Alphabeta Math
CounterexampleConstruction: 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 phase requires a convention

Statement refuted

“The displayed Gauss-sum phase is independent of the additive-character convention.”

Facts & Assumptions

Given: Under e(x)=exp⁡(2πix), τ(χ4)=2i (Gauss sum for the character modulo 4).

Counterexample

technique · direct
1.1

Replace e(x) by e(−x). The two terms become e(−1/4)−e(−3/4)=−i−i=−2i.

givenalgebra
2.1

This is the conjugate phase, not 2i, although its squared modulus remains 4.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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