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
1.1

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

F1givenconstruct
2.1

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

step 1.1algebra
3.1

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.

F1step 2.1discharge-construct∎

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