Proposition: Regular-origin certificate

Status: Conditional; implication proved for displayed mathematical data.

Suppose the complete ${{\Omega }_{\Phi }}^{\zeta }$ has a continuous representative $W_{\Phi }$ on a common origin neighborhood for tests supported in each compact $H$, with

Then ${t_{\delta }}^{\zeta }$ converges strongly to $W_{\bullet }(0)$, and both its raw difference and logarithmic derivative are $O_H({\delta }^a)\Vert \Phi {\Vert }_{j,H}$.

Proof. At small scale, ${t_{\delta }}^{\zeta }(\Phi )=\int \rho (s)W_{\Phi }(\delta s)\mathrm{d}s$. For the derivative use $\int \kappa (s)[W_{\Phi }(\delta s)-W_{\Phi }(0)]\mathrm{d}s$. Normalization and zero mean give the estimates. Uniqueness of continuous representatives gives linearity; the test bound gives distributional continuity.