RemarkRemark: AI-adaptedProof: Not applicable

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.

Long-range, threshold and model-scope limits of the scattering page

Physical scope

The explicit limits of the scattering page: the exclusion of the Coulomb tail from the short-range classes and its separate modified-comparison treatment, the threshold obstruction for uniform compact-defect statements at zero energy and the fact that completeness itself is unaffected, the absence of any zero-energy stationary statement, and the conditional status of the resonance model. This records scope; it asserts no new mathematical consequence.

Remarks

Long-range tails. The short-range classes of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units clause (b) exclude the pure Coulomb tail: the decaying-tail criterion of Asymptotic completeness for the resolvent-tail short-range class, with the decaying-tail example and the explicit exclusion of the Coulomb tail clause (c) requires a positive power $δ > 0$ , and the Coulomb quotient $(1 + ∣ x ∣ / L_{*})^{1 + δ} ∣ V (x) ∣$ is unbounded for every $δ > 0$ . The general resolvent-tail exclusion follows from clause (d) of that same theorem: its translated compact-bump input gives $h_{*} (r) \geq C / (2 r + R)$ with $C > 0$ and dimensionless radii $r, R$ (physical cutoff radius $L_{*} r$ ), which is not integrable. The Coulomb case is treated by the modified Dollard comparison (The declared long-range Coulomb model and its Dollard-modified free comparison), for which the ordinary short-range comparison is not claimed.
Threshold. The completeness statements of this page are made for the declared classes and do not require a zero-energy nonresonance hypothesis (Resolvent-tail short-range completeness). A different uniformity statement fails at zero energy: for compactly supported potentials the scattering defects are not compact uniformly when the energy cutoff does not vanish at $0$ (Compact scattering defects and the zero-energy obstruction). The general stationary beam and amplitude theorems use $k > 0$ and give no uniform zero-energy amplitude statement. The companion's declared square-well computation separately proves its explicit zero-energy resonance profile.
Positive eigenvalues. The absence of positive eigenvalues is part of Existence of the model's wave operators, isometry and intertwining, and the purely absolutely continuous free spectrum clause (d) for the compact-support class; no statement is made about negative or zero eigenvalues of the interacting operator, and no general bound-state inventory belongs to this page. The companion's explicit well has a separately proved local threshold branch, without a bound-state count for arbitrary potentials.
Resonances. The resonance model (The declared single-channel resonance model: pole-shaped channel factor and its cross-section reading) is a declared conditional model; the page proves no existence of a resonance for any declared potential and makes no analytic-continuation claim for the resolvent or the scattering operator.

Long-range, threshold and model-scope limits of the scattering page

Remarks

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