DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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.

Normalized finite Gaussian shell transform

Physical scope

Pure mathematics; no physical premises or physical claim.

Definition

Assume AC and let $d \geq 1$ . For a real symmetric positive-semidefinite covariance matrix $C$ , write $γ_{C} = N_{d} (0, C)$ , allowing singular $C$ . For a nonnegative Borel function $f : R^{d} \to [0, \infty]$ , define the normalized shell transform $T_{C} f (x) : = \int_{R^{d}} f (x + z) d γ_{C} (z) \in [0, \infty] .$ This is a Borel function of $x$ . If $V : R^{d} \to [0, \infty)$ is finite-valued and Borel, define $V_{C} (x) : = - \log (T_{C} (e^{- V}) (x)) .$ Then $V_{C}$ is a finite nonnegative Borel function, and $0 < T_{C} (e^{- V}) (x) \leq 1$ for every $x$ . The covariance convention is $E [Z Z^{T}] = C$ . The coordinates $x, z$ use the same finite-regulator field-coordinate convention, so $C$ has the square units of those coordinates. The scalar $V (x)$ is dimensionless in $e^{- V (x)}$ ; if a physical interpretation identifies $V$ with an action, it is normalized as $S / ℏ$ . The measures $γ_{C}$ are normalized probabilities, with no determinant/vacuum factor in $T_{C}$ . The transform is defined at finite regulator only.

Facts & Assumptions

Given: AC, finite $d \geq 1$ , symmetric positive-semidefinite $C$ , and the functions in the Definition.

[A1]

The Axiom of Choice: the finite-dimensional Gaussian-law definition assumes AC.

[F1]

Multivariate normal law, including singular covariance: for positive-semidefinite $C$ , the Borel probability $γ_{C} = N_{d} (0, C)$ exists, including singular $C$ .

[F2]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: for product-measurable nonnegative functions, parameter integrals are measurable and iterated integrals agree.

[F3]

The natural logarithm as the inverse of the exponential function: $\log$ is defined on $(0, \infty)$ as the inverse of $\exp$ .

[F4]

The exponential function is strictly increasing: $\exp$ is continuous and strictly increasing on $R$ .

Proof

technique · direct measurability and positivity check

Given: The objects and hypotheses in the Definition.

1.1A1F1F2given

Write $B_{d}$ for the Borel sigma-algebra of $R^{d}$ and $λ_{B}$ for Lebesgue measure restricted to $B_{d}$ . The countable basis of Euclidean open rectangles gives $B (R^{d} \times R^{d}) = B_{d} \otimes B_{d}$ . Hence $(x, z) \mapsto f (x + z)$ is product-measurable because addition is continuous and $f$ is Borel. The measure space $(R^{d}, B_{d}, λ_{B})$ is sigma-finite because $R^{d} = ⋃_{n \geq 1} [- n, n]^{d}$ and each cube has finite measure; $(R^{d}, B_{d}, γ_{C})$ is a probability space by [F1]. Applying [F2] to these two spaces makes $x \mapsto T_{C} f (x)$ measurable for $B_{d}$ , hence Borel, including when the integral is infinite.

2.1F1F2step 1.1algebra

For finite nonnegative Borel $V$ , $g_{x} (z) : = e^{- V (x + z)}$ is Borel and satisfies $0 < g_{x} (z) \leq 1$ . Its integral against the probability $γ_{C}$ is at most one. It is strictly positive: if the integral were zero, each set ${z : g_{x} (z) \geq 1 / n}$ would have measure zero, while their countable union is all of $R^{d}$ , contradicting $γ_{C} (R^{d}) = 1$ . Thus $T_{C} (e^{- V}) (x) \in (0, 1]$ .

3.1F1F3F4step 2.1algebra∎

By [F3], the logarithm is defined at these values. Its continuity follows directly from [F4]: for $y_{0} > 0$ and $ϵ > 0$ , strict increase and continuity of $\exp$ give $\exp (\log y_{0} - ϵ) < y_{0} < \exp (\log y_{0} + ϵ)$ ; put $δ = \min {y_{0} - \exp (\log y_{0} - ϵ), \exp (\log y_{0} + ϵ) - y_{0}} > 0$ . If $y > 0$ and $∣ y - y_{0} ∣ < δ$ , then $\exp (\log y_{0} - ϵ) < y < \exp (\log y_{0} + ϵ)$ , and strict monotonicity gives $∣ \log y - \log y_{0} ∣ < ϵ$ . Hence $V_{C}$ is finite, nonnegative, and Borel. If $C = 0$ , then for each coordinate $Z_{i}$ , $E Z_{i}^{2} = C_{i i} = 0$ . For every $n \geq 1$ , $0 = E Z_{i}^{2} \geq n^{- 2} P (∣ Z_{i} ∣ \geq n^{- 1})$ , so $Z_{i} = 0$ almost surely; the finite intersection gives $Z = 0$ almost surely and $γ_{0} = δ_{0}$ ; hence $T_{0} f = f$ and $V_{0} = V$ .

Normalized finite Gaussian shell transform

Definition

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources