ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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.

Deciding satisfiability by a NumberSAT oracle query

Statement

For $φ (x, y) = (x \lor y) \land (\neg x \lor y)$ with declared variables $(x, y)$ , one NumberSAT query returns $2$ (binary $10$ ), so a nonzero test decides that $φ$ is satisfiable.

Facts & Assumptions

Given: the displayed formula and declared-variable list.

[L1]

NumberSAT is the exact number of satisfying declared assignments (NumberSAT).

[L2]

A $P^{# P}$ machine may query NumberSAT and receive its exact binary integer value (Polynomial time with a Sharp-P oracle).

Verification

technique · direct

1.1L1givenalgebra

The formula is equivalent to $y$ : the assignments $(0, 1)$ and $(1, 1)$ satisfy it, while $(0, 0)$ and $(1, 0)$ do not. Thus [L1] gives $N u m b e r S A T (φ) = 2$ , returned as $10_{2}$ .

2.1step 1.1

Testing the returned integer against zero accepts, which is correct because the displayed satisfying assignments exist. The single exact query summarizes all four assignments.

3.1L2step 1.1step 2.1∎

By [L2], this is a finite instance of deciding satisfiability in $P^{# P}$ with one NumberSAT query.

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