Proposition: Region ledger for fixed preparation/switch

Assume Gate V. Distinguish four regions.

1. Full switched/prepared region. Comparison classes: formal universal reading, and the time-only class, whose Helmholtz analysis is now exact (Theorem thm:to-timeonly). Status: sufficient criteria (Proposition prop:fixed-crit); on $(0,\epsilon )\times {\mathbb{R}}^3$ with a universal switch the time-only table applies under the explicit hypotheses of Proposition prop:to-actual; actual dimensions where ${\chi }_{\mathrm{I}}$ is non-constant Unresolved.

2. Preparation plateau with variable switch, universal reading. On $\{{\sigma }_{\epsilon }\equiv 1\}$ with formal ${\chi }_{\mathrm{I}}$-jets (reading prep_plateau): support kills $C_2$/$M_2$; $\dim \ker \mathrm{Collapse}=21$ exactly; $\mathrm{SA}=(26,0,0,4,17)$ by $D$; the $D\ge 1$ SA directions lie in $\ker \mathrm{Collapse}$, so the variational image is exactly the $26$-dimensional ultralocal family. Exact for the universal reading; for the actual fixed ${\chi }_{\mathrm{I}}$ it holds under (ii)–(iii) only.

3. Double plateau. On $\{{\sigma }_{\epsilon }\equiv 1,{\chi }_{\mathrm{I}}\equiv 1\}$: recover Theorem thm:M4 exactly ($\dim \ker \mathrm{Collapse}=22$, variational image $1325$).

4. Transition regions near plateau boundaries: exact dimensions Unresolved. Only the domain-size bound $0\le \dim \ker {\mathrm{Collapse}}_{\mathrm{fixed}}\le 3321$ is asserted. (The previously stated upper bound $22$ was unsupported: rank is only lower semicontinuous, so special points of the fixed data could have larger kernels. No sharper bound is proved.)

Vanishing by support, additional relations among fixed coefficient functions, and genuine Gate V self-adjointness are kept distinct: no concrete ramp identity enlarging $\ker {\mathcal{G}}_{\mathrm{fixed}}$ is certified. {Status: fixed-data criteria, universal region comparisons and the time-only classification constructed; actual fixed-data dimensions unresolved where ${\chi }_{\mathrm{I}}$ is non-constant} (status table at the end of Section sec:to-timeonly). Archives: fixed_data_reading.py, fixed_data_reading.json, prep_plateau_sa.json; wpC/.