Terminal Defects, Growing Multiplicity, and Variance Extremality in the Double Dixie Cup Problem

We develop a terminal-defect method for the double Dixie cup problem and use it to prove the finite-variance extremality conjecture of Doumas and Papanicolaou. For every $m\ge1$ and $N\ge2$, among all positive coupon probability vectors $p=(p_1,\ldots,p_N)$, the variance of the time $T_m(N)$ to collect $m$ complete sets is uniquely minimized at the uniform vector. We prove the stronger radial statement that the variance is strictly increasing along every ray from the uniform vector. The proof is finite-$N$ and exact: after Poissonization, the completion time is a maximum of independent Erlang variables, and the radial derivative of its distribution is compared to a size-biased law using a monotone-likelihood-ratio argument based on a log-scale monotonicity property of the Gamma reverse hazard. The same framework gives a growing-multiplicity Gumbel theorem in the equal-probability case, with expectation and variance asymptotics on the inverse gamma-tail scale. This recovers the fixed-$m$ equal-probability variance asymptotic stated as Conjecture 1 by Doumas and Papanicolaou, classically known for $m=1$, and extends the mechanism to $m=m_N$. We also illustrate the unequal-probability theory with endpoint-Laplace limits for power-law probabilities.