Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12
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.

Repeating a randomized test with the same coin string gives no amplification

Statement refuted

Reusing the same random bits amplifies success.

Facts & Assumptions

Given: a randomized test whose output depends only on its first random bit.

[L1]

The false statement being refuted is exactly the claim above (FALSE: reusing the same random bits amplifies success).

Counterexample

technique · direct
1.1

Consider the randomized test that accepts exactly when its first random bit is 1. Its success probability is 1/2, so it is a legitimate witness against the statement in [L1] if repetition with the same randomness leaves that probability unchanged.

L1given
2.1

If the same random string is reused in two, ten, or one hundred repetitions, every repetition produces the same answer as the first one. So the majority outcome is still accept with probability 1/2 and reject with probability 1/2. No amplification occurs at all, and the statement is refuted.

step 1.1algebra∎

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