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.

Prime number theorem in a small progression

Example

Modulo four, $χ_{0} (1) = χ_{0} (3) = 1$ , $χ_{4} (1) = 1$ , $χ_{4} (3) = - 1$ , and both vanish on even integers. Hence $ψ (x; 4, 1) = \frac{1}{2} (Ψ_{χ_{0}} (x) + Ψ_{χ_{4}} (x)), ψ (x; 4, 3) = \frac{1}{2} (Ψ_{χ_{0}} (x) - Ψ_{χ_{4}} (x)) .$ Both corresponding prime counts are asymptotic to $Li (x) / 2$ .

Facts & Assumptions

Given: The data and hypotheses of the statement.

[F1]

Prime number theorem arithmetic progressions: For every fixed integer $q \geq 1$ and integer a with $\gcd (a, q) = 1$ , define $ψ (x; q, a)$ , $θ (x; q, a)$ and $π (x; q, a)$ by restricting their defining sums to integers, respectively primes, congruent to a modulo q. Then $ψ (x; q, a) \sim \frac{x}{φ (q)}, θ (x; q, a) \sim \frac{x}{φ (q)}, π (x; q, a) \sim \frac{Li (x)}{φ (q)} .$ No uniformity in a growing modulus is asserted.

[F2]

Orthogonality relations for Dirichlet characters modulo q: Let $G = (Z / q Z)^{\times}$ , and let the sum range over all Dirichlet characters modulo $q$ . 1. For unit classes $a, b \in G$ , $\sum_{χ m o d q} χ (a) \overline{χ (b)} = {\begin{cases} φ (q), & a = b, \\ 0, & a \neq b . \end{cases}$ 2. For Dirichlet characters $χ, ψ$ modulo $q$ , $\sum_{a \in G} χ (a) \overline{ψ (a)} = {\begin{cases} φ (q), & χ = ψ, \\ 0, & χ \neq ψ . \end{cases}$

Verification

1.1

The displayed character values give the two residue indicators as $(χ_{0} + χ_{4}) / 2$ and $(χ_{0} - χ_{4}) / 2$ , including the zero values on even numbers. Multiply by Lambda and sum to obtain both psi identities.

F2given

2.1

The principal sum is $Ψ_{χ_{0}} (x) = ψ (x) - \sum_{2^{k} \leq x, k \geq 1} \log 2 = ψ (x) - ⌊ \log x / \log 2 ⌋ \log 2$ for x at least one. Its correction is O(log x). Since $φ (4) = 2$ , the fixed-progression theorem gives each asserted prime-count asymptotic, including exclusion of the single prime two.

F1step 1.1∎

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

§4.4, Theorem 4.12 (standard reference, not scraped)