Alphabeta Math
False statementConstruction: Literature-sourcedVerification: 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.

Not every real number is normal

Statement

False claim: every real number is normal in every base.

Facts & Assumptions

Given: The binary number x=0.101010…2.

[F1]

Base-two normality requires every word of length 2, including 00, to have limiting frequency 2−2=1/4 (Canonical base-b expansions and normal numbers).

Refutation

technique · constructive periodic witness
1.1F1givenconstruct

The displayed digit string is not eventually one and is therefore the canonical binary expansion under [F1]'s convention.

2.1step 1.1algebra

Its adjacent pairs alternate between 10 and 01. The word 00 never occurs, so its frequency is 0.

3.1F1step 2.1discharge-construct∎

Since 0≠1/4, [F1] shows that x is not normal in base two and hence is not normal in every base. This single real number refutes the universal claim.

Depends on

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