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We present a rigorous mathematical demonstration that cosmic evolution is governed by optimal informational transport (Wasserstein geometry) on the Riemann Oscillatory Network (RON), not by metric expansion. We show that thermal entropy is the Radon–Nikodým derivative of RON informational entropy, that redshift emerges as cumulative spectral drift along transport geodesics, and that the Rényi family of entropies completely describes the multiscale, non-ergodic nature of the system. We demonstrate that RON, indexed on the zeros of the Riemann ζ function and governed by the Dynamic Zero Operator (DZO), produces all current cosmological observations (Hubble tension, CMB low-ℓ anomalies, JWST premature structure, BAO drift) as natural emergences, without requiring expansion. Central result: The Universe does not expand and does not die thermally — it self-organizes informationally through progressive compaction onto dynamically viable structures. We propose a decisive test: redshift dependence on spectral band (Δz/z ~ 10⁻⁴–10⁻³), measurable with DESI+Euclid+JWST, which can definitively falsify the ΛCDM paradigm. We explicitly show that ΛCDM emerges as an observational limit of NMSI in the coarse-resolution regime.