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.

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Sigma k equals pi k implies ph collapse

Statement

For fixed $k \geq 1$ , $Σ_{k}^{p} = Π_{k}^{p}$ implies $P H = Σ_{k}^{p} = Π_{k}^{p}$ . Also $P = N P$ implies $P H = P$ .

Facts & Assumptions

Given: the objects and hypotheses in the statement above.

[F1]

If $k \geq 1$ and $Σ_{k}^{p} = Π_{k}^{p}$ , then $Σ_{k + 1}^{p} = Π_{k + 1}^{p} = Σ_{k}^{p}$ . (Collapse at one level propagates).

Proof

1.1

Start at level $k$ and repeatedly apply the propagation lemma. After each application the next two polarities equal the same class, so the hypothesis needed for the following application holds. By finite induction all levels at least $k$ equal $Σ_{k}^{p}$ ; lower levels embed by unused quantifier blocks. Taking their union proves the first assertion.

2.1

If $P = N P$ , a one-block existential verifier defines a P language, and every P language has an unused existential witness. Deterministic P is complement closed by flipping a halting decider's bit, so $Σ_{1}^{p} = Π_{1}^{p} = P$ . Apply the first assertion at level one. Empty inputs and unused witnesses cause no exception.

step 1.1given∎

Depends on

Collapse at one level propagates

Used by

Ph complete language forces collapse Corollary

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; §5.2.1 Theorem5.6, pp92–93. (standard reference, not scraped)