The Leaky Integrator Bound: Exact Continuous-Input Capacity and Discrete-Alphabet Regimes

We consider a discrete-time leaky integrator of the form H_n = λ H_{n-1} + x_n with 0 < λ < 1, x_n ∈ {−1, 0, 1}, and a hard state constraint |H_n| ≤ K for all n. This system models a finite-capacity accumulator with exponential forgetting: past inputs decay but cannot accumulate beyond K. We prove that the long-term average input A = limsup_{N→∞} (1/N) Σ x_n is bounded above by A_max = ((1−λ)/(1+λ)) K, and that this bound is sharp. The proof uses the telescoping identity Σ x_n = (1−λ) Σ H_n + H_N − λ H_{-1} together with the state constraint |H_n| ≤ K. The result is equivalent to the capacity of a token-bucket regulator in communication networks. We provide a self-contained derivation and discuss analogous structures in biological, physical, and mathematical systems where finite capacity limits sustainable throughput.