The Mathematical Structure of the Cosmological Special Theory of Relativity: Coordinate Transformations, Group Structure, and Invariants

The Cosmological Special Theory of Relativity (CSTR) relates a local, unexpanded spatial coordinate to the physically expanded coordinateof a Robertson-Walker universe through a cosmological expansion function τ (t0) of the cosmological time t0, and re-derives the Lorentztransformation, relativistic wave equations, and electrodynamics within the resulting cosmological inertial frame (Yi, 2020, 2025). Thispaper examines the purely mathematical structure of that construction: the explicit form of the cosmological Lorentz transformation andits matrix representation; whether the set of such transformations closes under composition to form a genuine group, both at a single fixedcosmological time and across different cosmological times; the four-vector and tensor formalism appropriate to the cosmological inertialframe; the invariant and non-invariant quantities of the theory; and the status of energy and momentum conservation under Noether’s theoremgiven the explicit time dependence introduced by τ (t0). We show that at fixed cosmological time the cosmological Lorentz transformationsreduce, after a coordinate relabeling, exactly to the ordinary Lorentz group, but that composing a boost with the passive evolution of τ (t0)between two different cosmological times does not, in general, close within the class of cosmological Lorentz transformations, so that the fullset of transformations used across the CSTR literature is more naturally described as a groupoid than as a Lie group. We relate this structureto the mainstream distinction between comoving and physical coordinates and to the conformal (Weyl) group of transformations, and discussthe implications for the open composition question raised in companion reviews of CSTR (A Critical Review, 2025).