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.

Even and odd theta kernels

Example

For the even character modulo $1$ , the kernel is $\sum_{n} e^{- π n^{2} t}$ . For odd $χ_{4}$ , it is $\sum_{n} n χ_{4} (n) e^{- π n^{2} t / 4}$ ; the extra $n$ produces the $- i$ in its transformation.

Facts & Assumptions

Given: Twisted Poisson summation (Twisted Poisson summation) and the functional equation (Functional equation for primitive Dirichlet L-functions).

Verification

technique · direct

1.1

The parity- $a$ Mellin kernel is $\sum_{n} χ (n) n^{a} e^{- π n^{2} t / q}$ , so the two displayed kernels are its $a = 0, 1$ cases.

given

2.1

Applying the given transformation yields phase $(- i)^{a}$ , agreeing with the stated functional equations.

step 1.1given∎

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

Kedlaya, Chapter 6 (standard reference, not scraped)