Toward a Quantitative c_eff(a): Parameterization, Constraints, and Degeneracy-Breaking Tests

We introduce a quantitative parameterization for a possible state-dependent cosmological propagation limit within Energy-Flow Cosmology (EFC). Following earlier results showing that EFC’s late-time coupling operates through the background expansion sector rather than the Poisson equation, consistency of the framework suggests that the same vacuum-state modification affecting H(a) may also weakly affect the effective signal propagation speed. We therefore define a minimal first-order modelcₑff(a) = c₀[1 + ε β T(a)],where β is the empirically constrained background coupling amplitude (β ≈ 0.08), T(a) is the established EFC transition function that vanishes when Ωₘ → 1, and ε is a new dimensionless parameter describing how strongly propagation responds to the vacuum-sector change.We analyze observational consequences under two physically distinct implementations: (1) a metric (geometric) implementation, where cₑff modifies the conformal structure and remains degenerate with H(z) in distance-only probes; and (2) an optical-medium implementation, where the vacuum acts as an effective refractive medium, potentially violating the cosmic distance duality relation (CDDR) with a characteristic sign change near the redshift where T(a) peaks.We identify degeneracy-breaking tests that can constrain or falsify the parameter ε. The most robust are: • a β-split test comparing distance-derived and growth-derived coupling amplitudes, • standard sirens comparing gravitational-wave and electromagnetic luminosity distances, and • (conditionally) CDDR measurements if the optical-medium interpretation is realized.Current cosmological data are consistent with ε ≲ O(1) and provide no evidence for a varying propagation speed. The framework therefore represents a testable phenomenological extension, not a detection of new physics. The primary outcome of this work is to translate a qualitative consistency requirement into a quantitative, falsifiable parameterization and to map the observational pathways by which future surveys (DESI, Euclid, Rubin/LSST, and next-generation gravitational-wave detectors) can decisively probe it.