Sigma-Time Quantum Dynamics I: Unitary BV-Clock Evolutions with Gates and Pulses

Quantum dynamics is usually written relative to a smooth external time parameter, while actual control schedules often contain waits, pulses, gates, resets, and event-like interventions. This paper develops the first unitary part of a sigma-time quantum dynamics framework. We define a bounded-Hamiltonian quantum evolution over a finite-horizon bounded-variation sigma clock with a continuous clock component, flat clock intervals, and a locally finite set of atom times carrying unitary gates. Under these hypotheses the sigma-chronological product exists, is unitary, composes as an evolution family, conserves wave-function norm, is strongly continuous away from atom times, and obeys a sharp atom/gate insertion law. We also prove a fixed-clock perturbation estimate and a bounded isolated-pulse corridor in which shrinking finite-width pulses converge in operator norm to declared atomic gates. The result is deliberately scoped: it does not claim an unbounded Schroedinger-operator theorem, a measurement theorem, an open-system theorem, or a Zeno limit theorem. A finite qubit benchmark with two bounded waits and one atomic gate is included as a reproducibility appendix and verifies unitary defect, trace preservation, Choi positivity, and output-state validity to machine precision