Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-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.

Duplicate sites make the kernel Gram matrix singular

Example

Let K(x,y)=ϕ(∣x−y∣) be any radial kernel, and let a site list contain x1=x2.

Facts & Assumptions

Given: A radial kernel and a finite site list with x1=x2.

[F1]

A radial kernel has Gram entries Kij=ϕ(∣xi−xj∣), and ordinary RBF interpolation is defined on distinct sites. (Radial kernels, polynomial augmentation, and site geometry)

Verification

1.1F1givenalgebra

For every row index i, the equal sites give Ki1=ϕ(∣xi−x1∣)=ϕ(∣xi−x2∣)=Ki2. Thus the first two columns, and by symmetry the first two rows, coincide.

2.1step 1.1F1algebra∎

The nonzero vector e1−e2 is therefore in the nullspace of the Gram matrix, so the matrix is singular. This example violates the distinct-site hypothesis in [F1] and is not a counterexample to the interpolation theorem.

Depends on

Used by

Nothing in the library uses this result yet.

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Sources