TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge 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.

An oracle collapses p and np

Statement

For $A = T Q B F$ , $P^{A} = N P^{A} = P S P A C E$ .

Facts & Assumptions

Given: the objects and hypotheses in the statement above.

[F1]

$N P^{T Q B F} \subseteq P S P A C E \subseteq P^{T Q B F}$ . (Polynomial space oracle absorption).

Proof

1.1

Oracle absorption supplies $P S P A C E \subseteq P^{A}$ and $N P^{A} \subseteq P S P A C E$ . A deterministic oracle computation is a nondeterministic one with no choices, so $P^{A} \subseteq N P^{A}$ .

2.1

The inclusions form a closed chain, giving equality of all three classes. The same fixed total TQBF language is used throughout; it is not required to belong to P.

step 1.1algebra∎

Depends on

Polynomial space oracle absorption

Used by

Relativization alone cannot resolve p versus np Corollary
Relativized separations prove unrelativized separations False statement

Dependency tree · two levels

2 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–Barak, Computational Complexity, 2007 draft; §3.5 Theorem3.9 and Claim3.8(3), pp70–72; explicit TQBF adaptation. (standard reference, not scraped)