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.

Gauss sum for the character modulo 4

Example

For $χ_{4} (1) = 1$ , $χ_{4} (3) = - 1$ , one has $τ (χ_{4}) = 2 i$ and $ε (χ_{4}) = 1$ .

Facts & Assumptions

Given: The displayed Gauss normalization (Gauss sum of a Dirichlet character) and odd parity convention (Parity of a Dirichlet character).

Verification

technique · direct

1.1

The two nonzero terms are $e (1 / 4) - e (3 / 4) = i - (- i) = 2 i$ .

givenalgebra

2.1

Its squared modulus is $4$ , agreeing with the primitive norm theorem; with $a = 1$ , $(- i) τ (χ_{4}) / 2 = 1$ .

step 1.1algebra∎

Depends on

Used by

Gauss-sum phase requires a convention Counterexample
Trivial zeros of the beta function Example

Dependency tree · two levels

5 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 16.2 (standard reference, not scraped)