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 Υexc ≡ (Mdyn-Mbar)/Mbar 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, Υexc(Ncoh,T) = c₁ max[0, ln(Ncoh/N₀)] with Ncoh = M_* + βgas(T)Mgas and βgas(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 (Υexc ≈ 0.2), Milky-Way-class galaxies (Υexc ≈ 7-10), cool X-ray groups at T ≈ 1 keV (Υexc ≈ 13), the Bullet Cluster dynamical-vs-baryonic mass partition (Υexc ≈ 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 Dₘₐₓ = 0.98 ln N - 0.10 (R²=0.98) while Minkowski causal-set controls at 3+1D and 2+1D follow N1/4 and N1/3 respectively, matching continuum theory to 10⁻². We additionally show that DAGI's high-density saturation ρeff(ρ) is consistent with all current neutron star observational constraints for ρₛₐₜ ∈ [8 × 10¹⁴, 10¹⁶] 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).