A Combinatorial Proof Of The Time-Ordered Exponential Using The Heaviside Representation Of Time-Ordering Operator

We present a novel combinatorial proof of the Time-Ordered Exponential formula for the Time-Evolution Operator based on the Heaviside Formulation Of The Time-Ordering Operator. Unlike conventional derivations that rely on induction our approach decomposes the Symmetric Group into permutations that produce decreasing time sequences and those that do not using a simple discriminant function. This decomposition together with an Interchange Rule between time-ordering and multiple integrals yields the Dyson–Chen Identity directly. The proof is self-contained and avoids recursion entirely. It appears in the literature as a problem in Advanced Quantum Mechanics (Springer, 2012, Problem 13. 1).