Derivation of the Cosmic Dark Matter Ratio from Shannon Entropy and Born's Rule

A conditional derivation of the cosmic dark matter-to-baryon density ratio from Shannon's entropy uniqueness theorem and Born's rule. The normalized baryonic probability amplitude sources a logarithmic scalar field whose entropy critical point yields the ratio 2e ≈ 5.437, consistent with Planck at 1.1σ. Landauer erasure and the Euclidean on-shell action amplitude yield a fully derived, zero-parameter interpolation function that outperforms the standard MOND baseline on 175 SPARC galaxies at 5.9σ. The thermodynamic equilibrium prediction for cluster lensing mass ratios matches observations to 3%, and the background CMB cosmology is indistinguishable from ΛCDM at current precision.