Cosmological Predictions of the GTE/Phi_MDL Framework: Dark Energy, the CMB Spectral Tilt, and Gravitational Signatures from First Principles

We derive the principal cosmological observables of the Generative Triple Evolution (GTE) / Phi_MDL framework from zero-parameter first principles and certify the central algebraic steps in Lean 4. The observed dark-energy fraction is fixed by two structurally independent routes: a Perfect Self-Containment (PSC) reflexive-closure count giving Omega_Lambda = (2/3)_2(2000/3) = 0.6899 (0.71 from Planck 2018), and a three-tape holographic mode count giving Omega_Lambda = (/2) = 3/14 = 0.6732, with the proper-time rate = 3/7 derived from Rule-110 ether dynamics. The two routes bracket the observed 0.6889 from above and below; the 2.4 transcendentals). The associated energy density obeys rho_Lambda = (9/112) M_Pl^2H_0^2, agreeing with observation at the few-percent level; its dimensional structure M_Pl^2H_0^2 coincides with the Cohen–Kaplan–Nelson holographic scaling, but a PSC boundary condition fixes the equation of state at w = -1, evading the w = 0 obstruction that defeats holographic dark energy. Holographic mode counting (3L modes, not L^3) suppresses one-loop corrections to the constant by 10^-42 relative to the standard estimate, addressing the quantum-protection question. We further derive the CMB scalar spectral index n_s = 1 - 2/(2^2) = 0.96488 (0.004 from Planck 2018) from a binary holographic running rate, certified by fourteen zero-sorry Lean theorems; the Newton-constant normalization from a discrete Ollivier–Ricci (Gorard) curvature chain (k_SD = 10/13); Hawking emission as kink radiation near the BPS critical mass M_kink^ crit = 290.10 MeV; a strong-field ultraviolet bound V = 2m^2/49; the neutrino sum m = 59.4 meV; and the CKM CP phase _CP = /2 - 3/8 = 68.51^ (0.017 Lean 4 with zero sorry. We give an explicit falsifiability profile.