Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26
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.

FALSE: the growth function is independent of the generating set pointwise

Statement

The growth function of a finitely generated group is independent of the generating set pointwise.

Facts & Assumptions

Given: The group Z with generating sets S={±1} and T={±1,±2}.

[L1]

The growth function counts elements inside a word-length ball (The growth function of a finitely generated group).

[L2]

Growth type is independent of the finite generating set, but only up to the comparison relation ≃ (Growth type is independent of the finite generating set).

Refutation

technique · direct
1.1

With respect to S, the radius-one ball is {−1,0,1}, so βZ,S(1)=3. With respect to T, the radius-one ball is {−2,−1,0,1,2}, so βZ,T(1)=5.

L1algebra
2.1

Thus the two growth functions are not equal pointwise, even though [L2] says they have the same growth type. This refutes the statement.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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