= Solution
Let $M=M_1+M_2$, and let $\mu=M_1M_2/M$ be the <reduced mass>. The stellar distances from the <center of mass> are $a_1=aM_2/M$ and $a_2=aM_1/M$. For a <circular Kepler orbit>, $\Omega^2=GM/a^3$, so summing the two orbital contributions gives the \b[<circular-binary orbital angular momentum>]
$$
J_{\mathrm{orb}}=(M_1a_1^2+M_2a_2^2)\Omega
=\mu a^2\Omega
=\boxed{\frac{M_1M_2}{M}\sqrt{GMa}.}
$$
The separation $a$ is inside the square root. This also has the required dimensions of <angular momentum>.
For the <homologous rotating collapse>, label a fluid element by $s=r/R$ and its polar angle $\theta$. Its enclosed mass $m(s)$ is constant during <stellar homology>, as are the dimensionless density profile and the inertia coefficient $\alpha$. <Conservation of angular momentum> and <solid-body rotation> give
$$
\Omega=\frac{J}{\alpha MR^2}.
$$
The magnitude of the <centrifugal acceleration> is $\Omega^2r\sin\theta$, while the <Newtonian gravitational field> has magnitude $Gm(s)/r^2$. Therefore
$$
\boxed{\frac{F_{\mathrm{cent}}}{F_{\mathrm{grav}}}=\frac{\Omega^2r^3\sin\theta}{Gm(s)}
=\frac{J^2s^3\sin\theta}{\alpha^2GM^2m(s)}\frac1R\ \propto R^{-1}.}
$$
Using only the radial component of centrifugal force replaces $\sin\theta$ by $\sin^2\theta$ and leaves the scaling unchanged. The centrifugal force vanishes on the rotation axis; the central point is understood through the limiting field rather than a ratio of two zero forces.
At the equator of the cloud, <critical rotation of a spherical cloud> means $\Omega_c^2R_c=GM/R_c^2$, where $R_c=R_{\mathrm{crit}}$. Consequently
$$
J=\alpha M\sqrt{GMR_c},\qquad R_c=\frac{J^2}{\alpha^2GM^3}.
$$
Immediately after the first fission, each daughter has mass $M/2$ and radius $a/2$ because the spheres touch. Their orbital <moment of inertia> is $Ma^2/4$, while their combined spin <moment of inertia> is $\alpha Ma^2/4$. With a common spin and orbital frequency $\Omega_f=\sqrt{GM/a^3}$, the <synchronous fission model for binary formation> gives
$$
J=\frac{1+\alpha}{4}M\sqrt{GMa}.
$$
The common frequency can change during fission; corotation requires equal instantaneous frequencies, not an unchanged frequency from before the rearrangement. Equating the two values of $J$ yields
$$
\boxed{\frac a{R_c}=\frac{16\alpha^2}{(1+\alpha)^2}.}
$$
For a nonzero positive inertia coefficient, the \b[necessary and sufficient condition in this algebraic model for $a<R_c$] is
$$
\boxed{0<\alpha<\frac13.}
$$
For example, a <moment of inertia of a uniform solid sphere> has $\alpha=2/5$ and gives $a/R_c=64/49>1$; it fails the required compact-fission condition. A sufficiently centrally concentrated cloud can have smaller $\alpha$.
For the next collapse, neglect exchange of spin with the unchanged outer orbit, so each daughter's spin <angular momentum> is separately conserved. Immediately after the first fission that spin is
$$
S_d=\alpha\frac M2\left(\frac a2\right)^2\Omega_f
=\frac{\alpha M}{8}\sqrt{GMa}.
$$
For a daughter of mass $m=M/2$ to reach <critical rotation of a spherical cloud> at radius $R_{c,d}$, the same relation gives $S_d=\alpha m\sqrt{GmR_{c,d}}$. Hence
$$
\boxed{R_{c,d}=\frac a8.}
$$
Equivalently, its centrifugal-to-gravity ratio starts at $1/4$ when its radius is $a/2$ and reaches one after contraction by a factor four. Applying the same fission rule to this daughter produces an inner pair with separation
$$
\boxed{\frac{a'}a=\frac{16\alpha^2}{(1+\alpha)^2}\frac18
=\frac{2\alpha^2}{(1+\alpha)^2}.}
$$
Corotation at this second fission is local to each newly formed inner pair; it is not a single common frequency for the entire four-star hierarchy.
This spin assumption matters. If <tidal locking> instead kept each shrinking daughter synchronized to the fixed outer <orbital period> throughout the intervening collapse, its <angular velocity> would remain fixed. Its centrifugal-to-gravity ratio would then scale as $R_d^3$ and decrease, so it would never reach the proposed second fission. This is a counterexample to extending the corotation assumption through the whole contraction. The printed ratio describes the separately spin-conserving interpretation of the <repeated fission hierarchy>.
\b[The model can generate a compact hierarchy algebraically, but is not a general physical account of multiple-star formation.] In the allowed range $\alpha<1/3$, one has $a'/a<1/8$, suggesting well-separated inner and outer scales. However, rapidly rotating gas deforms, touching daughters are tidally distorted, and pressure, gas flows, dissipation and spin-orbit torques cannot generally be ignored. The assumed identical profiles, equal mass splits, instantaneous corotation and later torque-free contractions are restrictive. <Star formation> can involve <gravitational fragmentation> and redistribution of <angular momentum>; the toy budget neither proves that fission occurs nor predicts the population of real multiple systems.
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