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.

The monopole symbol of the exact spherical DtN map

Example

Since $h_{0}^{(1)} (z) = - i e^{i z} / z$ , its logarithmic derivative is $h_{0}^{'} (z) / h_{0} (z) = i - z^{- 1}$ . Therefore the degree-zero exact spherical DtN multiplier at radius $R$ is $τ_{0} (k, R) = i k - 1 / R$ .

Facts & Assumptions

Given: $k, R > 0$ and the exact modal map [F1].

[F1]

Exact modal Dirichlet-to-Neumann map on a sphere: $τ_{0} (k, R) = k h_{0}^{'} (k R) / h_{0} (k R)$ .

[F2]

The spherical modal DtN map is well-defined and exact for smooth data: the multiplier is the exact outward radial derivative of the outgoing degree-zero extension.

[F3]

Spherical Bessel and outgoing Hankel function conventions: $h_{0}^{(1)} (z) = - i e^{i z} / z$ .

Verification

Given: $k, R > 0$ and [F1]–[F2].

1.1F1F3algebra

By [F3], $h_{0} (z) = - i e^{i z} / z$ , so $h_{0}^{'} (z) = (- i) e^{i z} (i / z - 1 / z^{2}) = h_{0} (z) (i - 1 / z)$ .

2.1F1F2F3step 1.1algebra∎

As $h_{0} (k R) \neq 0$ by [F3], substitute $z = k R$ into step 1.1 and multiply by $k$ : $τ_{0} = k (i - 1 / (k R)) = i k - 1 / R$ . Theorem [F2] identifies this multiplier with the exact outgoing normal derivative.

Depends on

Used by

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Dependency tree · two levels

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Sources

NIST Digital Library of Mathematical Functions, Chapter 10: Bessel Functions (standard reference, not scraped)