Manuscript V@1 · V.tex · figure · e7ddf37d
Support classes and restriction
What the full-space classification does not settle › Support classes · lines 1612–1644
V@1#fig:support-and-restriction
metricscalarrepresented quantumdefect / unresolvedsupport / scope
Full-space classification
spacelike-compact vacuum solutions; compact smooth Cauchy data; residual pure gauges includedRestricted TT plane
Fourier-gap Schwartz data; spatial TT; generally not spacelike compact; no nonzero Schwartz pure gaugeA nonzero datum cannot be both. Compact spatial support makes the Fourier transform entire; an open Fourier gap then forces zero.
The full-space witness has a nonzero component and is not spatial TT. Failure on the larger space does not decide a smaller TT restriction.
Support classes and restriction. Compact-Cauchy residual witnesses and Fourier-gap Schwartz TT waves belong to different declared classes. A full-space obstruction transfers to a restriction only when its witness belongs to that restriction. On a fixed TT plane, representability is instead an extension problem modulo the plane's annihilator.