What question did this study set out to answer?

To create a substitution layer in Sigma Transform Calculus for integration involving bounded-variation clocks.

May 24, 2026Open Access

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

Key Points

To create a substitution layer in Sigma Transform Calculus for integration involving bounded-variation clocks.
Developed a framework for integrating in mass coordinates instead of physical time.
Addressed the behaviors of absolutely continuous clock advance, atomic clock advance, and flats.
Established a narrow spine for the theorem focusing on pushforward identities.
Outlined key features affecting integration, emphasizing the simultaneous treatment of diverse clock behaviors.
Proved important identities but did not provide results on measure-driven differential equations or numerical aspects.

Abstract

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.

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

Key Points

Abstract

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