Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)
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.

Dual reflection evaluator query complexity

Statement

For a supplied balanced dual certificate of cost A≥1, the evaluator uses exactly 2(M−1)queries,M=2m=least power of two with M≥512A. Thus its query cost is less than 2048A. The phase-estimation and input-independent witness-span reflections carry no asserted gate bound.

Facts & Assumptions

Given: The bounded-error evaluator and its phase-estimation schedule.

[F1]

Bounded-error evaluation from a dual adversary certificate: the schedule uses controlled powers with exponents 1,2,…,2^{m−1} and each controlled consistency reflection uses two bit queries.

[F2]

Dual reflection evaluator has bounded-error correctness: the phase-estimation procedure has the required bounded-error correctness.

[F3]

Phase-estimation query and gate costs: the repeated-power schedule contains M−1 controlled-U calls.

[F4]

Controlled consistency reflection uses two bit queries: one controlled consistency reflection uses exactly two XOR queries.

[F5]

Quantum query complexity: query count excludes nonoracle gates.

Proof

1.1givenF1F3algebra

Let M=2^m be the schedule length. The QPE powers use 2^j controlled U_x calls for j=0,…,m−1, whose total is Σ_j2^j=M−1 by [F3].

2.1givenF1F4step 1.1

Each controlled U_x has one controlled input-consistency reflection and one input-independent controlled Δ reflection. The first costs exactly two queries by [F4], while the second costs zero oracle calls. Therefore the total is exactly 2(M−1) queries.

3.1givenF2F3F5step 2.1algebra∎

Minimality of M implies M<1024A unless M=512A exactly, and hence 2(M−1)<2048A. Correctness is [F2]; preparations, equality tests, phase-estimation gates, and the Δ reflection are nonoracle operations and are not included by [F5]. This proves the O(A) query bound without a gate bound.

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Dependency tree · two levels

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Sources