DTA Distributed Task Scheduler Formal Mathematical Analysis: Correctness, Progress, Performance, and Competitive Bounds

This paper presents a unified formal mathematical analysis of the DTA distributed task scheduler. Using a second-quantisation-inspired operator formalism, we establish: (I) correctness properties --- no task loss, deadlock freedom, livelock freedom, and linearizability; (II) progress properties --- bounded waiting time and starvation freedom; (III) quantitative performance bounds --- load balance, kinetic-theory corrections to mean-field, makespan bounds via extreme-value theory, and a four-level stability ladder; and (IV) competitive and empirical analysis --- online competitive ratio relative to work stealing, adversarial stability comparison, and correspondence with Criterion benchmark measurements on GitHub Actions CI. All theoretical predictions are numerically validated and, where applicable, matched against empirical wall-clock data.