DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 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.

Multilinear extension of a Boolean-cube table

Definition

Let $F$ be a field, let $n \geq 0$ , and let $f : {0, 1}^{n} \to F$ be a table. An extension of $f$ is a formal polynomial $p \in F [X_{1}, \dots, X_{n}]$ satisfying $p (b) = f (b)$ for every Boolean vector $b$ . It is multilinear if every monomial has exponent at most one in each variable; the zero polynomial is included. Fields and formal polynomial rings are as in Field and Polynomial rings in finitely many commuting indeterminates by iteration.

For $n = 0$ , the cube contains the empty tuple, and a polynomial in no variables is a field constant. This definition specifies an extension relation; existence and uniqueness are separate assertions.

Depends on

Used by

Boolean-cube interpolation Theorem

Dependency tree · two levels

4 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

Justin Thaler, Proofs, Arguments, and Zero-Knowledge (2023), §3.5, Definition 3.4 and Fact 3.5, pp.28–29 (standard reference, not scraped)