Hierarchical Control of Compositional Systems: Three-Tier Architecture with Structural Transformation

This work establishes c-DME (Control Descriptive Medium Eta), a hierarchical architecture for robust regulation of compositional systems through indirect admissibility modulation. While the Phase Transitions in Descriptive Medium Eta (PTDME) framework identified dual collapse boundaries—λ_c = 0.5 (diversity loss, structurally reproducible) and λ_Babel = 3.1 (coherence breakdown, accessible only in temporal formulations)—defining the viable Compositional Regime, operational steering mechanisms enabling systems to reach and maintain safe regions remained unaddressed. We resolve this gap by validating the γ→λ causal relationship—demonstrating that looseness parameter γ governing threshold functions provides effective indirect control over compositional dynamics—and revealing fundamental architectural constraints through comprehensive parameter space analysis. The central discovery establishes the scale-dependent geometry of Structural Closure, where control derivatives |∂λ/∂γ| vanish, rendering continuous regulation ineffective. Minimal systems (n=5 expressions) exhibit 96% Structural Closure coverage, necessitating discrete structural transformation (Tier 3) as the dominant operational mode. However, realistic systems (n=50) invert this hierarchy: Structural Closure decreases to 64% while viable corridors—parameter regions supporting continuous admissibility modulation (Tier 2)—expand from 4% to 35.9% of parameter space. This scale-dependent inversion challenges conventional control paradigms treating structural change as exceptional backup rather than recognizing it as the primary necessity in resource-constrained environments while continuous regulation dominates in complexity-rich settings. Rigorous Lyapunov stability analysis guarantees exponential convergence within viable corridors, achieving near-perfect theoretical-empirical agreement (R²=1.000 to six significant figures, decay rate κ=2.181) distinguishing mathematically certified regulation from heuristic approaches.