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ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Interior constant and affine reproduction by a symmetric kernel

Example

Let f(y)=a+b·y be affine on R^d. Full-space smoothing by an admissible symmetric kernel reproduces f exactly: S_εf(x)=f(x) for every ε>0 and x∈R^d.

Facts & Assumptions

Given: An admissible kernel W, ε>0, a∈R, and b∈R^d.

[F1]

Scaled symmetric compactly supported kernel family: S_εf(x)=∫f(x−z)W_ε(z)dz.

[F2]

Scaling identities for kernel moments and derivatives: ∫W_ε=1.

[F3]

Symmetry cancels the first kernel moment: ∫zW_ε(z)dz=0.

Verification

1.1givenalgebra

Expand the affine function at x: f(x−z)=a+b·(x−z)=f(x)−b·z.

2.1F1F2F3step 1.1∎

Substitute this expansion into [F1]. The constant term integrates to f(x) by [F2], and the linear term vanishes by [F3]. Thus S_εf(x)=f(x).

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources