ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-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.

Interpolating a two-bit table over F_5

Example

Over $F_{5}$ , let $f (0, 0) = 1$ , $f (0, 1) = 2$ , $f (1, 0) = 1$ , and $f (1, 1) = 4$ . Its unique multilinear extension is $\tilde{f} (X, Y) = 1 + Y + 2 X Y$ . In particular $\tilde{f} (2, 3) = 1$ in $F_{5}$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

Every field-valued Boolean table has the unique multilinear extension given by the product-basis interpolation sum (Boolean-cube interpolation).

Verification

1.1

The interpolation formula gives $(1 - X) (1 - Y) + 2 (1 - X) Y + X (1 - Y) + 4 X Y$ . Expanding and combining coefficients yields $1 + Y + 2 X Y$ , which has degree at most one in each variable.

F1algebra

2.1

Its values at $(0, 0), (0, 1), (1, 0), (1, 1)$ are respectively $1, 2, 1, 4$ , so it agrees with the table; uniqueness follows from the interpolation theorem. At $(2, 3)$ its integer expression is $1 + 3 + 2 \cdot 2 \cdot 3 = 16$ , which reduces to one modulo five.

F1step 1.1algebra∎

Depends on

Boolean-cube interpolation

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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), Figure 3.2 p.31 (standard reference, not scraped)