ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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 Hausdorff–Young exponent arithmetic

Example

For any sigma-finite-space complex-linear core operator with bounds $L^{1} \to L^{\infty}$ of constant A and $L^{2} \to L^{2}$ of constant B, the value $p = 4 / 3$ gives target exponent 4 and bound $\sqrt{A B}$ . The endpoint targets at p=1 and p=2 are respectively infinity and two.

Facts & Assumptions

[F1]

The abstract endpoint bound uses theta=2-2/p and the target conjugate exponent Interpolate L1 to Linfinity and L2 to L2 bounds.

Verification

Given: The objects and hypotheses in the statement.

1.1

At $p = 4 / 3$ , $1 / p = 3 / 4$ , $θ = 2 - 2 / p = 2 - 3 / 2 = 1 / 2$ , and $1 / p^{'} = 1 - 3 / 4 = 1 / 4$ , so $p^{'} = 4$ . The bound in F1 becomes $∥ T f ∥_{4} \leq A^{1 / 2} B^{1 / 2} ∥ f ∥_{4 / 3} = \sqrt{A B} ∥ f ∥_{4 / 3}$ . For instance A=B=1 gives coefficient one.

2.1

At p=1 the reciprocal target exponent is $1 - 1 = 0$ , giving infinity and the hypothesis $∥ T f ∥_{\infty} \leq A ∥ f ∥_{1}$ . At p=2 it is $1 - 1 / 2 = 1 / 2$ , giving two and $∥ T f ∥_{2} \leq B ∥ f ∥_{2}$ . These direct endpoint statements remain meaningful for A=0 or B=0, while the interior square-root coefficient is zero if either vanishes. No Fourier operator or its endpoint bounds are being presumed.

F1step 1.1∎

Depends on

Interpolate L1 to Linfinity and L2 to L2 bounds

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Laugesen Remark C.7(1), p.169; explicit specialization (standard reference, not scraped)