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Manuscript V@1 · V.tex · figure · e7ddf37d

Direct smoothing and admissibility

Classification on the spacelike-compact solution spaces › The two classification theorems · lines 1342–1371

V@1#fig:smoothing-classification

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metricscalarrepresented quantumdefect / unresolvedsupport / scope
CC compact, K∗C=0K^*C=0
→
q=E2C♭♭‾q=\mathsf E_2\overline{C^{\flat\flat}} smooth
→
JC=2(LE(χq))♯♯J_C=2(L_{\mathrm E}(\chi q))^{\sharp\sharp}
χ = 0
transition slab
χ = 1

Spacelike compactness keeps the slab contribution compact.

JC∈E,FJC(h)=⟨C,h⟩ (h∈S)J_C\in\mathscr E,\quad F_{J_C}(h)=\langle C,h\rangle\ (h\in\mathscr S)
Equality on solutions, not off-shell equality of distributions. Direct Green identities; no radical exhaustion.
raw on S\mathscr S ⟺H=0\Longleftrightarrow \mathbf H=0
BR, B0, Braw∣SdDB_R,\ B_0,\ B_{\mathrm{raw}}|_{\mathscr S_{\mathrm{dD}}}⟺E1H=0\Longleftrightarrow \mathsf E_1\mathbf H=0

Direct smoothing and admissibility. Smooth causal propagation of a compact conserved distribution produces a smooth compact conserved representative through a temporal transition slab. The resulting equality is on vacuum solutions. Raw representability requires the complete defect to vanish; harmonic-prescription representability requires its causal propagation to vanish.

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