ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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.

Numerics for the third Mertens theorem

Example

For $x = 10, 30, 100$ the finite Euler product and its main term $e^{- γ} / \log x$ are

$\begin{array}{ccc} x & \prod_{p \leq x} (1 - 1 / p) & e^{- γ} / \log x \\ 10 & 0.228571 & 0.243839 \\ 30 & 0.157947 & 0.165077 \\ 100 & 0.120317 & 0.121919 \end{array}$

Facts & Assumptions

Given: The cutoffs $x = 10, 30, 100$ .

[L1]

The third Mertens theorem gives $\prod_{p \leq x} (1 - \frac{1}{p}) = \frac{e^{- γ}}{\log x} (1 + O (1 / \log x))$ (Mertens' third theorem for primes).

[L2]

The constant $γ$ is the Euler-Mascheroni constant (The Euler-Mascheroni constant).

Verification

technique · direct

1.1

Multiplying the Euler factors over the primes up to each cutoff and evaluating the comparison term $e^{- γ} / \log x$ gives the displayed table.

L1L2givenalgebra

2.1

The agreement improves across this short range, but the table is numerical evidence only. The exact constant $e^{- γ}$ on the A page comes from the Gamma-side analytic computation, not from the data itself.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Terence Tao, Mertens' theorems (standard reference, not scraped)