UDCT v2.9.1 Extension Note: Geometric Reinterpretation of the Hubble Constant, Cosmological Distances, and Effective Cosmic Timescale in a Non-Expanding Framework

This preliminary extension note builds directly on the effective phenomenological framework of UDCT v2.8 (Paik, June 10, 2026), which reinterprets the Hubble constant as an intrinsic geometric relaxation timescale and introduces the higher-dimensional screening correction Δ_geo(z, ξ₀) to resolve the Hubble tension without dark energy or physical spacetime expansion. Following up on v2.8, this note provides explicit operational formulas for: (1) Cosmological luminosity distances (d_{L,UDCT}) for standard candles and high-z galaxies, and (2) The effective geometric lookback time (t_{L,geo}) and a redefined cosmic timescale (τ_UDCT), rendering the traditional Big Bang singularity conceptually unnecessary. These formulas bridge the cross-scale unification from galactic dynamics (v2.6 SPARC analysis) to macroscopic cosmology. This note is part of an ongoing research program and will be followed by successive refined versions (v2.10 onward).