PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Successive finite Gaussian shells compose

Physical scope

Pure finite-dimensional Gaussian probability identity; no physical model realization or continuum limit is asserted.

Statement

Assume the Axiom of Choice. Let $d \geq 1$ , let $C_{1}, C_{2}$ be real symmetric positive-semidefinite $d \times d$ covariance matrices, and let $f : R^{d} \to [0, \infty]$ be Borel. For a covariance matrix $C$ , let $γ_{C}$ denote the centered Gaussian probability with covariance $C$ , including when $C$ is singular, and define

T_{C} f (x) : = \int_{R^{d}} f (x + z) d γ_{C} (z) .

For a finite nonnegative Borel potential $V$ , $0 < T_{C} (e^{- V}) (x) \leq 1$ ; define the finite nonnegative potential $V_{C} (x) : = - \log T_{C} (e^{- V}) (x)$ . Then

T_{C_{2}} (T_{C_{1}} f) (x) = T_{C_{1} + C_{2}} f (x) (x \in R^{d})

as an equality in $[0, \infty]$ , and

e^{- (V_{C_{1}})_{C_{2}} (x)} = e^{- V_{C_{1} + C_{2}} (x)} .

The Gaussian measures are normalized probabilities, so no determinant or vacuum constant appears. This is a finite-regulator composition identity; it asserts no derivative equation along a covariance path and no continuum limit.

Statement references

Normalized finite Gaussian shell transform

Successive Gaussian shell integration composes

Facts & Assumptions

Given: The Axiom of Choice; finite dimension $d \geq 1$ ; real symmetric positive-semidefinite $C_{1}, C_{2}$ ; and the Borel functions in the Statement.

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function; the Gaussian-law suppliers state this assumption.

[F1]

Normalized finite Gaussian shell transform: for a positive-semidefinite covariance $C$ , $T_{C} f (x) = \int f (x + z) d γ_{C} (z)$ , and $e^{- V_{C} (x)} = T_{C} (e^{- V}) (x)$ for finite nonnegative Borel $V$ .

[F2]

Successive Gaussian shell integration composes: for every nonnegative Borel $f$ , $T_{C_{2}} (T_{C_{1}} f) (x) = T_{C_{1} + C_{2}} f (x)$ , including singular and zero covariances.

Proof

technique · direct substitution into the normalized Gaussian semigroup identity

Given: The hypotheses and conventions in the Statement.

1.1givenF1

Put $f_{V} = e^{- V}$ . Since $V$ is finite, nonnegative, and Borel, $f_{V}$ is a nonnegative Borel function. By [F1], $e^{- V_{C_{1}} (x)} = T_{C_{1}} f_{V} (x)$ and $e^{- V_{C_{1} + C_{2}} (x)} = T_{C_{1} + C_{2}} f_{V} (x)$ .

2.1A1F1F2step 1.1∎

Apply [F2] to $f_{V}$ to obtain $T_{C_{2}} (T_{C_{1}} f_{V}) (x) = T_{C_{1} + C_{2}} f_{V} (x)$ , with the value $+ \infty$ allowed for the first equality. Substituting the two identities from step 1.1 proves both displayed conclusions. Normalization is part of [F1], and no limit or differentiation is used.

Depends on

Dependency tree · 2 levels

Sources

Nima Moshayedi, Quantum Field Theory and Functional Integrals