PropositionStatement: 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.

Signed weights extend the nonnegative adversary certificate

Statement

For every finite Boolean query problem, ${Adv}^{\pm} (f) \geq Adv (f)$ under the same mismatch masks and normalization.

Facts & Assumptions

Given: The two adversary optimizations under the common Boolean mismatch convention.

[F1]

General adversary matrix and spectral bound: Adv± optimizes over all real symmetric admissible matrices, including those with nonnegative entries.

[F2]

Nonnegative-weight adversary bound: Adv uses the same ratio and masks but restricts all entries to be nonnegative.

Proof

1.1givenF1F2

Every nonzero nonnegative admissible matrix for Adv is also an admissible matrix for Adv±, with the identical numerator and denominator because both use the same masks.

2.1givenstep 1.1∎

Taking the maximum over the larger signed feasible set cannot decrease the value, so Adv±(f)≥Adv(f). This proves only monotonicity; no strict separation is claimed here.

Depends on

Used by

A nonnegative-only adversary estimate need not capture the general bound Counterexample

Dependency tree · two levels

7 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

Andrew M. Childs, Lecture Notes on Quantum Algorithms (17 April 2025) (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (standard reference, not scraped)