Proposition: Sufficient criteria for fixed data

Assume Gate V. Always $\ker {\mathcal{G}}_{\mathrm{formal}}\subseteq \ker {\mathcal{G}}_{\mathrm{fixed}}$ and $\ker {\mathrm{Collapse}}_{\mathrm{formal}}\subseteq \ker {\mathrm{Collapse}}_{\mathrm{fixed}}$ (specialization). Let $U$ be an open set meeting the relevant support, and let ${\mathrm{ev}}_U$ be evaluation of formal external-jet monomials (the ${\sigma }_{\epsilon }$- and ${\chi }_{\mathrm{I}}$-jets occurring) on the actual fixed data over $U$. Consider the finite-dimensional coefficient spaces spanned by the external-jet coefficients occurring in

1. (i)

   the canonical graph map $\mathcal{G}$;

2. (ii)

   the collapse map $\mathrm{Collapse}$;

3. (iii)

   the adjoint-defect map $C\mapsto {E_C}^{†}-E_C$ (whose coefficients include the external-jet derivatives produced by integration by parts).

If ${\mathrm{ev}}_U$ is injective on the space in (i), then $\ker {\mathcal{G}}_{\mathrm{fixed}}{|}_U=\ker {\mathcal{G}}_{\mathrm{formal}}=0$; if it is injective on the space in (ii), then $\ker {\mathrm{Collapse}}_{\mathrm{fixed}}{|}_U=\ker {\mathrm{Collapse}}_{\mathrm{formal}}=0$; if it is injective on the space in (iii), then ${\mathrm{SA}}_{\mathrm{fixed}}{|}_U={\mathrm{SA}}_{\mathrm{formal}}$, the $26$-dimensional ultralocal family. Condition (iii) is needed for the SA statement: injectivity in (i) or (ii) alone does not control the defect map.

• Time-only caveat. The actual ${\sigma }_{\epsilon }$ is a time-only ramp, so every spatial ${\sigma }_{\epsilon }$-derivative vanishes identically and the full spacetime preparation jets are not independent; conditions (i)–(iii) cannot hold for the formal coefficient spaces. A time-only comparison class (spatial ${\sigma }_{\epsilon }$-jets set to zero, ${\chi }_{\mathrm{I}}$ formal) needs its own Helmholtz analysis, with its own collapse and adjoint-defect maps; injectivity of the time-only graph map alone is insufficient. That analysis is carried out in Section sec:to-timeonly (Theorem thm:to-timeonly, Proposition prop:to-chiplateau); its transfer to the actual ramp is Proposition prop:to-actual, which has explicit hypotheses.

• Universal versus actual. Readings with formal ${\chi }_{\mathrm{I}}$-jets (formal; preparation plateau) are universal variable-switch readings: their dimensions are exact for that universal reading. They coincide with the actual fixed ${\chi }_{\mathrm{I}}$ only under (i)–(iii) for the actual ${\chi }_{\mathrm{I}}$ on $U$, which are not verified here.

These are sufficient criteria, not a certification of the concrete XII-C ramp. Formal universal-jet vanishing at $D\ge 1$ does not by itself settle any fixed-data case.