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.

Specified chiral collapse channel

Definition

Fix an upper endpoint $U_{H} \in R$ and a scale $A > 0$ , both measured in seconds, and let $k > 0$ be measured in seconds $^{- 1}$ . Let $u \in R$ be the outgoing coordinate, measured in seconds, and define $U (u) = U_{H} - A e^{- k u} .$ Then $U$ is a smooth increasing diffeomorphism from $R$ onto $(- \infty, U_{H})$ , with derivative $U^{'} (u) = A k e^{- k u} > 0$ ; $U_{H}$ is only the upper-endpoint label in this channel. Let $χ > 0$ be a fixed conversion factor with units of length per second, and put $ℓ (u) = χ U (u)$ and $ℓ_{H} = χ U_{H}$ . For $g \in C_{c}^{\infty} (R; C)$ , let $u_{ℓ}$ be the smooth inverse of $ℓ$ and define $f_{g} (x) = g (u_{ℓ} (x))$ for $x < ℓ_{H}$ , and $f_{g} (x) = 0$ for $x \geq ℓ_{H}$ . The support of $g$ is compact, so $f_{g} \in C_{c}^{\infty} (R; C)$ . Define the channel current $j (g) = J (f_{g})$ using the Fock-vacuum chiral current $J$ , and set $W (g, h) = ⟨ Ω, J (\overline{f_{g}}) J (f_{h}) Ω ⟩_{2} .$ The associated two-point distribution is the diffeomorphism pullback of the supplied length-coordinate kernel $C_{ℓ} (X - Y) = - \frac{1}{4 π} (X - Y - i 0)^{- 2}$ ; in seconds coordinates its kernel is $K (u, v) = U^{'} (u) U^{'} (v) [- \frac{1}{4 π} (U (u) - U (v) - i 0)^{- 2}] .$ Here $i 0$ is the lower boundary obtained by testing the regularizations $- \frac{1}{4 π} (s - i ϵ)^{- 2}$ as $ϵ ↓ 0$ , and the pullback is defined by the smooth diffeomorphism, not by pointwise evaluation on the diagonal. The tested form is of positive type: $W (g, g) \geq 0$ for every $g$ .

For real-valued tests $g_{1}, \dots, g_{n}$ , write $M_{n} (g_{1}, \dots, g_{n}) = ⟨ Ω, J (f_{g_{1}}) \dots J (f_{g_{n}}) Ω ⟩_{2} .$ This Fock-vacuum field is Gaussian in the precise sense that $M_{n} = 0$ for odd $n$ , while for even $n$ it is the sum over pair partitions $π$ of ${1, \dots, n}$ of $\prod_{(i, j) \in π, i < j} M_{2} (g_{i}, g_{j})$ , with each contraction ordered as displayed. This is an exact one-dimensional test-coordinate channel; it does not assert existence of four-dimensional collapse scattering.

Definition references

Positive chiral current and exponential test-coordinate model

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Schwartz/distribution spaces and Hilbert Fock conventions

Bosonic and fermionic Fock operators with exact domains

A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions

Facts & Assumptions

Given: $U_{H} \in R$ , $A, k, χ > 0$ , the seconds coordinate $u$ , compact smooth tests, and Countable Choice.

[A1]

Assume the Axiom of Countable Choice, and no stronger choice principle. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

Positive chiral current and exponential test-coordinate model defines the positive-frequency Fock current on compact smooth tests, gives its vacuum covariance $C_{ℓ} (s) = - \frac{1}{4 π} (s - i 0)^{- 2}$ as a positive-type distribution, and constructs the canonical coordinate $U_{κ} (τ) = - κ^{- 1} e^{- κ τ}$ from $R$ onto $(- \infty, 0)$ , with smooth inverse $U_{κ}^{- 1} (X) = - κ^{- 1} \log (- κ X)$ and derivative $U_{κ}^{'} (τ) = e^{- κ τ} > 0$ . Its proof establishes this diffeomorphism and its lower-boundary pullback in step 4.1.

[F2]

Schwartz/distribution spaces and Hilbert Fock conventions fixes the bosonic Fock vacuum, the finite-particle domain, and the creation and annihilation formulas; its construction assumes Countable Choice.

[F3]

Bosonic and fermionic Fock operators with exact domains gives the bosonic commutation relations on the finite-particle domain, including $[a (h), a^{†} (q)] = ⟨ h, q ⟩_{2} I$ .

[F4]

A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions gives the Lebesgue change-of-variables formula for a $C^{1}$ diffeomorphism between open sets.

Proof

1.1givenA1F1F2constructalgebra

Put $τ = χ u$ , $κ = k / χ$ , $x_{H} = χ U_{H}$ , and $λ = A k > 0$ . In the length coordinate $τ$ , the map $x (τ) : = ℓ (τ / χ)$ satisfies $x (τ) = x_{H} + λ U_{κ} (τ)$ , since $λ U_{κ} (τ) = - χ A e^{- κ τ}$ . By [F1], $U_{κ}$ is a smooth increasing diffeomorphism from $R$ onto $(- \infty, 0)$ ; the positive affine map $y \mapsto x_{H} + λ y$ is a smooth diffeomorphism from $(- \infty, 0)$ onto $(- \infty, x_{H})$ . Since $u \mapsto τ = χ u$ is also a positive linear diffeomorphism, $ℓ$ is a smooth increasing diffeomorphism from $R$ onto $(- \infty, ℓ_{H})$ , with smooth inverse $u_{ℓ} (x) = χ^{- 1} U_{κ}^{- 1} ((x - x_{H}) / λ)$ . Differentiating the supplied coordinate formula gives $ℓ^{'} (u) = χ λ e^{- κ χ u} = χ A k e^{- k u}$ and hence $U^{'} (u) = A k e^{- k u} > 0$ . If $supp g \subseteq [a, b]$ , then $supp f_{g} \subseteq [ℓ (a), ℓ (b)] \subset (- \infty, ℓ_{H})$ , so the zero extension is smooth and compactly supported. Thus $J (f_{g})$ is defined by [F1]–[F2].

2.1A1F1F4step 1.1algebra

The map $(u, v) \mapsto (ℓ (u), ℓ (v))$ is a smooth diffeomorphism, so [F4] transports the two-variable test pairing with Jacobian $ℓ^{'} (u) ℓ^{'} (v)$ . Since $ℓ = χ U$ , the regularized identity $χ^{2} [- (4 π)^{- 1} (χ s - i ϵ)^{- 2}] = - (4 π)^{- 1} (s - i ϵ / χ)^{- 2}$ shows that the positive unit scale cancels in the boundary distribution, giving the stated seconds-coordinate kernel. Positivity is the supplied Fock norm identity $W (g, g) = ∥ h_{f_{g}} ∥_{2}^{2} \geq 0$ from [F1].

3.1A1F1F2F3step 1.1step 2.1algebra∎

On the finite-particle domain, the left vacuum annihilates every creation term and the right vacuum is annihilated by every annihilation operator. Commuting the annihilation part of the leftmost $J (f_{g_{1}})$ through the later fields, [F3] replaces each commutator by its scalar two-point contraction; the term with the annihilator reaching the right vacuum vanishes. Hence $M_{n} (g_{1}, \dots, g_{n}) = \sum_{j = 2}^{n} M_{2} (g_{1}, g_{j}) M_{n - 2} (g_{2}, \dots, \hat{g_{j}}, \dots, g_{n}) .$ The base cases are $M_{0} = 1$ and $M_{1} = 0$ . Induction gives zero for odd $n$ and the stated sum over pair partitions for even $n$ .

Specified chiral collapse channel

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources