The coupling α* = 1/(4 ln 2), fixed by Bekenstein–Hawking in P1 and used without recalibration to derive nine fermion masses (mean 2.4%, P2) and m_H/m_W = u⁵ (+0.73%, P3), is extended to the cosmological network. Six new predictions are established from α* alone: (i) δG/G = (c−b)/C(5,2) = 1.353% (Lamine et al. 2025: 0.001σ); (ii) a₀ = αcH₀/2 = 1.2002 × 10⁻¹⁰ m s⁻²; (iii) H₀^K₅ = 68.49 km s⁻¹ Mpc⁻¹ (DESI 2024: <0.02%); (iv) η^K₅ = 5.929 × 10⁻¹⁰ (−2.87% Planck); (v) a dark-matter acceleration law g_DM = (a₀/α) z e⁻ᶻ with z = α* x^γ(x) and γ = 1/2 + (1/|E(K₅)|) e^−√x, derived from the face-excitation ontology of P2–P3; (vi) a composition modulation and network smoothing whose six coefficients are all combinatorial invariants of K₅. The same α* has now traversed 61 orders of magnitude in length — from Planck-scale entropy to the Hubble horizon — without adjustment. No other framework predicts entries in more than one row of the master table. Result (v) surpasses the empirical MOND interpolation (χ̃² = 2.39 vs 2.82 on 171 SPARC galaxies, same degrees of freedom). Result (vi) reduces χ̃² to 2.11 (102/171 wins, 17% gap closure toward NFW); a sensitivity scan confirms that the composition coefficient c_comp = 3/10 = |F₃|/|E(K₅)| falls within the broad empirical minimum (Δχ̃² < 0.08). A uniqueness theorem establishes that K₅ is the only complete graph for which the face–edge exponent gap equals 1/|E| (Theorem VII.1). The companion verification script (105 tests, all PASS) provides a reproducible audit trail from axioms to predictions. Zero free parameters, 61 orders of magnitude, one coupling.