Mathematical Formalization of Zero-Distance Interaction: An Optimization and Control-Theoretic Reformulation of Fitts’s Law

This paper presents a mathematical formalization of human–computer interaction under a zero-distance constraint, introducing a degenerate formulation of Fitts’s Law. In classical models, movement time depends logarithmically on spatial distance and target size. By enforcing D → 0, the Index of Difficulty converges to zero, and movement time reduces to a constant equal to the physiological intercept, yielding a constant-time interaction model. A rigorous ε–δ limit analysis proves convergence, while an optimization formulation shows that zero-distance interaction achieves the global minimum of latency. From a control-theoretic perspective, the model eliminates nonlinear dependencies and produces a time-invariant system. The framework is empirically validated on a teleoperated mobile robotic platform using a haptic Touch–Release protocol. Experimental results show a reduction in total response latency from approximately 1040 ms to 450 ms (≈56%). Cryptographically secured telemetry (AES-256) ensures data integrity and reproducibility. The proposed model establishes a new paradigm of constant-time human–computer interaction, with implications for optimization and control in cyber–physical systems and safety-critical applications.