Product Integrals and Time-Ordered Exponentials for BV Clocks with Atoms and Flats

This paper is the STC “product-integral” layer: we define and analyze the σ-chronological exponential (time-ordered exponential) for BV clocks σ with atoms and flats, in a form that respects the frozen STC semantics. In the kernel regime dσ = w(t) dt + P k αkδtk with locally finite atoms, we build a propagator by interleaving (i) the continuous ordered exponential driven by w(t) dt and (ii) explicit jump operators at atomic tick times. We prove: existence, evolution-family composition, sharp ope