Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07
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.

Private-coin IP equals public-coin IP

Statement

Every polynomial-round private-coin interactive proof has a polynomial-round public-coin interactive proof for the same language. Thus private-coin IP equals public-coin IP.

Proof

Given: a private-coin polynomial-round verifier with polynomial communication and constant gap.

1.1

The Goldwasser--Sipser conversion replaces the hidden continuation at each verifier move by a public set-lower-bound subprotocol: Arthur announces a pairwise-independent hash and Merlin supplies a compatible accepting continuation in the announced hash fibre. The fibre estimate of Pairwise-independent hashing controls fibre size is the variance calculation used by that subprotocol.

given
2.1

Arora--Barak's stated public-coin conversion establishes IP[k]⊆AM[k+2] for every polynomial-time computable round bound k. Its construction has polynomial-length hashes, challenges, and witnesses, hence preserves polynomial verification and produces a public-coin protocol for the same language.

step 1.1given
3.1

A public-coin verifier is already a private-coin verifier which reveals its coins. Therefore both inclusions hold.

step 2.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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