LemmaStatement: 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.

Collapse at one level propagates

Statement

If $k \geq 1$ and $Σ_{k}^{p} = Π_{k}^{p}$ , then $Σ_{k + 1}^{p} = Π_{k + 1}^{p} = Σ_{k}^{p}$ .

Facts & Assumptions

Given: the objects and hypotheses in the statement above.

[F1]

For each fixed $k \geq 0$ , both $Σ_{k}^{p}$ and $Π_{k}^{p}$ are closed under polynomial-time many-one preimages and polynomially many uniformly indexed conjunctions and disjunctions. Precisely, if the language of pairs $(x, i)$ lies in the class and $0 \leq i < q (∣ x ∣)$ for a polynomial-time computable polynomial bound $q$ , then both “every such $i$ ” and “some such $i$ ” define languages in that class. Complements interchange $Σ_{k}^{p}$ and $Π_{k}^{p}$ , and each is contained in both classes at level $k + 1$ . These assertions also hold with one fixed base oracle. (Ph quantifier block closure).

Proof

1.1

A language in $Σ_{k + 1}^{p}$ has a leading existential block followed by a $Π_{k}^{p}$ predicate on the pair consisting of the input and that block. By hypothesis this single language of pairs has a uniform $Σ_{k}^{p}$ definition. Substitute it and merge its first existential block with the leading one. Because $k \geq 1$ , this gives $k$ blocks with polynomial bounds. No input-dependent selection of machines is made.

givenalgebra

2.1

The reverse containment follows by dummy-block padding. Complement the containment just proved; complements interchange the two polarities, and the assumed level is complement closed. Consequently both classes at level $k + 1$ equal $Σ_{k}^{p}$ . Zero-length witness blocks remain nonempty domains.

F1step 1.1∎

Depends on

Ph quantifier block closure

Used by

Sigma k equals pi k implies ph collapse Theorem

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

Goldreich, Computational Complexity lecture notes; Proposition9.2.3 and appendix proof, pp108,113–114. (standard reference, not scraped)