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 operator-norm growth bounds, and quantitative perturbation estimates separating continuous mismatch from jump mismatch. A link theorem shows the pure jump-product decay law used in the K-series is a special case (A ≡ 0) of this Σ4 formalism. Throughout, STC Paper 0 (clock/selector governance) and STC Papers Σ1–Σ3 are treated as frozen dependencies: we do not redefine clocks, the plateau selector, or σ-differentiation conventions; we only build reusable operator objects from them.