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TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)
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Local support from one-sided vanishing and visible conormal data

Statement

Let R be an analytic double-fibration transform, let (z0,ζ0;x0,η0)∈CZ, and let Σ be a C2 hypersurface through x0 with (x0,η0)∈N∗Σ∖0. Let f∈E′(X) vanish on one side of Σ near x0. If the pointwise Bolker condition holds at (z0,ζ0;x0,η0) and Rf=0 in a neighborhood of z0, then f=0 in a neighborhood of x0.

Facts & Assumptions

Given: The transform R, a relation point satisfying pointwise Bolker, a C2 hypersurface Σ, and a compactly supported distribution f with the stated vanishing and data hypotheses.

[F1]

Analytic double-fibration transform and canonical relation: Rf is a distributional density on the data manifold.

[F2]

Analytic wavefront recovery under Bolker: if the data covector is absent from WF⁡A(Rf) under pointwise Bolker, then the paired object covector is absent from WF⁡A(f).

[F3]

One-sided support forces its analytic conormal wavefront: one-sided vanishing plus absence of either oriented analytic conormal implies local vanishing.

[F4]

Analytic wavefront set by Gaussian FBI packets: exponential decay of the localized Gaussian FBI pairings is the criterion for absence from WF⁡A.

Proof

Given: The data in the statement.

1.1F1F4given

Choose a data cutoff supported inside the neighborhood on which Rf=0 and equal to one near z0. Its localized FBI test functions are supported where the distribution vanishes, so all the pairings are exactly zero for every frequency. By [F4], this gives (z0,ζ0)∉WF⁡A(Rf).

2.1F2F3step 1.1given∎

By [F2], step 1.1 gives (x0,η0)∉WF⁡A(f). Applying the one-sided continuation implication [F3] then gives f=0 near x0. All their hypotheses hold: the relation point has pointwise Bolker, f has compact support, and the same nonzero object covector is a conormal to the supplied C2 hypersurface.

Depends on

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