Foundations of Sigma Transform Calculus: Clocks, Observables, Gauge, and Morphisms

Sigma Transform Calculus studies systems whose evolution is driven by a bounded-variation clock rather than by a smooth time parameter alone. Such clocks may have an absolutely continuous component, atomic event mass, and physical-time intervals carrying no clock mass. This paper fixes the foundational interface for the series. It defines the clock measure, the mass-coordinate selector, sigma-adapted trajectories, gauge representatives, physical observables, units, model identifiers, and dictionary-preserving morphisms. The main point is not to prove a particular evolution theorem; it is to make later substitution, derivative, energy, transform, numerical, and PDE results stateable over one stable object language. The paper also records necessity examples showing that selector choices, gauge choices, measurement maps, and data-split rules are mathematical parts of the model identity rather than cosmetic presentation choices.