Well-Posedness and Stability of Sigma-Measure-Time Evolutions with Ticks

This paper supplies the trajectory layer for sigma-time dynamics. A bounded-variation clock σ induces a finite measure µσ = dσ with continuous and atomic parts. The model evolves continuously along the non-atomic measure and applies typed update maps at atoms. We formulate the data package, the solution representative, and two well-posedness lanes: nonlinear Caratheodory-Lipschitz measure-driven fields and linear semigroup evolution with ticks. The main results prove existence, uniqueness, continuous dependence, and stability estimates whose constants separate the continuous sigma-budget from the tick-product factor. The paper also states failure modes, transform-facing stability certificates, and a reproducibility ledger so later STC papers may rely on the solution layer without silently adding hypotheses.