ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

The three four one trigonometric inequality

Example

The weight in the three-four-one inequality is nonnegative term by term: $3 + 4 \cos u + \cos (2 u) = 2 (1 + \cos u)^{2} .$ It vanishes exactly when $u$ is an odd multiple of pi.

Facts & Assumptions

Given: The data and hypotheses of the statement.

[F1]

Zeta three four one logarithmic derivative inequality: For $σ > 1$ and $t \in R$ , $- 3 \frac{ζ^{'} (σ)}{ζ (σ)} - 4 Re \frac{ζ^{'} (σ + i t)}{ζ (σ + i t)} - Re \frac{ζ^{'} (σ + 2 i t)}{ζ (σ + 2 i t)} \geq 0.$

Verification

1.1

Using $\cos (2 u) = 2 \cos^{2} u - 1$ , the left side is $2 + 4 \cos u + 2 \cos^{2} u = 2 (1 + \cos u)^{2}$ . The square vanishes precisely for $\cos u = - 1$ .

algebra

2.1

In the logarithmic-derivative expansion the nth contribution is $2 Λ (n) n^{- σ} (1 + \cos (t \log n))^{2}$ . It is zero for non-prime-powers and nonnegative otherwise. At t=0 its weight is eight, and at t log n equal to an odd multiple of pi its weight is zero, exactly as required by the inequality.

F1step 1.1∎

Depends on

Zeta three four one logarithmic derivative inequality

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Theorem 6.6 proof (standard reference, not scraped)