TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Exact spherical-shell coordinate marginal and fixed-block Gaussian limit

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $D \geq 2$ , $1 \leq k < D$ , and $R > 0$ . Let $Z_{1}, \dots, Z_{D}$ be independent real random variables with common Lebesgue probability density $ϕ (x) : = (2 π)^{- 1 / 2} e^{- x^{2} / 2}$ ; its integral is one by the Gaussian integral in Gaussian, gamma and beta product integrals, and this is the standard-normal coordinate law. Set $Z = (Z_{1}, \dots, Z_{D})$ and define $P : = R Z / ∣ Z ∣$ when $Z \neq 0$ , and $P : = 0$ when $Z = 0$ . The joint density of $Z$ makes $P (Z = 0) = 0$ . The distribution of $P$ is rotationally invariant, meaning that for every orthogonal map $Q \in O (D)$ and Borel set $A \subseteq R^{D}$ , $P (Q P \in A) = P (P \in A)$ . It is concentrated on the sphere in the precise sense that $P (∣ P ∣ = R) = 1$ . Its first $k$ coordinates have Lebesgue density

$f_{D, k} (y) = \frac{Γ (D / 2)}{Γ ((D - k) / 2) (π R^{2})^{k / 2}} {(1 - \frac{∣ y ∣^{2}}{R^{2}})}^{(D - k - 2) / 2} (∣ y ∣ < R),$

and zero for $∣ y ∣ \geq R$ . This density is integrable even when it diverges at the boundary. Also $E [P_{i}] = 0$ , $E [P_{i} P_{j}] = 0$ for $i \neq j$ , and $E [P_{i}^{2}] = R^{2} / D$ .

For fixed $k$ and $σ > 0$ , set $R_{D} = \sqrt{D} σ$ and let $D \to \infty$ through integers $D > k$ . If $g : R^{k} \to R$ is bounded and Lipschitz in the Euclidean norm, meaning that finite constants $B_{g}, L_{g}$ satisfy $∣ g (x) ∣ \leq B_{g}$ and $∣ g (x) - g (y) ∣ \leq L_{g} ∣ x - y ∣$ for all $x, y \in R^{k}$ , then

$E g ((P_{1}, \dots, P_{k})) ⟶ E g (σ (Z_{1}, \dots, Z_{k})) .$

This is a fixed-block bounded-Lipschitz test limit; it makes no claim uniform in $k$ or for unbounded test functions. If $k = D$ , the projection is singular on the sphere and the displayed Lebesgue density is not asserted.

Facts & Assumptions

Given: The Axiom of Countable Choice ${A C}_{ω}$ , integers $D, k$ with $D \geq 2$ and $1 \leq k < D$ , radius $R > 0$ , and the product probability law of $D$ independent coordinates with density $ϕ (x) = (2 π)^{- 1 / 2} e^{- x^{2} / 2}$ .

[F1]

Gaussian, gamma and beta product integrals: $Γ (a)$ is finite for $a > 0$ , $Γ (1 / 2) = \sqrt{π}$ , positive gamma densities with the same scale convolve by adding their shape parameters, and the Gaussian second and fourth moments are $1$ and $3$ for a standard normal coordinate.

[F2]

A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions: assuming ${A C}_{ω}$ , for a $C^{1}$ diffeomorphism $T : U \to V$ between open subsets of Euclidean space and nonnegative Lebesgue-measurable $f$ on $V$ , the Lebesgue integral transforms by $∣ \det D T ∣$ .

[F3]

On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}: assuming ${A C}_{ω}$ , product Lebesgue measure and Euclidean Lebesgue measure agree on Borel subsets of $R^{m + n}$ for positive integers $m, n$ .

[F4]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: for a nonnegative product-measurable function on a product of sigma-finite measure spaces, either order of integration agrees with the product integral.

[F5]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ says every sequence of nonempty sets has a choice function; it is the explicit set-theoretic hypothesis used by [F2].

Proof

1.1givenF2F3F5algebra

Under Countable Choice [F5], the Borel product-measure identity [F3] and independence give the joint Lebesgue density $ϕ_{D} (z) = (2 π)^{- D / 2} e^{- ∣ z ∣^{2} / 2}$ . A singleton has Lebesgue measure zero, so $Z \neq 0$ almost surely. For any orthogonal $Q$ and Borel $A \subseteq R^{D}$ , the map $Q^{- 1} : R^{D} \to R^{D}$ is a $C^{1}$ diffeomorphism with $∣ \det Q^{- 1} ∣ = 1$ ; applying [F2] to $1_{A} ϕ_{D}$ and using $∣ Q^{- 1} z ∣ = ∣ z ∣$ gives $P (Q Z \in A) = P (Z \in A)$ . Since $Q (R Z / ∣ Z ∣) = R Q Z / ∣ Q Z ∣$ off the null event, the law of $P$ is invariant under every orthogonal map, and $∣ P ∣ = R$ almost surely.

1.2F1F2algebra

Write $Z = (A, B)$ with $A \in R^{k}$ and $B \in R^{D - k}$ . The vectors are independent. On each open half-line $(0, \infty)$ and $(- \infty, 0)$ , the map $z \mapsto z^{2} / 2$ is a $C^{1}$ diffeomorphism onto $(0, \infty)$ ; applying [F2] to the Gaussian density times the indicator of any Borel target set on each branch shows that each $Z_{j}^{2} / 2$ has density $w^{- 1 / 2} e^{- w} / Γ (1 / 2)$ . Repeated gamma convolution from [F1] then shows $T : = ∣ B ∣^{2}$ has density $t^{b - 1} e^{- t / 2} / (2^{b} Γ (b))$ for $t > 0$ , where $b = (D - k) / 2$ , independently of $A$ .

1.3givenconstructalgebra

For fixed $t > 0$ , the first $k$ coordinates $y = R A / \sqrt{∣ A ∣^{2} + t}$ range over $∣ y ∣ < R$ . The inverse is $A = \sqrt{t} y / \sqrt{R^{2} - ∣ y ∣^{2}}$ . Its derivative has radial eigenvalue $\sqrt{t} R^{2} / (R^{2} - ∣ y ∣^{2})^{3 / 2}$ and $k - 1$ tangential eigenvalues $\sqrt{t} / (R^{2} - ∣ y ∣^{2})^{1 / 2}$ , hence Jacobian $t^{k / 2} R^{2} / (R^{2} - ∣ y ∣^{2})^{k / 2 + 1}$ .

2.1F1F2F3F4step 1.2step 1.3algebra

The map $Ψ (y, t) : = (A (y, t), t)$ from $U : = {(y, t) : ∣ y ∣ < R, t > 0}$ to $V : = R^{k} \times (0, \infty)$ is a $C^{1}$ diffeomorphism, with inverse $(a, t) \mapsto (R a / \sqrt{∣ a ∣^{2} + t}, t)$ . Apply [F2] to the nonnegative joint density of $(A, T)$ and this diffeomorphism. The product Lebesgue measure on $R^{k} \times (0, \infty)$ agrees with Euclidean Lebesgue measure on Borel sets by [F3], and Tonelli's theorem [F4] gives the marginal density by integration in $t$ . Substituting the Gaussian density of $A$ , the gamma density of $T$ , and the Jacobian from step 1.3 yields $f_{D, k} (y) = \frac{R^{2}}{(2 π)^{k / 2} 2^{b} Γ (b) (R^{2} - ∣ y ∣^{2})^{k / 2 + 1}} \int_{0}^{\infty} t^{D / 2 - 1} \exp ⁣ (- \frac{t R^{2}}{2 (R^{2} - ∣ y ∣^{2})}) d t$ . The substitution $u = t R^{2} / [2 (R^{2} - ∣ y ∣^{2})]$ evaluates the integral as $Γ (D / 2) [2 (R^{2} - ∣ y ∣^{2}) / R^{2}]^{D / 2}$ . Since $D / 2 = k / 2 + b$ , simplification gives the displayed density. Its boundary singularity, when present, has exponent $b - 1 > - 1$ and is integrable in the radial variable; the pushforward construction shows total mass one. Because $T = ∣ B ∣^{2} > 0$ almost surely, the marginal has no boundary atom at $∣ y ∣ = R$ .

2.2step 1.1givenalgebra

Sign changes of one Gaussian coordinate show $E [P_{i}] = 0$ and $E [P_{i} P_{j}] = 0$ for $i \neq j$ . Permutation invariance makes all $E [P_{i}^{2}]$ equal, while $\sum_{i = 1}^{D} P_{i}^{2} = R^{2}$ almost surely; summing expectations gives $E [P_{i}^{2}] = R^{2} / D$ .

3.1F1step 1.1algebra∎

For each finite $D$ , take the product Gaussian vector $Z = (Z_{1}, \dots, Z_{D})$ in that dimension and set $S_{D} : = D^{- 1} \sum_{i = 1}^{D} Z_{i}^{2}$ . By [F1] and independence, $E S_{D} = 1$ , $E [(S_{D} - 1)^{2}] = 2 / D$ , and $E ∣ Z_{1 : k} ∣^{2} = k$ . Fix $0 < δ < 1 / 2$ and $M > 0$ . On $E_{D} = {∣ Z_{1 : k} ∣ \leq M, ∣ S_{D} - 1 ∣ \leq δ}$ , $S_{D} \in [1 / 2, 3 / 2]$ and the identity $∣ u^{- 1 / 2} - 1 ∣ = ∣ u - 1 ∣ / [\sqrt{u} (1 + \sqrt{u})]$ gives $∣ S_{D}^{- 1 / 2} - 1 ∣ \leq \sqrt{2} δ$ ; hence $∣ σ Z_{1 : k} / \sqrt{S_{D}} - σ Z_{1 : k} ∣ \leq \sqrt{2} σ M δ$ . If $g$ is bounded Lipschitz with bound $B_{g}$ and Lipschitz constant $L_{g}$ , then $M^{2} 1_{{∣ Z_{1 : k} ∣ > M}} \leq ∣ Z_{1 : k} ∣^{2}$ and $δ^{2} 1_{{∣ S_{D} - 1 ∣ > δ}} \leq (S_{D} - 1)^{2}$ imply $P (E_{D}^{c}) \leq k / M^{2} + 2 / (D δ^{2})$ . Therefore $∣ E g (P_{1 : k}) - E g (σ Z_{1 : k}) ∣ \leq L_{g} \sqrt{2} σ M δ + 2 B_{g} (\frac{k}{M^{2}} + \frac{2}{D δ^{2}})$ . Choose $M$ large, then $δ$ small, then $D$ large. The right side becomes arbitrarily small, proving the stated test convergence without assuming independence between $Z_{1 : k}$ and $S_{D}$ .

Remarks

This is a specified rotational Gaussian-direction preparation. It is not a general coarea formula, a proof that every energy-shell measure has this marginal, or a dynamical equilibration result.

Exact spherical-shell coordinate marginal and fixed-block Gaussian limit

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources