A Substitution Theorem for Bounded-Variation Clocks with Atoms and Flats

This paper supplies the substitution layer for Sigma Transform Calculus. It starts with a physical time interval 0, T, a nondecreasing right-continuous bounded-variation clock σ, and the associated Lebesgue-Stieltjes measure µσ = dσ. The central task is to replace integration in physical time against µσ by ordinary integration in a mass coordinate. The required selector must handle three features simultaneously: • absolutely continuous clock advance, where mass time behaves like a change of variables; • atomic clock advance, where a physical-time jump becomes a positive-length mass plateau; • flats, where physical time moves but clock mass does not. The theorem spine is deliberately narrow. It proves a pushforward identity and its substitution consequences. It does not prove existence or uniqueness for measure-driven differential equations, semigroup generation, energy decay, transform inversion, numerical convergence, or PDE well-posedness.