ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Two comparable and two incomparable carleson tiles

Example

Let $s = [0, 1) \times [0, 1)$ and $t = [0, 2) \times [0, 1 / 2)$ . Then $s \leq t$ . The tiles $u = s$ and $v = [1, 2) \times [0, 1)$ are incomparable.

Facts & Assumptions

Given: The four explicitly specified rectangles in the example.

[F1]

The finite interval and tile-order conventions are those of Carleson tiles wave packets and tile order. Only these combinatorial clauses are used; no Fourier or choice-dependent construction is used.

Proof

1.1F1algebra

All four displayed intervals are half-open dyadic intervals. The area products are $1 \cdot 1 = 1$ for s,u,v and $2 \cdot (1 / 2) = 1$ for t. Moreover $[0, 1) \subset [0, 2)$ and $[0, 1 / 2) \subset [0, 1)$ , so $s \leq t$ by the two defining inclusions. The reverse order fails because $1 \in [0, 2)$ but $1 \notin [0, 1)$ .

1.2F1algebra

For u,v the frequency intervals agree, but neither spatial interval contains the other: $0 \in I_{u} ∖ I_{v}$ and $1 \in I_{v} ∖ I_{u}$ . Thus both possible order relations fail, while every tile is comparable to itself.

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 verify the claimed comparable pair and the two failures for the incomparable pair, with midpoint and endpoint membership fixed by the half-open convention.

Depends on

Carleson tiles wave packets and tile order

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Lacey, Carleson’s Theorem: Proof, Complements, Variations (standard reference, not scraped)