Alphabeta MathHow 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.
Scope of the scattering examples page
The scope of the four examples of the scattering examples page: their status as idealised conditional model calculations with complete verifications, the parameters they leave symbolic, and the absence of any measured data, apparatus or statistical inference on the companion page.
Remarks
- Status of the examples. Every item on this companion page is an idealised setup with a complete verification inside the declared model: the square well of S-wave phase shift of a declared spherical square well, with its low-energy limit and hard-sphere comparison, the declared resonance factor of The declared single-channel resonance: Lorentzian factor, half-height points and resonance-value cross section, the uniform ball Born amplitude of Born amplitude of a declared uniform ball potential with an explicit weak-coupling condition, and the Yukawa class check of A Yukawa tail satisfies the resolvent-tail class while the Coulomb limit does not. None is a reported experiment: there is no apparatus, calibration, sample, count or uncertainty budget on this page, and no number appearing here is a measured value.
- Symbolic parameters. The depth and radius of the well, the resonance energy and width, the ball strength and radius, and the Yukawa coupling and screening parameter remain symbolic throughout; the examples verify identities and class memberships, and assert no numerical value of or .
- No extrapolation. The hard-sphere comparison of the square-well example is a limit inside the declared family of wells; the low-energy limit is taken for the exact phase shift of the declared well and is not asserted as a universal low-energy law. The failures and limits recorded in the Yukawa example are statements about the declared potential class, not about any material target.
Depends on
S-wave phase shift of a declared spherical square well, with its low-energy limit and hard-sphere comparison
The declared single-channel resonance: Lorentzian factor, half-height points and resonance-value cross section
Born amplitude of a declared uniform ball potential with an explicit weak-coupling condition
A Yukawa tail satisfies the resolvent-tail class while the Coulomb limit does not
The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units