Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: AI-generatedPipeline-generated
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.

Query-address projectors for bit-oracle algorithms

Definition

Let an N-address bit-query algorithm act on H=CN⊗Hresponse⊗Hwork, where the first factor has computational basis {∣i⟩:i∈[N]}. The query-address projector is Πi=∣i⟩⟨i∣⊗Iresponse⊗Iwork. It projects onto every state whose query address is i, regardless of response or work state.

Facts & Assumptions

Given: An N-address algorithm with the declared address, response and work tensor factors.

[F1]

Finite Boolean query problem and error model for adversary bounds: the address set is [N]={1,…,N}.

[F2]

Finite registers and their ordered computational bases: the address register has orthonormal basis {|i⟩:i∈[N]} and all other registers are finite tensor factors.

[F3]

The computational basis is orthonormal: distinct address basis states are orthogonal.

Proof

1.1givenF2F3algebra

On H=C^N⊗H_response⊗H_work define Π_i=|i⟩⟨i|⊗I_response⊗I_work. Since ⟨i|j⟩=δ_{ij}, Π_iΠ_j=0 for i≠j and Π_i²=Π_i=Π_i*.

2.1givenF1F2step 1.1algebra∎

The ordered address basis resolves identity: Σ_i|i⟩⟨i|=I_{C^N}. Tensoring with the two identity factors gives Σ_iΠ_i=I_H. This includes every response and work state and proves the asserted resolution.

Depends on

Used by

Dependency tree · two levels

15 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