CounterexampleConstruction: 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.

Quasienergy has no canonical ground value

Statement refuted

The one-period unitary of a finite-dimensional periodic system determines a distinguished least real quasienergy that is independent of the chosen Hermitian logarithm.

Statement references

Finite Floquet unitary normal form

Facts & Assumptions

Given: Countable Choice, $T > 0$ , $ℏ > 0$ , $ω = 2 π / T$ , and the one-dimensional state space $C$ .

[A1]

Countable Choice is assumed by the finite Floquet normal-form result (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Finite Floquet unitary normal form).

[F1]

Finite Floquet unitary normal form defines a quasienergy as a real eigenvalue of a chosen Hermitian logarithm of $U (T, 0)$ , with representatives taken modulo $ℏ ω$ .

[F2]

The finite matrix exponential is defined by Finite-dimensional matrix exponential, and equality of scalar exponential values is characterized by $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ .

[F3]

Norm Dyson series and uniqueness supplies the unique propagator for the constant zero Hamiltonian.

Counterexample

Given: [A1], [F1]–[F3], $T > 0$ , $ℏ > 0$ , and $ω = 2 π / T$ .

1.1A1F1F2F3algebra

Take the constant zero Hamiltonian $H (t) = 0$ on $C$ . Its propagator is $U (t, 0) = 1$ by [F3]. For each integer $n$ , the Hermitian scalar $H_{F}^{(n)} = n ℏ ω$ satisfies $e^{- i T H_{F}^{(n)} / ℏ} = e^{- i 2 π n} = 1 = U (T, 0)$ , so each is a valid logarithm of the same monodromy.

2.1F1F2step 1.1∎

In dimension one the only eigenvalue of $H_{F}^{(n)}$ , and hence its least quasienergy representative, is $n ℏ ω$ . It varies with $n$ , while all representatives have the same class modulo $ℏ ω$ by [F1, F2]. Therefore the period unitary determines a quasienergy class but no branch-independent least real quasienergy.

Quasienergy has no canonical ground value

Statement refuted

Statement references

Facts & Assumptions

Counterexample

Depends on

Dependency tree · 2 levels

Sources