ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

The binary number 0.1010... is not normal

Example

The canonical binary number

$x = 0.101010 \dots_{2} = \frac{2}{3}$

is not normal in base two: the word $00$ has frequency $0$ , not $1 / 4$ .

Facts & Assumptions

Given: The periodic binary string $0.101010 \dots_{2}$ .

[F1]

The canonical convention excludes expansions eventually equal to one, and base-two normality requires each length-two word to have frequency $1 / 4$ (Canonical base-b expansions and normal numbers).

Verification

technique · direct periodic computation

1.1algebra

The digit-one positions are $1, 3, 5, \dots$ , so the represented value is $\sum_{j = 0}^{\infty} 2^{- (2 j + 1)} = \frac{1 / 2}{1 - 1 / 4} = \frac{2}{3} .$

2.1F1step 1.1

The string is not eventually one, so [F1] says it is the canonical binary expansion of $2 / 3$ .

3.1step 2.1algebra

Adjacent pairs alternate between $10$ and $01$ . Thus $00$ occurs zero times among every collection of starting positions, and its limiting frequency is $0$ .

4.1F1step 3.1∎

Since base-two normality would require frequency $2^{- 2} = 1 / 4$ by [F1], $2 / 3$ is not normal in base two.

Depends on

Canonical base-b expansions and normal numbers

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

Charles Walkden, Ergodic Theory lecture notes (standard reference, not scraped)