Ramp premises: comparison only

The separate ramp main theorem supplies the first-order coordinate identity

The source establishes this identity off shell and on the switched solution. Its observable-invariance conclusion is quoted only within the domain and perturbative order of that theorem Ramp (Theorem (main)). In particular, a first-order identity is not here promoted to an all-orders assertion.

The ramp edge proposition supplies a parameter supported in $0<t<\epsilon$ for which (eq:copied-ramp) fails. In the source's discrepancy convention, its nonzero spatially integrated component is

The notation ${\alpha }_{\mathrm{edge}}$ identifies the edge test profile in that proposition. No order-${\kappa }^2$ term repairs this first-order discrepancy Ramp (edge proposition). The edge parameter is outside the frozen ramp algebra.

The convolution $S_{\epsilon }$ is a spacetime mollifier of the metric argument; the ramp ${\sigma }_{\mathrm{R}}=\chi (t/\epsilon )$ multiplies the deviation of a metric density. Neither operator is the other. The subscript $\mathrm{R}$ also prevents confusion with the amplitude-encoding scale $\sigma >0$ below. The ramp signature is $(+,-,-,-)$. The VII-A and VI materials quoted inside the ramp manuscript use the opposite signature and are not inputs here. No ramp kernel, scalar-vertex calculation, or handoff calculation is imported into Options A–D.