Abstract

We establish the sharp-width limit of the completed scalar-linear observable of X-C-ST on the supported host already fixed by X-A-ST. The calculation takes place on the declared Minkowski completion; the conical host, its worldsheet, signature, and future relation remain prescribed inputs.

For fixed smooth classical backgrounds, a fixed interaction switch, and observation and interaction supports separated from the initial surface, both completed coefficient tests converge at rate $O(\epsilon )$ in every seminorm of one regulator-independent compact-test space. Their classes therefore converge in the original free-scalar quotient topology. The proof first integrates the coordinate normal flux and then estimates the shrinking layer. It derives zero limiting positive-order positions and the velocities

The quadratic velocity survives, whereas profile-dependent position moments vanish. A conformal example shows that the $O(\epsilon )$ rate cannot generally be improved to $o(\epsilon )$.

The limit is the continuous-coordinate, continuous-normal-flux transmission prescription. Its observation-smeared linear background kernel can jump at the initial surface but contains no hypersurface delta. It is generally different from the unchanged zero-velocity multiplied-equation prescription. The completed background identity passes to the limit on the gauge algebra frozen before the calculation.

The principal limit theorem is credited to Cheung's X-D. The contribution here is its proved supported transfer, an explicit topology and defect audit, and a finite complete-channel encoding of supplied evaluations. Its prospective usefulness is a reproducible sharp scalar baseline that does not confuse test convergence with quantum metric admission or characteristic composition. No time-ordered hierarchy, loop regulator, continuum-field acquisition, or spacetime-emergence construction is supplied.