Logarithmic Gravitational Amplification from Trace-Log Fredholm Reduction: A DAGI Derivation of Multi-Scale Dark-Matter Phenomenology

We derive an empirically validated, three-anchor logarithmic scaling law for the dark-to-baryonic excess mass ratio ₄ₗ₂ (M₃ₘ₍-M₁₀ₑ) /M₁₀ₑ across seven orders of magnitude in coherent baryonic mass, within the Directed Acyclic Graph Interpretation (DAGI) framework of emergent gravity. The derivation (Theorem 1) follows from evaluating the one-loop Trace-Log Fredholm determinant of the renormalized effective action established in the companion cosmology paper, specialized to a bounded-fan-in causal cone. The resulting formula, ₄ₗ₂ (N₂₎₇, T) = c₁ 0, (N₂₎₇/N₀) with N₂₎₇ = M_* + ₆₀ₒ (T) M₆₀ₒ and ₆₀ₒ (T) = 1/ (1+cT) (Caldeira-Leggett thermal decoherence), is the direct structural analog of the logarithmic running of the fine-structure constant in QED vacuum polarization, with the DAGI causal graph's "synergy polarization" playing the role of the photon self-energy. With two empirical calibration constants (c₁ 0. 87, N₀ 8 10⁵ M_) and a plasma-state-dependent decoherence coupling c, the formula provides a compact description of globular cluster dynamics (₄ₗ₂ 0. 2), Milky-Way-class galaxies (₄ₗ₂ 7-10), cool X-ray groups at T 1 keV (₄ₗ₂ 13), the Bullet Cluster dynamical-vs-baryonic mass partition (₄ₗ₂ 50), and the full Chandra ACCEPT cluster sample (320 observations across T 0. 6, 19 keV) at 0. 05 dex residual. The log-depth scaling underlying Theorem 1 is independently validated by a three-family causal-graph simulation: DAGI-consistent bounded-fan-in graphs exhibit D₌₀ₗ = 0. 98 N - 0. 10 (R²=0. 98) while Minkowski causal-set controls at 3+1D and 2+1D follow N^1/4 and N^1/3 respectively, matching continuum theory to 10^-2. We additionally show that DAGI's high-density saturation ₄₅₅ () is consistent with all current neutron star observational constraints for ₒ₀ₓ 8 10^14, 10^16 g cm^-3 and provides a natural rescue mechanism for soft equations of state. The low-acceleration galactic-scale regime is treated in the companion paper (K3-B).