Conservative pre-giant period-ratio minimum ($\min (P/m_d^2)\text{ occurs at }{m}_d=5m/8$)

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.