Sigma-Clocks, BV-in-Sigma Calculus, and the Jump Product Contraction Law

This paper fixes the sigma-clock kernel used by the later stability, numerical, and certifiedPDE papers in the Sigma Transform Calculus suite. A sigma-clock is a nondecreasing rightcontinuous bounded-variation time change whose clock measure may contain continuous mass, flats, and atomic ticks. The paper defines BV-in-sigma trajectories, records the one-sided bookkeeping needed at atoms, and proves the Jump Product/Contraction Law: a finite sequence of contractive ticks multiplies the energy by the ordered product of the tick contraction factors. The result is deliberately local. It supplies the atomic part of the STC energy machinery, not the full mixed continuous-plus-atomic decay theory, the SBP/SAT discretization theory, or a PDE existence theorem.