Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

Expected duration of simple gambler's ruin

Statement

Assume AC. A simple symmetric random walk started at i∈{1,…,N−1} and stopped on first hitting 0 or N has expected duration i(N−i). From the midpoint of an even interval, the mean duration is N2/4.

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

Expected duration of symmetric gambler's ruin gives the general duration formula.

[F2]

The Axiom of Choice states AC, assumed here because F1 requires it.

Proof

1.1

Apply F1 to obtain Eτ=i(N−i).

F1
2.1

If N is even and i=N/2, direct substitution gives Eτ=(N/2)(N−N/2)=N2/4. This calculation illustrates that almost-sure exit can have a quadratic mean duration. AC has exactly the inherited role in F2.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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