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.

IP is contained in PSPACE

Statement

$I P \subseteq P S P A C E$ .

Facts & Assumptions

Given: $L \in I P$ and its constant-gap protocol.

[L1]

The value of the protocol's public-information game can be computed in polynomial space, by A polynomial-round protocol value is computable in polynomial space.

[L2]

Completeness gives value at least $2 / 3$ on members and soundness bounds the value by $1 / 3$ on nonmembers, by The class IP.

Proof

technique · direct

1.1

On input $x$ , compute the public-information protocol root value in polynomial space. The prover maxima are indexed by public transcripts, while verifier randomness is averaged over the compatible private histories, so this is the value of a legal prover strategy rather than a full-information strategy.

L1given

2.1

Accept precisely when that value exceeds $1 / 2$ . It is at least $2 / 3$ on $x \in L$ and at most $1 / 3$ on $x \notin L$ , so this test decides $L$ .

L2step 1.1given∎

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

Arora and Barak, §8.5 (standard reference, not scraped)