ExampleConstruction: AI-generatedVerification: 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.

Commuting longitudinal drive

Example

Assume Countable Choice. On $C^{2}$ let $σ_{z} = diag (1, - 1)$ , choose real $E_{0}, A$ , and choose $ℏ > 0$ , $ω > 0$ , and $T = 2 π / ω$ . For $H (t) = (E_{0} + A \cos (ω t)) σ_{z}$ , the unique propagator from time $0$ is $U (t, 0) = \exp ⁣ [- \frac{i}{ℏ} (E_{0} t + \frac{A}{ω} \sin (ω t)) σ_{z}] .$ One Floquet normal form is $H_{F} = E_{0} σ_{z}$ and $P (t) = \exp [- i A \sin (ω t) σ_{z} / (ℏ ω)]$ .

Example references

Norm Dyson series and uniqueness

Finite Floquet unitary normal form

Finite-dimensional matrix exponential

Facts & Assumptions

Given: Countable Choice, $E_{0}, A \in R$ , $ℏ > 0$ , $ω > 0$ , $T = 2 π / ω$ , and $σ_{z} = diag (1, - 1)$ .

[A1]

Countable Choice is assumed by the bounded-dynamics theorem used for uniqueness (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The scalar derivatives are $(\sin x)^{'} = \cos x$ and $(\cos x)^{'} = - \sin x$ , and sine and cosine have period $2 π$ (The derivatives of sine and cosine are cosine and minus sine, The zero sets of sine and cosine and the least positive common period 2 pi).

[F2]

Norm Dyson series and uniqueness gives uniqueness for a norm-continuous bounded Hermitian matrix family; Finite-dimensional matrix exponential and Unitarity and differentiation of the finite Hermitian matrix exponential give the finite exponential's definition, derivative, and group law.

[F3]

Finite Floquet unitary normal form defines the Floquet representation as $U (t, 0) = P (t) e^{- i t H_{F} / ℏ}$ with $P$ unitary and $T$ -periodic.

[F4]

The chain rule and the scalar sine derivative give $(\sin (ω t))^{'} = ω \cos (ω t)$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The derivatives of sine and cosine are cosine and minus sine).

Verification

Given: [A1], [F1]–[F4], and the matrices and parameters in the Statement.

1.1A1F1F2F4algebra

Put $F (t) = E_{0} t + (A / ω) \sin (ω t)$ . By [F1] and [F4], $F^{'} (t) = E_{0} + A \cos (ω t)$ . The diagonal entries of $\exp [- i F (t) σ_{z} / ℏ]$ are $e^{- i F (t) / ℏ}$ and $e^{i F (t) / ℏ}$ ; differentiating them using [F2] gives $i ℏ \partial_{t} U (t, 0) = F^{'} (t) σ_{z} U (t, 0) = H (t) U (t, 0)$ . At $t = 0$ the exponential is $I_{2}$ , so uniqueness in [F2] identifies it with the propagator.

2.1F1F2F3algebra∎

The finite exponential group law in [F2] splits $U (t, 0) = e^{- i A \sin (ω t) σ_{z} / (ℏ ω)} e^{- i t E_{0} σ_{z} / ℏ}$ . Thus $H_{F} = E_{0} σ_{z}$ and the stated $P (t)$ give the required factorization. The matrix exponential is unitary by [F2], and $P (t + T) = P (t)$ by the $2 π$ period in [F1]. Hence this is a Floquet normal form with quasienergies $E_{0}$ and $- E_{0}$ , each understood modulo $ℏ ω$ .

Commuting longitudinal drive

Example

Example references

Facts & Assumptions

Verification

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