Time-only universal Helmholtz classification (work package C)

of the time-only analysis.

• Constructed (Gate V; on the current coherent presentation): the intertwining lemma and its tests; exact route 1 / route 2 agreement; the time-only classification (Theorem thm:to-timeonly): ${\mathrm{Collapse}}_t$ injective, variational image $=$ the $26$ ultralocal directions, all regression-preserving, no $D\ge 1$ direction; the time-only $\chi$-plateau classification $(26,0,156,0,390)$ with its exact spatial-operator identification; the matching probes and divergence identities (Proposition prop:to-match); the linear independence of ${{\sigma }_{\epsilon }}^{(k)}$ on open subintervals of $(0,\epsilon )$ (Lemma lem:to-indep).

• Proposition with explicit hypotheses (not Constructed): the transfer to the actual XII-C ramp with a universal switch on $(0,\epsilon )\times {\mathbb{R}}^3$, and to open parts of $\{{\chi }_{\mathrm{I}}=1{\}}^{\circ }$ in the slab $(0,\epsilon )\times {\mathbb{R}}^3$ (Proposition prop:to-actual, hypotheses (H1)–(H3)).

• Unresolved: the actual fixed pair $({\sigma }_{\epsilon },{\chi }_{\mathrm{I}})$ wherever ${\chi }_{\mathrm{I}}$ is non-constant, because ${\chi }_{\mathrm{I}}$ is not given explicitly; transition neighbourhoods of $\partial \{{\chi }_{\mathrm{I}}=1\}$ and of $t\in \{0,\epsilon \}$. Sufficient test (recorded): on an open $U$, a full-rank exact evaluation matrix, on $U$, of the external-data coefficient functions ${{\sigma }_{\epsilon }}^{(k)}{\partial }^{\rho }{\chi }_{\mathrm{I}}$ that occur in (i) the graph map, (ii) ${\mathrm{Collapse}}_t$ and (iii) the adjoint-defect map $H_t$. Full rank for (iii) gives ${\mathrm{SA}}_{\mathrm{fixed}}{|}_U=\ker H_t$. The condition concerns the joint family $\{{{\sigma }_{\epsilon }}^{(k)}(t){\partial }^{\rho }{\chi }_{\mathrm{I}}(x)\}$; it is not verified for any concrete ${\chi }_{\mathrm{I}}$, and it does not follow from separate linear independence of the ${{\sigma }_{\epsilon }}^{(k)}$ (Lemma lem:to-indep) and of the ${\partial }^{\rho }{\chi }_{\mathrm{I}}$ (Remark rem:to-joint).

• Non-claims: the $546$ spatial $\chi$-plateau directions are not gravitational (Lorentz-scalar) counterterms and are not matched to any curvature functional; no statement is made after other diagrams are added, and none about a completed gravitational sector; no $\epsilon \to 0$ limit is taken; preimage coefficients are not Wilson coefficients; the double-plateau $R^2/{\mathrm{Ric}}^2$ matchings (Theorem thm:M5-plateau-R2) are unchanged as double-plateau statements, but they do not admit the stated universal time-only completion (Proposition prop:to-match(3)); the actual-ramp conclusion holds under Proposition prop:to-actual.