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.

Balancing density and size levels in the carleson sum

Example

The nonnegative two-sided series satisfies $\sum_{n \in Z} \min (2^{n}, 2^{- n}) = 3$ .

Facts & Assumptions

Given: The displayed explicitly indexed nonnegative series, interpreted as the supremum of its finite subsums.

Proof

1.1givenalgebra

For $n \geq 0$ the minimum is $2^{- n}$ , whereas for $n \leq - 1$ it is $2^{n}$ . Finite geometric cancellation therefore gives, for integers $M \geq 1$ and $N \geq 0$ , $\sum_{n = - M}^{N} \min (2^{n}, 2^{- n}) = (1 - 2^{- M}) + (2 - 2^{- N}) = 3 - 2^{- M} - 2^{- N}$ . In particular the index zero is counted once and contributes one.

2.1step 1.1algebra∎

Every finite set of integer indices lies in an interval $[- M, N]$ of the above form, and all summands are nonnegative. Thus the supremum of finite subsums is bounded above by three by step 1.1. Taking $M = N \to \infty$ in that same formula gives the lower bound three, since $2^{- M} \to 0$ . The series is therefore three.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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