False statementConstruction: Literature-sourcedVerification: AI-adaptedjudge 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.

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False: normality implies algorithmic randomness

Statement

Every binary-normal sequence is Martin-Löf random.

Refutation

Given: the computable binary Champernowne sequence $B$ , which is binary normal.

1.1

For each $n$ , enumerate the single cylinder $[B ↾ n]$ as $U_{n}$ . Computability of $B$ makes $(U_{n})$ uniformly effectively open and its measure is $2^{- n}$ .

given

2.1

Thus $(U_{n})$ is a Martin-Löf test and $B \in ⋂_{n} U_{n}$ , so $B$ is not Martin-Löf random by Martin-Löf tests and random sequences despite normality.

step 1.1given∎

Depends on

Martin-Löf tests and random sequences

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

A zoo of computable binary normal sequences, The Peculiarity of BinChamp (standard reference, not scraped)