The coupon collector problem is classical: draw coupons uniformly at random until you have one of each type. With n types, this takes Θ(n ln n) draws on average. The problem assumes perfect retention — once collected, a coupon stays collected.
Janson and Kuszmaul (arXiv:2602.20705) introduce the careless collector, who loses each coupon independently with probability p after each round. The completion time depends on p, and the dependence reveals a phase structure.
When p is small enough — p = o(ln n / n²) — losses are rare enough that the collector still finishes in Θ(n ln n) time. The losses don't matter. When p = c/n for constant c, something different happens: the collection fraction stabilizes around 1/(1+c) and stays there for exponentially long — e^Θ(n) rounds. The collector is trapped in a metastable state, perpetually close to but never reaching completion. The losses and gains balance at a fixed fraction below one.
The metastability is the interesting feature. The collector isn't failing catastrophically or succeeding slowly — they are stuck in a dynamic equilibrium where each round's progress is exactly canceled by each round's losses. The system has a non-trivial fixed point that is not the solution. Reaching the actual solution (all coupons collected) requires a fluctuation large enough to overcome the loss barrier, and that fluctuation has exponentially small probability.
For larger p, completion becomes exponentially unlikely and the system never escapes the metastable state in any reasonable time.
The general observation: when a process both accumulates and loses, the competition between gain and loss can create stable intermediate states that are neither success nor failure. The system reaches a dynamic equilibrium well short of its goal and stays there — not because progress is impossible, but because the loss rate exactly matches the gain rate at a sub-optimal level. Persistent effort produces persistent incompleteness, not eventual success.