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

This remark records the scope of the page's items. It is bookkeeping: every mathematical statement it repeats is proved in the item cited, and it adds no law, no observation and no numerical value of its own.

Adopted assumptions. The force branch is Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with its standing regularity and equation of motion; the interaction restrictions (external-plus-pair presentation, antisymmetry, centrality, the potential branch and the gravitational branch) are Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch; the page's central-force class, the reduced problem, the fixed-centre idealization and the reading conventions are Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions; and the gravitational model with its mass identification and the measured value of $G_{\mathrm{N}}$ is Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant. None of these is proved here.

Conditional deductions. The relative reduction, the uniform centre-of-mass motion and the reconstruction bijection are Two-body reduction of a static central pair-potential model: the relative coordinate is a central-force trajectory for the reduced mass, the centre of mass moves uniformly, and the reconstruction is a bijection; the planarity and area law are Planarity of the reduced orbit and the area law: the areal-velocity vector is the constant $j\mathrm{/}(2\mu )$, the swept-area rate is $J\mathrm{/}(2\mu )$, and both labels sweep equal areas in equal times about the centre of mass; the effective one-dimensional problem and its converse are Reduction of the central problem to the effective one-dimensional potential: $\mu {\rho }^{\mathrm{\prime }\mathrm{\prime }}=-(U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}{)}^{\mathrm{\prime }}(\rho )$, the energy identity, and the converse reconstruction of the planar trajectory from the radial data; the circular-orbit criterion and the linearised radial frequency are Circular relative orbits of the central model: the critical-point criterion $u^{\mathrm{\prime }}(r_0)=J^2\mathrm{/}(\mu r_0^3)$, uniform rotation of both labels about the centre of mass, and the linearised radial frequency; the conic classification, the barrier and the elliptic geometry are The inverse-square gravitational relative trajectory has the Kepler polar form: $\rho =p\mathrm{/}(1+e\cos (\theta -{\theta }_0))$, $E=(\mu {\kappa }^2\mathrm{/}2J^2)(e^2-1)$, elliptic completeness and barycentric scaled ellipses; and the period and third law are Kepler's third law for the gravitational two-label model: the relative period is $T=2\pi \mu ab\mathrm{/}J=2\pi \sqrt{\mu \mathrm{/}\kappa }{a}^{3\mathrm{/}2}$, so $T^2=4{\pi }^2{a}^3\mathrm{/}(G_{\mathrm{N}}M)$ under the mass identification, and both labels are periodic relative to the centre of mass. Each is a conditional consequence of the adopted model and the cited mathematics. The conic and period items assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); every scope statement below that invokes their conclusions carries that assumption.

• Collision and the domain. The configuration domain excludes $x_1=x_2$, i.e. $r=0$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clause (v)); the equations of motion and every orbit statement above are undefined at the collision. In the gravitational model with $j\mathrm{\ne }0$ the barrier of The inverse-square gravitational relative trajectory has the Kepler polar form: $\rho =p\mathrm{/}(1+e\cos (\theta -{\theta }_0))$, $E=(\mu {\kappa }^2\mathrm{/}2J^2)(e^2-1)$, elliptic completeness and barycentric scaled ellipses clause 2 keeps $\mid r\mid \ge {\rho }_{\min }(E)>0$ throughout the motion; no such bound is available for the radial case, so the conic, area and period statements must not be extended to it.

• Radial motion. For $j=0$ the nonzero-angular-momentum polar parametrization and its distinguished plane orthogonal to $j$ are unavailable; a constant polar angle can still be chosen in any plane containing the fixed radial direction. Planarity of the reduced orbit and the area law: the areal-velocity vector is the constant $j\mathrm{/}(2\mu )$, the swept-area rate is $J\mathrm{/}(2\mu )$, and both labels sweep equal areas in equal times about the centre of mass clauses 2-4, The inverse-square gravitational relative trajectory has the Kepler polar form: $\rho =p\mathrm{/}(1+e\cos (\theta -{\theta }_0))$, $E=(\mu {\kappa }^2\mathrm{/}2J^2)(e^2-1)$, elliptic completeness and barycentric scaled ellipses and Kepler's third law for the gravitational two-label model: the relative period is $T=2\pi \mu ab\mathrm{/}J=2\pi \sqrt{\mu \mathrm{/}\kappa }{a}^{3\mathrm{/}2}$, so $T^2=4{\pi }^2{a}^3\mathrm{/}(G_{\mathrm{N}}M)$ under the mass identification, and both labels are periodic relative to the centre of mass exclude this case explicitly. Clause 1 of Reduction of the central problem to the effective one-dimensional potential: $\mu {\rho }^{\mathrm{\prime }\mathrm{\prime }}=-(U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}{)}^{\mathrm{\prime }}(\rho )$, the energy identity, and the converse reconstruction of the planar trajectory from the radial data also assumes $J=\mid j\mid >0$, so it does not supply the radial equation. Instead, angular-momentum conservation in Planarity of the reduced orbit and the area law: the areal-velocity vector is the constant $j\mathrm{/}(2\mu )$, the swept-area rate is $J\mathrm{/}(2\mu )$, and both labels sweep equal areas in equal times about the centre of mass clause 1 and the relative equation of Two-body reduction of a static central pair-potential model: the relative coordinate is a central-force trajectory for the reduced mass, the centre of mass moves uniformly, and the reconstruction is a bijection show that, when $j=0$, $r^{\mathrm{\prime }}=\lambda r$ with $\lambda =⟨r,r^{\mathrm{\prime }}⟩\mathrm{/}\mid r{\mid }^2$. Setting $\rho =\mid r\mid >0$ gives ${\rho }^{\mathrm{\prime }}=\lambda \rho$ and $n^{\mathrm{\prime }}=r^{\mathrm{\prime }}\mathrm{/}\rho -r{\rho }^{\mathrm{\prime }}\mathrm{/}{\rho }^2=0$ for $n=r\mathrm{/}\rho$; hence the unit direction is constant, say $n_0$, and $r=\rho n_0$ with $\rho \in J_0$. Substitution into that relative equation yields the scalar equation $\mu {\rho }^{\mathrm{\prime }\mathrm{\prime }}=-u^{\mathrm{\prime }}(\rho )$ on $J_0$. No centrifugal term, equal-area, conic or period claim is made for radial trajectories.

• Point particles and finite size. The model retains only positions and fixed masses of the labels (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (vi)); size, shape, rotation, deformation, internal structure and mass loss are omitted. A real body whose radius, multipole moments, tides or spin matter for the question asked is not described by these items, and their applicability must be justified separately wherever they are used.

• External bodies and non-gravitational forces. The model has no external forces by construction (Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clause (i)), so third bodies, drag, radiation pressure or any other agent are outside it; a two-body computation with such influences is a different model, not a corollary of this page.

• Regime. The meanings of weak-field and non-relativistic use are those defined in Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (iv): for a trajectory segment and chosen tolerance $\eta \in (0,1)$, in the fixed comparison inertial frame of that clause, the maximum source-potential ratio ${\max }_i\mid {\mathrm{\Phi }}_i\mid \mathrm{/}{c}^2$ and the maximum squared label-speed ratio ${\max }_i\mid x_i^{\mathrm{\prime }}{\mid }^2\mathrm{/}{c}^2$ are each at most $\eta$ throughout the segment. Relativistic corrections (for instance the perihelion precession of the inner planets) require the general-relativistic framework and are neither derived nor bounded here; the inverse-square law itself is an adopted law of the model.

• The constant. Formula-level statements use the symbol $G_{\mathrm{N}}$ and are conditional on Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (i). Any numerical application that assigns the CODATA value of clause (iii) — relative standard uncertainty $2.2\times {10}^{-5}$, with the mutual inconsistency of the 16 input measurements recorded there — inherits that uncertainty and the conditions of every other input actually used. The framework does not derive the value of $G_{\mathrm{N}}$ and does not claim it is exact.

• The mass identification. The simplification of the third law to $T^2=4{\pi }^2{a}^3\mathrm{/}(G_{\mathrm{N}}M)$ in Kepler's third law for the gravitational two-label model: the relative period is $T=2\pi \mu ab\mathrm{/}J=2\pi \sqrt{\mu \mathrm{/}\kappa }{a}^{3\mathrm{/}2}$, so $T^2=4{\pi }^2{a}^3\mathrm{/}(G_{\mathrm{N}}M)$ under the mass identification, and both labels are periodic relative to the centre of mass clause 2 uses the empirical identification $m_i^g=m_i$ of Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (ii), which is an adopted assumption, not a consequence of the equation of motion. Statements that keep the couplings $m_i^g$ distinct remain valid without it.

• No local experiment. No item of this page reports an experiment or apparatus record. Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (iii) does report the CODATA 2022 recommended estimate and standard uncertainty for $G_{\mathrm{N}}$, the source's 16 measured input values, and the 3.9 uncertainty-expansion factor. Those are source-reported aggregate evidence, not a measurement performed or newly reported by this page; the individual experimental apparatus records are not reproduced here. The numerical checks on the companion examples page use declared model data. Comparing any prediction of this page with observations would test the combined model, parameter and apparatus assumptions; agreement would be evidence, not a proof of the model, and a discrepancy would identify no particular postulate by itself.

No global existence of arbitrary central-force trajectories, no nonlinear stability of circular orbits, no closed-form solution of the general non-Coulomb central problem, no asymptotes or scattering cross-sections for the unbounded conic branches, no relativistic or finite-size corrections, and no claim that any physical system is exactly a model of the adopted class.