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

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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.

Gaussian Poisson summation and theta inversion

Statement

Assume countable choice. For real $t > 0$ , define $θ (t) = \sum_{k \in Z} e^{- π t k^{2}}$ . Then $θ (t) = t^{- 1 / 2} θ (1 / t) .$

Facts & Assumptions

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

[F1]

Polynomial Gaussians with positive parameter are Schwartz (Polynomial Gaussians are Schwartz).

[F2]

The normalized Gaussian transform is $F (e^{- π t x^{2}}) (ξ) = t^{- 1 / 2} e^{- π ξ^{2} / t}$ (Euclidean Gaussian transform with the 2π normalization).

[F3]

Poisson summation applies to Schwartz functions, with both lattice sums absolutely convergent (Poisson summation for Schwartz functions).

Verification

1.1

The series defining $θ (t)$ converges: for $k \geq 1$ , $e^{- π t k^{2}} \leq e^{- π t k}$ and $0 < e^{- π t} < 1$ , so its positive and negative tails are bounded by geometric series; the zero term is one. The same proof applies to $1 / t > 0$ . By [F1], $g_{t} (x) = e^{- π t x^{2}}$ is Schwartz, and [F2] gives its transform with factor $t^{- 1 / 2}$ .

F1F2givenalgebra

2.1

Apply [F3] at $x = 0$ to $g_{t}$ : $\sum_{k} g_{t} (k) = \sum_{k} {\hat{g}}_{t} (k) = t^{- 1 / 2} \sum_{k} e^{- π k^{2} / t}$ . By step 1.1 these are the two absolutely convergent theta series, proving the identity. At $t = 1$ both sides agree termwise. The parameter is only real and positive; this example makes no complex modular-form assertion.

step 1.1F3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 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

Noam Elkies, Theta functions and weighted theta functions of Euclidean lattices (standard reference, not scraped)