CorollaryStatement: Literature-sourcedProof: 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.

Unit modulus of the Dirichlet root number

Statement

For primitive $χ$ , $∣ ε (χ) ∣ = 1$ .

Facts & Assumptions

Given: A primitive character $χ$ .

[F1]

$ε (χ) = (- i)^{a} τ (χ) / \sqrt{q}$ (Functional equation for primitive Dirichlet L-functions).

[F2]

$∣ τ (χ) ∣^{2} = q$ (Norm of a primitive Gauss sum).

Proof

technique · direct

1.1

Taking absolute values in [F1] gives $∣ ε (χ) ∣ = ∣ τ (χ) ∣ / \sqrt{q}$ .

F1given

2.1

Fact [F2] makes this quotient $1$ .

F2step 1.1algebra∎

Depends on

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

Andersen, Remark 16.9 (standard reference, not scraped)