ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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 von Mangoldt residue table

Example

For $x > 1$ , the contour integrand has contributions $x$ at $1$ , $- x^{ρ} / ρ$ at a nontrivial zero, $- \frac{1}{2} \log (1 - x^{- 2})$ from all negative even zeros, and $- ζ^{'} (0) / ζ (0)$ at $0$ .

Verification

Given: $x > 1$ and the residue ledger.

1.1

The pole at $1$ and each nontrivial zero have the listed local residues, with zero multiplicity multiplying the second entry.

givenalgebra

2.1

Summing the geometric-logarithmic trivial family and retaining the independent pole of $1 / s$ at $0$ gives the remaining two entries.

step 1.1algebra∎

Depends on

Residues in the von Mangoldt contour shift

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Kiran S. Kedlaya, Analytic Number Theory, §10.1 (standard reference, not scraped)