Abstract

We freeze one classical prescription. A minimally coupled scalar lives on the formal metric jet $q=\eta +\kappa h+{\kappa }^{2}k$; the curved-minus-flat scalar density is multiplied by one compact switch $g$ before variation. Four coordinate functions are defined by the divergence-form equation of an edge-regulated auxiliary density ${\mathfrak{A}}_{\epsilon }[q]={\eta }^{-1}+{\sigma }_{\epsilon }({\mathfrak{A}}_q-{\eta }^{-1})$, where ${\sigma }_{\epsilon }$ is an explicit smooth time ramp of width $\epsilon$ and ${\mathfrak{A}}_q$ is the metric density of $q$, with fixed Cauchy data on the initial surface. The gauge algebra consists of vector fields supported in a plateau $U$ whose closure lies inside $\{g=1\}\cap \{t>\epsilon \}$.

The regulator preflight is passed exactly: the regulated first coordinate coefficient returns $\xi$ on every pure gauge $K\xi$ of the frozen algebra. An explicit witness shows that the same identity fails for gauge parameters entering the edge zone $0<t<\epsilon$, so the plateau condition is part of the freeze and not removable.

The relational scalar $\Phi =\varphi \circ {{\mathsf{X}}_{\mathsf{q}}}^{-1}$, smeared by one fixed label-space test, is proved gauge invariant through ${\kappa }^2$ at fixed $\epsilon >0$, off shell in the metric and in the scalar, by two independent routes: covariance of the regulated coordinate operator and of the switched Euler operator under plateau-supported transformations together with coefficientwise Cauchy uniqueness; and an exact varied-action identity in which every switch term is displayed and cancels only through $g=1$ and $\mathrm{d}g=0$ on the gauge support. The switched scalar solution with fixed free input intertwines the same action, so the on-shell evaluation is invariant as well. The pure-gauge two-jet and a sourced diagnostic are checked. The linear coordinate kernel is smooth: the regulator replaces the hypersurface delta of the sharp prescription by a ramp term.

No quantum product, quantum Ward identity, sharp-regulator limit, boundary algebra, admission topology, or identification with the VII–IX or cone constructions is claimed.