ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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 rational number has exactly two finite regular continued-fraction expansions

Example

The rational number $8 / 5$ has the two finite regular continued-fraction expansions $\frac{8}{5} = [1; 1, 1, 2] = [1; 1, 1, 1, 1],$ and the normalized one is $[1; 1, 1, 2]$ because its last digit is at least $2$ .

Facts & Assumptions

Given: The rational number $8 / 5$ .

[F1]

Every rational number has a unique normalized finite regular continued fraction, and exactly one other finite expansion obtained by splitting the last digit $a_{n} \geq 2$ into $a_{n} - 1, 1$ (Normalized finite regular continued fractions are unique).

Verification

technique · direct

1.1

Direct calculation gives. [given, algebra] $[1; 1, 1, 2] = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{2}}} = 1 + \frac{1}{1 + \frac{2}{3}} = 1 + \frac{3}{5} = \frac{8}{5} .$

givenalgebra

2.1

Likewise. [F1, step 1.1, algebra] $[1; 1, 1, 1, 1] = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1}}}} = 1 + \frac{3}{5} = \frac{8}{5},$ so the same rational has two finite expansions. The last digit of $[1; 1, 1, 2]$ is $2$ , so [F1] identifies it as the normalized one and shows there are no further finite expansions.

F1step 1.1algebra∎

Depends on

Normalized finite regular continued fractions are unique

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Peter Hackman, Elementary Number Theory (standard reference, not scraped)
William Stein, Elementary Number Theory: Primes, Congruences, and Secrets (standard reference, not scraped)