Planck-Scale Smearing of the Causal Time Field: Boundary Regularisation of the Emergent Time Theory

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 $\ell_{P}=\sqrt{\hbar G/c^3}$, 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 $\ell_{P}\to0$.  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 $\Theta_{\ell_{P}}$ are finite and continuous in the regular sector $\mathcal{M}_{\rm reg}$.  All structural properties of the parent theory are preserved on $\mathcal{M}_{\rm reg}$; 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.