anda quantified verdict on Chebyshev post-processing after Laplacetransformation, including the exact parity mechanism behind theboundary-kink obstruction and a blind certification rule that correctlyrefuses all 54 kink configurations.