Cosmological Redshift Without Expansion

This paper develops a phenomenological framework in which cosmological redshift is described by a cumulative motion kernel $K(t)$ rather than the FRW scale factor $a(t)$. The framework introduces no scattering or absorption, conserves photon number, and reproduces the three classical background tests of cosmology: the Hubble redshift law, the $(1+z)$ scaling of supernova time dilation, and the $(1+z)^{-4}$ Tolman surface-brightness relation. Unlike tired-light hypotheses, which fail to predict time dilation, and unlike Milne's kinematic cosmology, which leaves surface-brightness scaling theoretically ambiguous, the present framework succeeds on all three counts. This makes it the first non-expansion model to match the full observational triplet that has historically distinguished expanding from static cosmologies. The key distinction is that $K(t)$ is defined operationally from observational invariants rather than imposed as a metric parameter. The analysis shows that background observables do not uniquely require an expanding-space ontology, establishing an observational degeneracy in redshift, distance, and flux relations. The scope is deliberately limited to background-level tests, leaving the question of kernel dynamics for future work, but within this range the framework provides a consistent and empirically grounded representation of cosmic phenomena. The contribution is methodological, it demonstrates that the same empirical laws can be formulated without invoking metric expansion, underscoring the underdetermination of cosmic ontology by background data.