Autor: A. Jaster | Datum: 2026/05/24 | DOI: 0.5281/zenodo.20362994
The causal-Machian theory of emergent time of \citet{Jaster2026a} employs a sharp top-hat averaging kernel over the causal horizon domain, identified there as an idealisation requiring a smooth replacement. This paper addresses that open task by replacing the sharp characteristic function of the causal domain boundary with a $C^\infty$ causal bump function of width $\lP=\sqrt{\hbar G/c^3}$, constructed from a locally smooth boundary-defining function $\chi_H$. A concrete form for $\chi_H$ encoding the full four-dimensional retarded causal structure is given for the flat FLRW case. The smeared kernel is a genuine volume-averaging kernel that recovers the original top-hat pointwise in the regular sector as $\lP\to0$. The interpolation function $S(r)$ of the parent theory is simultaneously replaced by a fully $C^\infty$ saturating function $S_\infty(r)$, ensuring that second and higher derivatives of the scalar time field $\Theta_{\lP}$ are finite and continuous in the regular sector $\Mreg$. All structural properties of the parent theory are preserved on $\Mreg$; inherited open tasks, in particular the explicit construction of the constraint stress tensor $T^{\rm C}_{\mu\nu}$, are identified and listed. The construction introduces a natural Planck-energy scale and suppresses ultraviolet contributions from causal-boundary gradients in the weak-field limit; full UV-finiteness of a quantised theory requires a separate analysis. No new dimensionful parameter is introduced; dimensionless functional choices remain part of the model definition.