Alphabeta Math
PropositionStatement: Literature-sourcedProof: 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.

NP is contained in P with a Sharp-P oracle

Statement

NP⊆P#P.

Facts & Assumptions

Given: L∈NP with polynomial-time verifier R(x,y) and a polynomial witness-length bound p.

[L1]

A #P oracle returns the exact binary integer value of its counting function, by Polynomial time with a Sharp-P oracle.

[L2]

#P functions count accepting paths, by Sharp-P and Gap-P functions.

Proof

technique · direct
1.1

Define h(x)=∣{y∈{0,1}p(∣x∣):R(x,y)=1}∣. A nondeterministic machine guesses the p(∣x∣) bits and accepts exactly when R does, so h∈#P by [L2].

L2givenconstruct
2.1

A deterministic polynomial-time oracle machine makes the single query h(x) and accepts iff the returned binary integer is nonzero. The answer has at most p(∣x∣)+1 bits, and h(x)>0 exactly when an NP witness exists.

L1step 1.1
3.1

This oracle machine decides L, proving the containment.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

8 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