Alphabeta Math
False statementConstruction: 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.

NL equals coNL follows by swapping accepting and rejecting states

Statement

False statement. For a nondeterministic machine, swapping accepting and rejecting states recognizes the complement language; therefore it proves NL=coNL.

Facts & Assumptions

Given: nondeterministic acceptance means that at least one branch accepts.

[L1]

NL uses the existential accepting-branch convention for nondeterministic read-only-input machines (The classes L and NL).

[L2]

The actual equality NL=coNL follows from inductive counting, not from a terminal-label change (Immerman-Szelepcsényi theorem: NL equals coNL).

Refutation

technique · direct branch analysis
1.1

Suppose a machine has on one input both an accepting branch and a rejecting branch. By [L1] it accepts that input.

L1given
2.1

After terminal labels are swapped, the former rejecting branch is accepting, so the swapped machine also accepts the same input. The complement language instead rejects it.

L1step 1.1contradiction
3.1

Hence state swapping expresses “there exists a rejecting branch,” not “there is no accepting branch.” [L2] identifies the additional inductive-counting argument that genuinely proves the class equality.

L2step 2.1∎

Depends on

Used by

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