PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge 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.

Approximation constraints apply only on the promise domain

Statement

For $f : D \to {0, 1}$ , the condition that a polynomial $p$ is an $ϵ$ -approximant means exactly $∣ p (x) - f (x) ∣ \leq ϵ$ for $x \in D$ ; it imposes no condition at inputs outside $D$ . If $D ⊊ {0, 1}^{N}$ , then for every $y \notin D$ and every polynomial $p$ , there is another polynomial agreeing with $p$ on $D$ and having a different value at $y$ .

Facts & Assumptions

Given: A finite promise domain $D \subseteq {0, 1}^{N}$ , a function $f : D \to {0, 1}$ , a real polynomial $p$ , and a point $y \in {0, 1}^{N} ∖ D$ when $D$ is proper.

[F1]

Acceptance polynomials and approximate degree on Boolean domains: approximation to a promise function is defined by inequalities only at points of $D$ .

[F2]

Polynomial rings in finitely many commuting indeterminates by iteration and Monomials, coefficients, degree in each variable and total degree in $F [x_{1}, \dots, x_{n}]$ : finite products and sums of coordinate polynomials are real polynomials.

Proof

1.1givenF1

The first assertion is the defining predicate in [F1], whose universal quantifier ranges over $x \in D$ only.

1.2givenF2constructalgebra

For $y = (y_{0}, \dots, y_{N - 1})$ , define $δ_{y} (X) = \prod_{i < N} (y_{i} X_{i} + (1 - y_{i}) (1 - X_{i}))$ , an empty product equal to $1$ when $N = 0$ . This is a polynomial by [F2]. At a Boolean point $x$ , its $i$ th factor equals $1$ if $x_{i} = y_{i}$ and $0$ otherwise, so $δ_{y} (x) = 1$ exactly when $x = y$ and equals $0$ at every other cube point.

2.1step 1.2algebra∎

Since $y \notin D$ , step 1.2 gives $δ_{y} (x) = 0$ on all $x \in D$ and $δ_{y} (y) = 1$ . Therefore $p + δ_{y}$ agrees with $p$ on $D$ but differs from $p$ at $y$ . This proves the off-promise freedom claim.

Depends on

Used by

Nothing in the library uses this result yet.

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

Andrew M. Childs, Lecture Notes on Quantum Algorithms (2025), Chapter 21 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes (arXiv:1907.09415), §11.2 (standard reference, not scraped)