The Expiring Coupon Collector: Sliding-Window Surjection Flux and Rare-Entry Laws

We study the coupon collector with deterministic expiration: one coupon is drawn at each time, and each coupon remains active for exactly $M$ draws. Completion occurs when all $n$ coupon types are simultaneously active. Equivalently, the current length-$M$ sliding window of draws must contain all $n$ types. The central object is not the one-time probability that a random window is onto, but the stationary flux of new entries into the onto-window set. We compute this flux exactly: $${\mu }_{n,M}=\text{Pbb}(W_{t-1}\text{ is not onto},W_t\text{ is onto})=\frac{(n-1)(n-1)!S(M-1,n-1)}{n^M},$$ where $S(\cdot ,\cdot )$ denotes a Stirling number of the second kind. Under a quantitative subcritical separation condition, satisfied in particular by every fixed integer scale $M=\text{floor}\alpha n\log n$, $0<\alpha <1$, we prove local declumping and obtain $${\mu }_{n,M}{T}_{n,M}\Rightarrow \text{Exp}(1)\mathrm{.}$$ For the fixed subcritical scale $M=\text{floor}\alpha n\log n$, $0<\alpha <1$, this gives the logarithmic scale $$\log T_{n,M}=n^{1-\alpha }+o_{\mathbb{P}}(n^{1-\alpha }),\log \text{Ebb}{T}_{n,M}=n^{1-\alpha }+o(n^{1-\alpha }),$$ and, when $\alpha >1\mathrm{/}2$, the sharper normalization $$n^{-\alpha }{e}^{-n^{1-\alpha }}{T}_{n,M}\Rightarrow \text{Exp}(1),\text{Ebb}{T}_{n,M}\sim n^{\alpha }{e}^{n^{1-\alpha }}\mathrm{.}$$ Thus the leading scale proposed in the Math StackExchange discussion is made rigorous; the exact finite-$n$ flux gives the canonical normalization throughout the subcritical range. The result is a sliding-window companion to rare-void entry-flux methods for nonmonotone coupon collectors.