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Manuscript XVI-M@1 · XVI-M.tex · section · 7346c909

Time-only universal Helmholtz classification (work package C)

M.4 Contact collapse and Helmholtz conditions › Time-only universal Helmholtz classification (work package C) · lines 1446–1470

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2 / fixed-data status after work package C.

Region / reading	Result	Status
Formal universal reading	$ker Collapse = 0$ ; $SA = (26, 0, 0, 0, 0)$	Constructed (Theorem thm:M4)
Time-only universal reading ( $σ (t)$ formal, $χ_{I}$ formal)	$ker Collapse_{t} = 0$ for all $D$ ; $ker H_{t} = (26, 0, 0, 0, 0)$ ; variational image $= 26$ ultralocals, regression-preserving	Constructed (Theorem thm:to-timeonly)
Time-only, $χ_{I} \equiv 1$	$ker H_{t} = (26, 0, 156, 0, 390)$ ; spatial-only, non-Lorentz, variational only by the time-only structure	Constructed (Proposition prop:to-chiplateau)
Preparation plateau, universal $χ_{I}$	$ker Collapse = 21$ ; $SA = (26, 0, 0, 4, 17)$ ; variational image $26$	Constructed for the universal reading (Proposition prop:fixed-regions)
Double plateau	$ker Collapse = 22$ ; variational image $1325$	Constructed (Theorem thm:M4)
Actual $σ_{ε}$ on $(0, ε) \times R^{3}$ , universal switch	time-only table exact	Proposition prop:to-actual under (H1)–(H3); not counted as Constructed
Actual $σ_{ε}$ and $χ_{I}$ on open parts of ${χ_{I} = 1}^{\circ}$ in the slab	time-only $χ$ -plateau table exact (vacuous if empty)	Proposition prop:to-actual under (H1)–(H3)
Actual $χ_{I}$ where non-constant	—	Unresolved ( $χ_{I}$ not explicit); sufficient test recorded
Preparation plateau with actual $χ_{I}$	universal numbers under criteria (ii)–(iii) only	Unresolved for the actual $χ_{I}$
Transitions ( $\partial {χ_{I} = 1}$ ; $t \in {0, ε}$ )	only $0 \leq dim ker Collapse_{fixed} \leq 3321$	Unresolved

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