Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

A nondeterministic branch tree where state swapping is not complementing

Statement refuted

State swapping complements a nondeterministic machine's language.

Facts & Assumptions

Given: a fixed input x and a nondeterministic machine which, from its start configuration on x, has exactly two choices: one moves directly to an accepting state and the other directly to a rejecting state.

[L1]

A nondeterministic machine accepts when at least one branch accepts (A read-only-input logarithmic-space machine).

[L2]

The state-swapping inference is the false statement refuted on the A page (NL equals coNL follows by swapping accepting and rejecting states).

Counterexample

technique · direct branch analysis
1.1

The original accepting leaf makes the original machine accept x by [L1].

L1given
2.1

After labels swap, the original rejecting leaf is accepting, so [L1] makes the swapped machine accept x as well.

L1step 1.1
3.1

Since x is in the original language, it is not in its complement; step 2.1 therefore contradicts the proposed complementing behavior, as asserted in [L2].

L2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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