Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-27
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.

IP admits perfect completeness

Statement

For every language L∈IP there is an interactive proof for L with perfect completeness and soundness error at most 1/3: an honest prover is accepted with probability one on every input of L, every prover is accepted with probability at most 1/3 on every input outside L, and the verifier is a probabilistic polynomial-time machine with polynomially many rounds and polynomially bounded communication. The protocol need not be the given protocol for L.

Facts & Assumptions

Given: A language L∈IP.

[A1]

Every language in IP lies in PSPACE (IP is contained in PSPACE, IP equals PSPACE).

[A2]

Every language in PSPACE has an interactive proof with a probabilistic polynomial-time verifier, polynomially bounded interaction, completeness one and soundness error at most 1/3; the proof is obtained by reducing the input to TQBF and running the arithmetized sum-check-style protocol there (PSPACE is contained in IP).

Proof technique: direct.

Proof

1.1

By [A1] the language L lies in PSPACE.

A1given
2.1

By [A2] the language L, being in PSPACE, has an interactive proof with perfect completeness and soundness error at most 1/3, with a probabilistic polynomial-time verifier and polynomially bounded interaction.

step 1.1A2given
3.1

That protocol is an interactive proof for L with perfect completeness and constant soundness error, as required; it is constructed from the deterministic polynomial-time reduction of L to TQBF and therefore may differ from any particular protocol for L that was used to witness L∈IP.

step 2.1A2given∎

Depends on

Used by

Dependency tree · two levels

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Sources