Modeling Attention Decay in Single-Session Cognitive Work: A Derivation of the Exponential Form With Simulated-Data Validation

Productivity tools typically track elapsed time and treat it as a stand-in for how much a person can still give a task. Sustained-attention research suggests this is the wrong unit: capacity itself falls over the course of a session, and not at a constant rate. Focusware is a productivity app built around modeling attention directly rather than timing it, which raises a concrete question: what shape should that decay curve take, and can the choice be justified rather than assumed? I derive the answer from one assumption — that the rate of attention loss is proportional to attention currently remaining — and show it forces an exponential solution over the naive linear alternative. From there I derive a half-life, a bounded productivity score, an optimal stopping time, and a multi-session model of output under repeated, imperfect breaks. I then fit both the exponential and linear forms to a simulated dataset built to resemble Focusware's self-reported session checkpoints, at the single-session level and across 60 simulated users. The exponential model fits better in both cases (mean R² 0.974 vs. 0.934, better fit in 93.3% of simulated users), consistent with what the derivation predicts. Every formula in the paper maps to a function already running in Focusware, and I close by stating what result on real user data would overturn the model rather than confirm it.