ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 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.

A Sigma_2 verification of a candidate SAT circuit

Statement

For the concrete $Π_{2}^{p}$ predicate $\forall y \in {0, 1}^{m} \exists z \in {0, 1}^{m} (z = y),$ the Karp--Lipton certificate is the search circuit $C (y) = y$ followed by the universal check that its output satisfies the formula encoding $z = y$ .

Facts & Assumptions

Given: the displayed predicate and the identity circuit $C$ .

[L1]

The Karp--Lipton theorem uses a candidate SAT search circuit with a local check. by The Karp--Lipton collapse.

Verification

technique · direct

1.1

For a fixed $y$ , let $F_{y} (z) = ⋀_{i = 1}^{m} (z_{i} \leftrightarrow y_{i})$ . The identity circuit has size $O (m)$ and outputs $C (y) = y$ , so $F_{y} (C (y)) = 1$ for every $y$ .

givenconstruct

2.1

Hence the predicate has the $Σ_{2}^{p}$ certificate form $\exists C \forall y F_{y} (C (y)) = 1$ . The deterministic check evaluates only the displayed assignment $C (y)$ in $F_{y}$ ; it does not try to verify a global claim that $C$ decides satisfiability.

L1step 1.1

3.1

This is the finite local-verification pattern used in the Karp--Lipton argument of [L1].

L1step 2.1∎

Depends on

The Karp--Lipton collapse

Used by

Nothing in the library uses this result yet.

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 and Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)
Lance Fortnow, Counting Complexity (standard reference, not scraped)