Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)
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 χ4(1)=1, χ4(3)=−1, one has τ(χ4)=2i and ε(χ4)=1.

Facts & Assumptions

Given: The displayed Gauss normalization (Gauss sum of a Dirichlet character) and odd parity convention (Parity of a Dirichlet character). The root number is ε(χ)=(−i)aτ(χ)/q for primitive χ of parity a (Functional equation for primitive Dirichlet L-functions).

Verification

technique · direct
1.1givenalgebra

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

2.1step 1.1givenalgebra∎

Its squared modulus is 4, agreeing with the primitive norm theorem; χ4 is primitive of parity a=1, so the cited root-number formula gives ε(χ4)=(−i)τ(χ4)/2=1.

Depends on

Used by

Dependency tree · two levels

10 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