False statementConstruction: 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.

Sharp-P is a language class

Statement

$# P$ is a class of languages, so every member has only a Boolean yes/no value on each input.

Facts & Assumptions

Given: formulas $φ (x, y) = x \lor y$ and $ψ (x, y) = (x \lor y) \land (\neg x \lor \neg y)$ , each with declared list $(x, y)$ .

[L1]

#P is defined as a class of numerical functions. by Sharp-P and Gap-P functions.

Refutation

technique · direct

1.1

By direct enumeration, $N u m b e r S A T (φ) = 3$ and $N u m b e r S A T (ψ) = 2$ . NumberSAT is a $# P$ function, while neither $2$ nor $3$ is a Boolean membership value.

L1givenalgebra

2.1

The associated decision language SAT retains only the predicate $N u m b e r S A T (θ) > 0$ ; it answers yes on both displayed formulas and forgets their distinct counts.

step 1.1

3.1

Thus, under [L1], $# P$ is a numerical function class rather than a language class.

L1step 1.1step 2.1∎

Depends on

Used by

NumberSAT is not a Boolean membership predicate Counterexample

Dependency tree · one level

2 results within one dependency step 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)