Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26‡ rests on unproved material
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.

‡ Rests on 1 statement not proved in this library. Every dependency marked ‡ below is recorded with a citation but is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

FALSE: every subexponential growth group has polynomial growth

Statement

Every subexponential growth group has polynomial growth.

Facts & Assumptions

Given: The existence result of Grigorchuk groups of intermediate growth ‡.

[A1]

There exist finitely generated groups of intermediate growth.

Refutation

technique · direct
1.1

By [A1], some finitely generated group has intermediate growth, meaning subexponential growth but not polynomial growth.

A1
2.1

Such a group satisfies the hypothesis of the statement and fails its conclusion, so the statement is false.

step 1.1∎

Depends on

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