Lemma 1: Optical Equivalence of Inward Scale Damping to Metric Expansion

We establish Lemma 1 of the Fundamental Pocket Model (FPM) framework, providing a mathematical proof that an inward recursive scale-damping metric across discrete iteration tiers produces an optical reference frame identical to FLRW metric expansion. We derive the exact cosmological redshift relation , the logarithmic tier differential, and the FPM luminosity distance formula . This demonstrates that outward cosmological expansion is an optical manifestation of an inward recursive geometry.