ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

A geometric series that is p-adically convergent and really divergent

Example

For any prime $p$ , the series

$1 + p + p^{2} + \dots$

diverges in the usual absolute value and converges in $Q_{p}$ to $1 / (1 - p)$ .

Facts & Assumptions

Given: A prime $p$ .

[L1]

The $p$ -adic absolute value satisfies $∣ p ∣_{p} = p^{- 1} < 1$ (The p-adic absolute value on the rationals).

[L2]

$Q_{p}$ is a field (The p-adic completion is a complete valued field).

Verification

technique · direct

1.1

The $N$ th partial sum is $s_{N} = (1 - p^{N + 1}) / (1 - p)$ . By [L1], $∣ p^{N + 1} ∣_{p} = p^{- (N + 1)} \to 0$ , so $s_{N} \to 1 / (1 - p)$ in $Q_{p}$ .

L1L2givenalgebra

2.1

In the usual absolute value, $∣ p^{N} ∣_{\infty} = p^{N}$ does not tend to $0$ , so the same series cannot converge there.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

J. S. Milne, Algebraic Number Theory, Chapter 7 (standard reference, not scraped)