ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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.

$B (1 / 2, 1 / 2) = π$

Example

$B (1 / 2, 1 / 2) = π .$

Facts & Assumptions

Given: The Beta-Gamma identity and the value $Γ (1 / 2) = \sqrt{π}$ .

[L1]

$B (p, q) = Γ (p) Γ (q) / Γ (p + q)$ on the right half-planes (The Beta-Gamma identity).

[L2]

$Γ (1 / 2) = \sqrt{π}$ (The value of Gamma at one half).

[L3]

$Γ (1) = 1$ (Gamma at the positive integers).

Verification

technique · direct

1.1

Substitute $p = q = 1 / 2$ into [L1] to obtain $B (1 / 2, 1 / 2) = Γ (1 / 2)^{2} / Γ (1)$ .

L1given

2.1

By [L2], $Γ (1 / 2)^{2} = π$ , and [L3] gives $Γ (1) = 1$ . Therefore step 1.1 simplifies to $B (1 / 2, 1 / 2) = π$ .

step 1.1L2L3algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 6 §1(vi)-(vii) (standard reference, not scraped)

B(1/2,1/2)=πB(1/2,1/2)=π