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

We present a unified formal mathematical analysis of the DTA distributed task scheduler (N = 4 workers, SPSC mailboxes, Vyukov MPMC warehouse, hop-bounded deflection with N/2 max hops). Using a second-quantisation operator algebra, we prove: (I) four correctness theorems — task conservation, deadlock freedom, livelock freedom, and linearizability; (II) bounded waiting time with an explicit ceiling Wmax(λ,µ,N,H) and k-bounded fairness; (III) quantitative performance bounds — mean-field load balance (M/M/1/H self-consistency, imbalance O(1/√N)), BBGKY kinetic-theory derivation with O(1/N) pair-correlator correction, a two-line makespan bound (deterministic + Gumbel extreme-value), a four-level stability ladder (Foster–Lyapunov, spectral gap, burst absorption, first-passage crash time), and fork-join DAG scheduling bounds via a sequential nucleation kinetic equation; (IV) competitive analysis — Bag-of-Tasks ratio versus work stealing and an adversarial worst-case bound ρDTA,worst c ≤ 1+NN/2δhop/W → 1; (V) empirical validation against Criterion CI benchmarks confirming all theoretical predictions within confidence intervals. All results are accompanied by Monte Carlo and ODE simulation scripts.