Scope of the scattering and impulse items: model boundaries, branch and cutoff qualifications, and the claims deliberately not made

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the scattering measures.

This remark records the boundaries of the page's claims. It adds no statement of its own and no empirical content; each clause refers to the items that do.

Finite-range central scattering in the model class: the scattering motion on all of time, the asymptotic velocities and impact parameter $b=J\mathrm{/}\sqrt{2\mu E}$, the advance $\mathrm{\Delta }$ with principal deflection $\arccos (\cos (\pi -2\mathrm{\Delta }))$, and the cross-section of a deflection branch is a conditional deduction inside the two-label central model class of Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions with a finite-range $C^2$ potential and a simple outer turning point: the relative motion extends to all of time, its asymptotic velocities and impact parameter are computed, the half-angle advance is a quadrature, the asymptotic directions satisfy the deflection relation with angle $\pi -2\mathrm{\Delta }$ when $0<\mathrm{\Delta }<\pi \mathrm{/}2$, and on a smooth deflection branch that records the model's deflections the outgoing angular measure has that branch's cross-section contribution $b\mathrm{\mid }{b}^{\mathrm{\prime }}\mathrm{\mid }\mathrm{/}\sin \theta$. The differential and truncated total cross-sections of a deflection branch: densities, units, branch sums and the empirical qualifications of the uniform-incident reading sums contributions from disjoint incident intervals, including where their angular images overlap. Repulsive Coulomb scattering in the model class: global non-colliding motion, the deflection map $\theta =2\arctan (\kappa \mathrm{/}(2Eb))$, the Rutherford density $(\kappa \mathrm{/}4E{)}^2{\csc }^4(\theta \mathrm{/}2)$, finite angular cutoffs and the divergent ideal total treats the separate long-range repulsive $\kappa \mathrm{/}r$ potential and adds the explicit deflection map, the Rutherford density, the finite truncated band values and the divergence of the ideal total.

• No monotonicity or completeness of the deflection map. Finite-range central scattering in the model class: the scattering motion on all of time, the asymptotic velocities and impact parameter $b=J\mathrm{/}\sqrt{2\mu E}$, the advance $\mathrm{\Delta }$ with principal deflection $\arccos (\cos (\pi -2\mathrm{\Delta }))$, and the cross-section of a deflection branch clause 5 states explicitly that the map $b\mapsto {\theta }_{\mathrm{s}}$ is not asserted to be decreasing, injective or free of stationary points, and that the branch images are not asserted to cover $(0,\pi )$. The counterexample Smooth finite-range free scattering refutes global strict decrease; as a separate contact comparison, an attractive refracting shell with index $3\mathrm{/}2$ has $\theta (b)=2(\arcsin (b\mathrm{/}a)-\arcsin (2b\mathrm{/}(3a)))$ strictly increasing, with a jump down to $0$ at grazing incidence ↗ shows that global strict decrease fails even in the smooth finite-range class because the deflection has a constant free tail; its separate adopted contact comparison also has an increasing branch. The cross-section formula of The differential and truncated total cross-sections of a deflection branch: densities, units, branch sums and the empirical qualifications of the uniform-incident reading therefore applies to specified invertible branches rather than assuming a global inverse. For a finite family of branches with disjoint incident intervals, the total density at an angle is the sum of all branch contributions whose images contain that angle; this total is distinct from any one branch's density.

• Cross-section and probability are distinct. The cross-section is a model quantity of the adopted uniform-incident preparation (Scattering data of the central model class: the scattering motion, the impact parameter, the asymptotic directions, the deflection conventions and the uniform-incident preparation clause (iv), The differential and truncated total cross-sections of a deflection branch: densities, units, branch sums and the empirical qualifications of the uniform-incident reading clause (iv)): it is not a count of detected events, not a probability per incident label, and not a property of any apparatus. For a specified branch and Borel angle window, the probability per incident label of an impact parameter in that branch's preimage is its incident preimage area within $D_{\max }=(0,b_{\max })\times (0,2\pi )$ divided by $\pi b_{\max }^{}$, with fixed energy and $J(b)=\sqrt{2\mu E}b$; this statement covers only the recorded branch portions. No experiment is authored on this page, and no measured rate, uncertainty or statistical claim about nature is made here. A comparison would require a specified apparatus, sampling model and analysis, as Scope of the two-body and central-force page: collision and radial exclusions, point-particle and finite-size limits, the empirical parameters and their uncertainty, and the conditional status of the orbit statements requires of the collision claims of the preceding page.

• Contact scattering uses separate example rules. The hard-sphere and refracting-shell models of the examples page adopt their fixed-wall and interface rules locally; neither full rule is supplied by Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting, whose restitution law covers a finite-mass collinear two-label contact. These contact trajectories are not instances of the $C^2$ finite-range hypothesis of Finite-range central scattering in the model class: the scattering motion on all of time, the asymptotic velocities and impact parameter $b=J\mathrm{/}\sqrt{2\mu E}$, the advance $\mathrm{\Delta }$ with principal deflection $\arccos (\cos (\pi -2\mathrm{\Delta }))$, and the cross-section of a deflection branch. In particular the finite total cross-section $\pi a^2$ of the hard-sphere contact model is computed from its locally defined uniform incident-area pushforward under the separately adopted reflection rule and does not contradict the forward divergence of the ideal Coulomb branch.

• No electromagnetic, screening, radiative or quantum content. The Coulomb item treats $\kappa$ as a parameter of the mechanical model and does not identify it with $q_1{q}_2\mathrm{/}(4\pi {\epsilon }_0)$ or any other electromagnetic quantity; it asserts no screening law for a medium of scatterers, no inelastic or radiative loss, and no relativistic or quantum modification. Attractive Coulomb capture, for which suitable data do allow capture, lies outside its statement.

• Relative data do not give label angles. The scattering angles and cross-section densities are computed from the reduced relative coordinate and its asymptotic velocities, as defined by Scattering data of the central model class: the scattering motion, the impact parameter, the asymptotic directions, the deflection conventions and the uniform-incident preparation. The two-label reconstruction $x_1=X+\frac{m_2}{M}r$, $x_2=X-\frac{m_1}{M}r$ shows that individual-label directions generally differ from the relative direction and from each other; this page asserts no individual-label angle.

Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting adopts three extensions of the force branch: the isolated reciprocal impulse, the collinear restitution contact law with $e\in [0,1]$, and the open-body material accounting. Consequences of the adopted impact and open-mass models: continuity of total momentum, the smooth-force exclusion, the restitution solution with $\mathrm{\Delta }K=-\frac{1-e^2}{2}\mu \parallel v_2^{-}-v_1^{-}{\parallel }_2^2$, and the open-body and rocket laws proves their conditional consequences — momentum continuity across the impulse, exclusion of a nonzero impulse for a candidate motion that also satisfies the smooth-force branch, the restitution velocities, the kinetic-energy change and the open-body and rocket laws — and records that momentum balance alone does not fix the outcome. The finite-regularity open-flow model: moving-control-region mass and momentum balances, their material and fixed special cases, and the inlet/outlet decomposition reproducing the open-body and rocket laws derives the moving-control-region identities from adopted material balances and identifies the lumped accounting of the impact postulate as their inlet/outlet specialisation.

These items assert no law of nature and no measured value: $e$, the impulse, the contact direction and the mass rates and velocities are model data (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework for the dimensions). They exclude contacts outside $0\le e\le 1$, simultaneous or tangential contacts, friction, rotation and deformation modes, and — for the open flow — constitutive laws, boundary conditions and incompressibility, which are separate models. Their empirical reading (for instance that a particular pair of bodies has a particular coefficient) requires a separately specified experiment, sampling model and uncertainty analysis, and obtains evidence about the model-plus-apparatus assumptions rather than a proof of the model.