ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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

1.1

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 $1 \notin [0, 1)$ .

F1algebra

1.2

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.

F1algebra

2.1

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.

step 1.1step 1.2∎

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)