Decay — the series
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The counter with no window in it
A counter that fades by half every H settles, on a steady stream, at exactly the count of a window of 1.44H. That correspondence holds in the mean, on a steady stream, and nowhere else — and it is the reason a decayed counter is not an estimate of a windowed count for any window.
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The fading nobody computes
Twelve thousand arrivals into thirty-two decayed counters cost 382,976 fade multiplications. An implementation that aged every counter on every tick would have cost 3,830,256, and the ratio is exactly the mean gap between arrivals — not a coincidence, and the reason the family is deployable.
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A decay measured from where it started
An exponentially decayed counter is one number because its weights fade by elapsed time alone. Forward decay keeps a polynomial weight in one number too, by measuring each arrival from a fixed landmark. Its memory is then a share of the time since that landmark: at β = 2 an arrival counts half at 29% of that time. Anchored at the start of a stream, it takes 5.3 seconds to register a fourfold rise twenty seconds in and 83 seconds when the rise comes at five minutes.