= Solution
A <cataclysmic variable> is a close interacting <binary star> with a <white dwarf> accretor and a low-mass <donor star>, usually a <red dwarf>, filling its <Roche lobe>. Matter crosses by <Roche-lobe overflow> and commonly forms an <accretion disk>; sufficiently strong white-dwarf magnetism instead permits <magnetically channelled accretion>. Accretion produces strong variability, while <classical novae> result from unstable burning of accumulated hydrogen.
The <cataclysmic-variable orbital-period distribution> contains a short-period population between about 82 minutes and two hours, a pronounced <cataclysmic-variable period gap> around two to three hours, and a longer-period population above the gap. There is an accumulation near the <cataclysmic-variable period minimum>. These describe ordinary hydrogen-rich systems; evolved donors and helium-rich systems can have shorter periods. Selection by luminosity and outburst activity affects the observed counts, so $N(P)$ is not simply the intrinsic distribution.
For the contact <red dwarf>, $R_2=R_L\propto a(M_2/M)^{1/3}$. Using its supplied thermal-equilibrium <mass-radius relation>, $R_2\propto M_2$, gives $a\propto M_2^{2/3}M^{1/3}$. <Kepler's third law> then gives
$$
\boxed{P\propto a^{3/2}M^{-1/2}\propto M_2},
$$
independent of $M_1$. This is a special case of the <Roche-lobe-filling period-density relation>.
To examine rapid mass loss, use the <stellar radius response exponent> $\zeta_{\rm ad}=d\log R_2/d\log M_2=-1/3$. During <conservative binary mass transfer>, $M$ and $J=M_1M_2\sqrt{Ga/M}$ are fixed, and $dM_1=-dM_2$. Therefore
$$
\frac{d\log a}{d\log M_2}=-2(1-q),\qquad
\zeta_L=\frac{d\log R_L}{d\log M_2}=-2(1-q)+\frac13=2q-\frac53.
$$
The change in overfill is $d\log(R_2/R_L)=(\zeta_{\rm ad}-\zeta_L)d\log M_2$. Since $d\log M_2<0$, overfill grows, and mass loss runs away, when $\zeta_{\rm ad}<\zeta_L$. The <dynamical stability of binary mass transfer> criterion here is consequently
$$
\boxed{q_{\rm crit}=\frac23,\qquad \text{unstable for }q>\frac23,\quad\text{stable for }q<\frac23}.
$$
\b[The printed instability inequality is reversed.] With the defined $q=M_2/M_1$, the specified radius laws imply the inequality above; $q<q_{\rm crit}$ is the stable side. Equality is marginal, and the supplied Roche-radius approximation restricts this calculation to its stated mass-ratio range.
Long-term contact is maintained by <angular momentum transport> out of the orbit. <Gravitational-wave emission from a binary system> provides one loss mechanism. <Magnetic braking of a binary star> uses the donor's <stellar wind>: its <magnetic field> enforces approximate corotation out to the <Alfvén radius>, so the wind carries much more <specific angular momentum> than a nonmagnetic surface outflow. At that radius, roughly
$$
\boxed{v_{\rm wind}\simeq v_A=\frac{B}{\sqrt{\mu_0\rho}},\qquad
\dot M_w\simeq4\pi R_A^2\rho v_{\rm wind}},\qquad
\dot J_w\sim-\dot M_w\Omega R_A^2.
$$
The geometry contributes order-one factors to the torque. Even when $\dot M_w\ll|\dot M_2|$, the large lever arm can remove substantial spin <angular momentum>. <Tidal locking> couples donor spin to orbital motion, so the wind torque ultimately brakes the orbit. In practice <classical novae> can also eject matter; the conservative calculations here isolate the stated idealization.
For the prescribed $\dot J/J=-\alpha$, now use the supplied equilibrium law $R_2\propto M_2$ for the slowly evolving donor. Contact implies $\dot a/a=(2/3)\dot M_2/M_2$, assuming negligible net wind mass loss. Taking the logarithmic derivative of $J$ gives
$$
-\alpha=(1-q)\frac{\dot M_2}{M_2}+\frac12\frac{\dot a}{a}
=\left(\frac43-q\right)\frac{\dot M_2}{M_2}.
$$
Since $P\propto M_2$ along this equilibrium sequence,
$$
\boxed{\frac{\dot P}{P}=\frac{\dot M_2}{M_2}=\frac{3\alpha}{3q-4}}.
$$
For the dynamically stable mass ratios, this is negative. The result uses thermal equilibrium as well as slow hydrostatic evolution; evolution slow compared only with the dynamical time does not by itself guarantee the radius law $R_2\propto M_2$.
In the <disrupted magnetic braking model>, the donor above the gap loses mass fast enough to be inflated relative to thermal equilibrium. When the donor becomes a <fully convective star>, the model assumes a substantial reduction in magnetic braking. The <red dwarf> then contracts toward equilibrium on its <Kelvin-Helmholtz cooling time>, becoming smaller than its <Roche lobe>; accretion largely stops. <Gravitational-wave emission from a binary system> continues to reduce the separation, carrying the detached binary through the gap until contact is restored near its lower edge.
This explanation requires strong braking immediately above the gap: the donor must already be out of equilibrium and inflated, so there is room to contract and detach when braking drops. With nearly fixed component masses during detachment, $R_L\propto a\propto P^{2/3}$, giving the illustrative inflation factor
$$
\boxed{\frac{R_{2,\rm upper}}{R_{2,\rm equilibrium}}
\simeq\left(\frac{P_{\rm upper}}{P_{\rm lower}}\right)^{2/3}
\simeq\left(\frac32\right)^{2/3}\simeq1.31}.
$$
A weak braking torque that kept the donor in thermal equilibrium just above the gap would not produce this detachment. The model requires a decrease in the torque, not disappearance of the donor's <magnetic field>.
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