Manuscript XX-C@1 · XX-C.tex · lemma · 7a8cab37

Lemma: Compact testing gives smoothness, not decay

Covariance tails, coordinate pullback, and failure of unconditional infrared control › What compact testing does supply · lines 1202–1219 · proof included or linked

XX-C@1#lem:smoothsmear

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The section $w_{f}$ is smooth and solves $B w_{f} = 0$ . No spatial decay follows from (eq:Wdata)–(eq:Wwavefront) alone.

Proof. The field equation is immediate. The wavefront relation has no point with zero covector in the tested slot. The standard pushforward criterion after compact testing therefore makes $w_{f}$ smooth. Spatial decay is not a microlocal conclusion. The counterexample below proves the final assertion on the native operator.

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section: What compact testing does supply
theorem: Theorem: Sufficient one-sided covariance-tail condition
section: What compact testing does supply