ExampleConstruction: Literature-sourcedVerification: 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.

Scaled and tensor Gaussian examples

Example

Assume countable choice. For $t > 0$ , $g_{t} (x) = e^{- π t ∣ x ∣^{2}}$ has transform $t^{- n / 2} e^{- π ∣ ξ ∣^{2} / t}$ on $R^{n}$ . The normalized density $t^{n / 2} g_{t}$ has mass one and transform $e^{- π ∣ ξ ∣^{2} / t}$ .

Facts & Assumptions

Given: $n \geq 1$ , $t > 0$ and The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

The normalized Gaussian transform formula holds in every positive dimension (Euclidean Gaussian transform with the 2π normalization).

[F2]

Linear scaling uses the absolute determinant (Translation, modulation, linear dilation and reflection laws).

Verification

1.1

Write $g_{t} = g_{1} \circ (\sqrt{t} I)$ . F2 and F1 give ${\hat{g}}_{t} (ξ) = t^{- n / 2} {\hat{g}}_{1} (ξ / \sqrt{t}) = t^{- n / 2} e^{- π ∣ ξ ∣^{2} / t}$ . Thus the spatial scale is $t^{- 1 / 2}$ , while the frequency scale is $t^{1 / 2}$ .

F1F2given

2.1

At zero frequency the formula gives $\int g_{t} = t^{- n / 2}$ . Multiplying both sides by $t^{n / 2}$ gives the mass-one formula. For t=1 the Gaussian is unchanged by Fourier transform; in n coordinates the product of n one-dimensional values gives the factor $t^{- n / 2}$ . Countable choice is inherited from F1 and F2.

F1F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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

Gerald Teschl, Topics in Real and Functional Analysis (2017) (standard reference, not scraped)