Abstract

We decide the sharp limit of the completed scalar-only observable of Landmark X-C-v3 for fixed observation test, interaction switch, classical metric coefficients, and plateau gauge algebra. The observation and interaction supports remain separated from the initial surface.

Both completed coefficient tests converge in the smooth topology of one regulator-independent compact test space, with rate $O(\epsilon )$ in every seminorm, and hence in the quotient test topology of the free scalar algebra. The proof integrates the coordinate normal flux before estimating the boundary layer. The limiting positive-order initial positions vanish. The limiting velocities are $-{a_1}^{0\rho }[h]{|}_{\Sigma }$ and $({a_1}^{00}[h]{a_1}^{0\rho }[h]-{A_2}^{0\rho }[h,k]){|}_{\Sigma }$. The quadratic contribution is retained; profile-dependent position moments vanish at order $\epsilon$.

These data identify the sharp action-level transmission problem: continuous coordinates and continuous normal flux. Its smeared linear background kernel is $\theta C^{\ast }({{\mathsf{E}}_0}^{\mathrm{A}}a)$, with a possible jump but no hypersurface delta. It differs from the multiplied-equation prescription with zero positive-order Cauchy velocities.

The fixed-algebra Ward identity passes to the limit. The supplied VII-C hypotheses are copied and audited: its quantum metric block, mixed insertion, normal-topology approximants, and mandatory triple product are absent here. VII-C therefore does not apply and is not contradicted. No new product, quantum metric, or subtraction is constructed. Neither the moving edge witness nor a crude inverse-regulator upper bound is used as the decision.