ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

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The continued fraction [1; overline 1] for the golden ratio

Example

If $x = [1; \overline{1}],$ then $x = 1 + \frac{1}{x},$ so $x$ is the positive root of $x^{2} - x - 1 = 0$ , namely the golden ratio $φ = \frac{1 + \sqrt{5}}{2} .$

Facts & Assumptions

Given: The purely periodic continued fraction $x = [1; \overline{1}]$ .

[F1]

Every eventually periodic regular continued fraction has quadratic- irrational value (Eventually periodic regular continued fractions are quadratic irrationals).

[F2]

The value of an infinite regular continued fraction is the common limit of its convergents; for $[1; 1, 1, \dots]$ the increasing even subsequence starts at $1$ (Every infinite regular continued fraction converges to a unique real number).

[F3]

Finite regular continued fractions are evaluated by the recursion $[a_{0}; a_{1}, \dots, a_{n}] = a_{0} + 1 / [a_{1}; \dots, a_{n}]$ (Finite and infinite regular continued fractions).

Verification

technique · direct

1.1

Let $x_{m} = [1; \underset{m further digits}{\underset{⏟}{1, \dots, 1}}]$ . By [F2], $x_{m} \to x$ and $x \geq 1$ , while [F3] gives $x_{m + 1} = 1 + 1 / x_{m}$ and $x_{m} \geq 1$ . Hence $∣ \frac{1}{x_{m}} - \frac{1}{x} ∣ = \frac{∣ x_{m} - x ∣}{x_{m} x} \leq ∣ x_{m} - x ∣ ⟶ 0,$ so taking limits in the recursion gives $x = 1 + \frac{1}{x} .$ Multiplication by $x$ now gives $x^{2} - x - 1 = 0,$ whose positive solution is $x = \frac{1 + \sqrt{5}}{2} .$

F2F3givenalgebra

2.1

The continued fraction is purely periodic, so [F1] says its value is a quadratic irrational. Step 1.1 exhibits the quadratic equation explicitly, and its positive root is the golden ratio $φ$ .

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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)
Bruce Ikenaga, Periodic Continued Fractions (standard reference, not scraped)