Proposition: Application reference and profile comparison

For this specialization:

1. $t_{\delta }$ is eventually zero on each compact subset of $U\setminus S_{\kappa }$, so the proved reference there is zero;

2. replacing $p,\psi ,\omega$ by fixed compact smooth normalized profiles of the same factorized form, using one common even nonnegative longitudinal $p$ for the two four-variable factors, preserves $\mathcal{T}$;

3. the inverse-scale coefficient depends on $Q_p=\int p^2$, and is not profile independent;

4. ${\delta }^{-1}\mathcal{A}$ is a mathematical subtraction but is not authorized by the declared class ${\mathcal{C}}_{\mathrm{allowed}}=\{0\}$.

No uniformity over all profiles is asserted.

Proof. The support of the exact expression satisfies

All tangential parameters stay in compact positive ranges. Thus the support approaches $S_{\kappa }$ uniformly and eventually misses every compact set in its complement. Distributional differentiation does not enlarge support.

The finite-part moment values are $1,1/2,0$, independent of the even normalized $p$. The other factors affect the convolution only through their normalized marginals. For a different fixed compact longitudinal support, the rescaled integration range remains compact, although its size and the remainder constants may change.

The inverse-scale formula contains $Q_p$; normalized even profile dilations change this number. Since $\mathcal{A}\ne 0$, the proposed subtraction is nonzero and is not in $\{0\}$.