ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Trivial zeros of the beta function

Example

The beta function $L (s, χ_{4})$ has forced zeros at $- 1, - 3, - 5, \dots$ . No assertion at $s = 0$ follows from this odd-parity rule.

Facts & Assumptions

Given: $χ_{4}$ is odd (Gauss sum for the character modulo 4) and the trivial-zero corollary (Parity-forced trivial zeros).

Verification

technique · direct

1.1

Odd parity means $a = 1$ .

given

2.1

The given corollary gives $L (- 1 - 2 m, χ_{4}) = 0$ for $m \geq 0$ , which is exactly the displayed list.

step 1.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Andersen, section 16.3.2 (standard reference, not scraped)