Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 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.

Elastic phase derivative

Example

Let I be an open energy interval and let δ:I→R be C1. The one-channel unitary matrix S(E)=e2iδ(E)∈U(1) has Wigner–Smith matrix Q(E)=2ℏ δ′(E), which is real and has units of seconds when E is measured in joules and ℏ in joule-seconds. This algebraic quantity is not, without further model assumptions, an operational arrival-time measurement.

Facts & Assumptions

Given: E∈I, δ∈C1(I;R), S(E)=e2iδ(E), and ℏ>0.

[F1]

For a C1 unitary channel matrix, Q=−iℏS∗S′ (Wigner–Smith matrix for differentiable finite channels).

Verification

Given: [F1] and the interval, differentiable phase, unitary matrix, and ℏ>0 in the Example.

1.1givenalgebra

Since δ(E) is real, S(E)∗S(E)=e−2iδ(E)e2iδ(E)=1; thus this is a one-channel unitary matrix on all of I. Differentiation gives S′(E)=2iδ′(E)e2iδ(E).

2.1givenF1step 1.1algebra∎

Substitution into [F1] gives Q(E)=−iℏe−2iδ(E)2iδ′(E)e2iδ(E)=2ℏδ′(E), which is real. Since δ is dimensionless and dδ/dE has units J−1, Q has units s. No pointwise value of Q is interpreted as a measured time.

Depends on

Dependency tree · 2 levels

Sources