ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge 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.

The powers of two form a Hadamard-lacunary sequence

Example

The sequence $λ_{j} = 2^{j}$ for $j \geq 1$ is Hadamard-lacunary with ratio $q = 2$ .

Facts & Assumptions

Given: The sequence $λ_{j} = 2^{j}$ .

Verification

technique · direct ratio calculation

1.1

For every $j \geq 1$ , $λ_{j + 1} = 2^{j + 1} = 2 λ_{j}$ .

givenalgebra

2.1

Thus $λ_{j + 1} \geq 2 λ_{j}$ with one fixed ratio $2 > 1$ , exactly [step 1.1] as required by Hadamard-lacunary sequences and lacunary trigonometric series.

step 1.1∎

Depends on

Hadamard-lacunary sequences and lacunary trigonometric series

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

Gilles Pisier, Sidon Sets in Uniformly Bounded Orthonormal Systems, More Examples (standard reference, not scraped)