Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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: a finite-index d subgroup of a rank n free group has rank dn

Statement

If H has finite index d in a rank-n free group, then rank⁡(H)=dn.

Facts & Assumptions

Given: The false claim above.

[L1]

The correct formula is rank⁡(H)=1+d(n−1) (The Schreier index-rank formula).

Refutation

technique · direct
1.1

Consider the index-two subgroup H≤F(a,b) consisting of words with even exponent sum in a. Here the ambient free group has rank n=2 and d=[F(a,b):H]=2.

givenconstruct
2.1

By [L1], rank⁡(H)=1+2(2−1)=3, whereas the false formula predicts dn=4.

L1step 1.1algebra
3.1

Since 3≠4, this subgroup refutes the statement.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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