V87: From Brownian Update Dynamics to Sector-Dependent Activity — A Multi-Scale Consistency Extension of the Unified ρ-Engine

V87 extends the YAGC (Yielding Analog-Generated Cosmos) cross-scale consistency framework by establishing a rigorous connection between local Brownian dynamics and cosmological parameters. We derive Ornstein-Uhlenbeck (OU) process parameters (θ ≈ 0.46, μ ≈ 0.62, σ ≈ 0.07) directly from V16 phase-space dynamics, and show that the activity coupling γ_local = 0.5 emerges mathematically from the boost coefficient—not as a fitted parameter. This derived value exactly matches the V85 cluster prediction, confirming theoretical consistency. Forcing a single γ across cluster and cosmological scales reveals a 326% tension: clusters require γ ≈ 0.5 for the 8.4% temperature enhancement, while cosmology requires γ ≈ 2.0 to reproduce the Planck DE/DM ratio (≈72/28). This "Detected Inconsistency" is not a failure but a diagnostic success predicted by V85, indicating that activity definitions must be sector-dependent. We resolve this tension by formalizing sector-dependent activity A_s and coupling γ(s) = γ₀ × (N_s/N₀)^β with β ≈ 0.01. This very weak scaling explains why local and cluster scales share γ = 0.5 while cosmological scales require γ = 2.0. Extrapolating to the micro-sector yields γ_micro ≈ 3.0 ± 0.4, with the top quark (y_t ≈ 1) serving as a saturation node—the particle-physics analogue of black holes at cosmological scales. We establish a mathematical identity between the V87 OU process and V51 heartbeat random walk, showing that primordial fluctuations (δρ/ρ ~ 10⁻⁵) and V16's phase-diagram "purupuru" share the same origin: fluctuations in information certainty. The YAGC axiom "static = non-existence" finds quantitative expression in α₀ > 0 (vacuum breathing value), meaning updates cannot stop. This work is intentionally phenomenological; first-principles derivation of β and the microscopic origin of fluctuations are deferred to V88.