Proposition: Sine limit and derivative check

On $R>0$,

Globally, with the Fourier definitions,

Proof. For a one-dimensional compactly supported smooth test function $\zeta$,

The integrable majorant $\Vert \zeta {\Vert }_{\infty }/(1+a^2)$ proves this delta-sequence limit. It is the standard distributional interpretation, not a pointwise limit on the cone NIST.

Applying it to the two translated terms of (eq:D-regulated) proves (eq:D-evaluated). The argument also holds under joint pairing in $(v,\mathbf{z})$ on compact subsets of $R>0$, where $1/R$ is smooth and bounded.

At finite regulator,

both by Fourier differentiation and by (eq:H-arctan). Distributional differentiation is continuous, so the global convergence of $H_{\epsilon }$ proves (eq:D-H-check), including the apex in the Fourier sense.