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

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Adopt the two-label gravitational model 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 (i): inertial masses $m_1,m_2>0$, $M=m_1+m_2$, reduced mass $\mu =m_1{m}_2\mathrm{/}M$, gravitational couplings $m_1^g,m_2^g>0$, $\kappa =G_{\mathrm{N}}{m}_1^g{m}_2^g$, and the potential $u(\rho )=-\kappa \mathrm{/}\rho$ on $(0,\mathrm{\infty })$. Let $x:I\to Q$ be a motion on a time interval $I$ with relative coordinate $r$, angular momentum $j=J{e}_3$, $J>0$, and energy $E<0$, and let $e\in [0,1)$, ${\theta }_0\in \mathbb{R}$, $p=\frac{J^2}{\mu \kappa },a=\frac{p}{1-e^2}=-\frac{\kappa }{2E},b=a\sqrt{1-e^2}$ be its conic data (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 clauses 1 and 3). Then:

1. (Period of the relative motion.) The relative motion extends uniquely to a central-force trajectory on all of $\mathbb{R}$ (as in 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 3); along that extension (with $\theta$ its polar angular coordinate, as supplied by the cited orbit theory), if $t_0\in \mathbb{R}$ and $T>0$ is determined by $\theta (t_0+T)=\theta (t_0)+2\pi$, then $r(t+T)=r(t)$ and $r^{\mathrm{\prime }}(t+T)=r^{\mathrm{\prime }}(t)$ for all $t\in \mathbb{R}$, and $T=\frac{2\pi \mu ab}{J}=2\pi \sqrt{\frac{\mu }{\kappa }}{a}^{3\mathrm{/}2},T^2=\frac{4{\pi }^2\mu a^3}{\kappa }\mathrm{.}$
2. (Gravitational parameters.) With $\mu =m_1{m}_2\mathrm{/}M$ and $\kappa =G_{\mathrm{N}}{m}_1^g{m}_2^g$, $T^2=\frac{4{\pi }^2{a}^3{m}_1{m}_2}{M{G}_{\mathrm{N}}{m}_1^g{m}_2^g};$ if in addition the model adopts the 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), this reduces to $T^2=\frac{4{\pi }^2{a}^3}{G_{\mathrm{N}}M}\mathrm{.}$
3. (Both labels.) On the global coordinate extension supplied by 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 3(a) (write it again as $x$, with centre of mass $X$), for each $i=1,2$ the position relative to the centre of mass is periodic with period $T$, $x_i(t+T)-X(t+T)=x_i(t)-X(t)$, while the absolute motions satisfy $x_i(t+T)-x_i(t)=VT(t\in \mathbb{R}),$ with the constant centre-of-mass velocity $V=X^{\mathrm{\prime }}$ 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 clause 2; the absolute motions are periodic exactly when $V=0$. The restriction of this extension to the model time window $I_0$ is a model motion; statements about $x_i(t+T)$ within that model require $t,t+T\in I_0$.
4. (Area per period.) The area swept by the relative radius vector in one period is $\pi ab$, and it equals $JT\mathrm{/}(2\mu )$; equivalently $T=2\pi \mu ab\mathrm{/}J$ as in clause 1.

• Which hypotheses carry the numerical form. Clauses 1, 3 and 4 use only the central model and the gravitational parameters symbolically; the simplification of clause 2 to $4{\pi }^2{a}^3\mathrm{/}(G_{\mathrm{N}}M)$ uses the empirical identification 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 not a theorem of the framework.

• No measured value. $a$, $T$, the masses and $G_{\mathrm{N}}$ are model data or symbols; no observation, calibration or uncertainty is used here, and no numerical prediction is made.

• Equal periods for both labels. Clause 3 states that both labels share the relative period about the centre of mass while the centre of mass may drift; the absolute motions are periodic only in the zero-total-momentum case.