ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generated

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 centered qubit-record displacement variance

Example

Assume full AC and initialize the selected energy-resolved record with the Gibbs law $π$ . Use the two energy states $∣ 0 ⟩, ∣ 1 ⟩$ of Connected Gibbs jump generators satisfy detailed balance with energies $E_{0} = Δ / 2$ and $E_{1} = - Δ / 2$ , where $Δ > 0$ is in joules and $β > 0$ is in joules $^{- 1}$ . Let $a > 0$ be the rate from $0 \to 1$ and $b > 0$ the rate from $1 \to 0$ , both in seconds $^{- 1}$ , with $b / a = e^{- β Δ}$ . The stationary mean is $z_{e q} : = (b - a) / (a + b)$ . Write $Z = diag (1, - 1)$ and let $u$ be a speed scale in metres per second; assign centered pointer velocities $v_{a} = u (Z_{a a} - z_{e q})$ . The monitored record displacement $Y_{T} = \int_{0}^{T} v_{a (s)} d s$ has

$C_{v} (s) = u^{2} (1 - z_{e q}^{2}) e^{- (a + b) s},$

and exact variance

$Var (Y_{T}) = 2 u^{2} (1 - z_{e q}^{2}) [\frac{T}{a + b} - \frac{1 - e^{- (a + b) T}}{(a + b)^{2}}] .$

Consequently

$\lim_{T \to \infty} \frac{Var (Y_{T})}{2 T} = \frac{u^{2} (1 - z_{e q}^{2})}{a + b} .$

This is a variance of the selected classical record with units of squared metres. The record is not asserted to be a continuous quantum position or an experimental velocity.

Facts & Assumptions

Given: Full AC, a two-state connected Gibbs jump model, its ideal energy-resolved record instrument, and $u \in R$ in metres per second.

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function, retained by the finite joint-path supplier.

[F1]

Stationary monitored-chain transport variance coefficient: for the stationary selected record and any centered real velocity labels, the covariance integral and variance formulas hold.

[F2]

Connected Gibbs jump generators satisfy detailed balance: for energies $E_{0} = Δ / 2, E_{1} = - Δ / 2$ , the Gibbs weights obey detailed balance with the stated rates, and diagonal populations follow the two-state generator.

Verification

1.1givenF2algebra

The Gibbs weights are $π_{0} = b / (a + b)$ and $π_{1} = a / (a + b)$ : their ratio is $b / a = e^{- β Δ} = e^{- β (E_{0} - E_{1})}$ , and they sum to one. Thus $π_{0} a = π_{1} b$ and $π (v) = 0$ . The two-state generator has centered eigenvector $Z - z_{e q} I$ with eigenvalue $- (a + b)$ ; therefore $e^{s W} v = e^{- (a + b) s} v$ and $⟨ v, v ⟩_{π} = u^{2} (1 - z_{e q}^{2})$ .

2.1A1F1step 1.1algebra∎

Substituting this covariance into [F1] gives

Var (Y_{T}) = 2 u^{2} (1 - z_{e q}^{2}) \int_{0}^{T} (T - s) e^{- (a + b) s} d s .

Evaluating the integral gives the displayed finite-time formula. Dividing by $2 T$ and letting $T \to \infty$ gives $u^{2} (1 - z_{e q}^{2}) / (a + b)$ .

Exact centered qubit-record displacement variance

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources