Pre-giant Roche-lobe-filling period limit ($P_i<2\pi \sqrt{R_{\mathrm{BGB}}^3/(C^{3}G{M}_d)}$)

For $R_L=Ca(M_d/M{)}^{1/3}$, the period when a donor reaches radius $R$ is $P=2\pi \sqrt{R^3/(C^{3}G{M}_d)}$. Requiring overflow before a specified evolutionary radius therefore bounds the initial period independently of companion mass in this lobe approximation. If that radius scales as $M_d^{5/3}$, the bound scales as $M_d^2$.

For fixed total solar-unit mass $m$ and conserved orbital angular momentum, conservative binary mass transfer preserves $K=P[m_d(m-m_d){]}^3$. Hence $P/m_d^2=K/[m_d^5(m-m_d{)}^3]$. Its logarithmic derivative is $-5/m_d+3/(m-m_d)$, vanishing at $5m/8$ with positive second derivative. The ratio diverges at both endpoints, so this is its global minimum. Comparing this minimum to a pre-giant bound $P_0$ tests whether any such conservative antecedent can overflow before the giant branch; satisfying it is necessary, not a proof of an entire evolutionary history.