The Intervention Paradox: Why Direct Governance Near Criticality Fails — A Coarse-Grained Potential Theory

Well-intentioned interventions frequently worsen the crises they seek to prevent: financial rescues can amplify systemic risk; organizational restructurings trigger departure cascades; ecological management interventions push ecosystems past tipping points. This empirical regularity — observed across management science, ecology, and financial economics under names such as the change paradox, intervention-induced tipping, and signaling-induced contagion — has lacked a unified structural explanation. This paper provides that unifying mechanism within a coarse-grained potential model. Systems operating near self-tuned criticality (Paper I of the DFG Trilogy) reside in shallow potential wells V (X) = −AX² + Bₚot·X⁴ whose barrier height vanishes at the critical manifold: ΔV ~ |ξ| → 0 as ξ → 0, since A ~ |ξ|^1/2. Direct interventions above a magnitude threshold act as energy injections enabling Kramers escape over the barrier — triggering the very cascade they aimed to prevent. Theorem 1 (Intervention Paradox): Above Dₜhreshold, the probability of collapse increases monotonically with intervention magnitude within the model. The stochastic operational threshold Dₜhreshold = 2·sqrt (A·Bₚot) ·G (Xc) ·σ_η shrinks as Dₜhreshold ~ |ξ|^1/4 → 0 as the system approaches criticality — meaning the safe intervention window collapses precisely when crises appear most imminent. Two distinct threshold formulas are derived from the same Kramers escape framework under different noise-scaling assumptions: Dgeom = A²/ (4Bₚot·|Xc|) is the barrier-elimination criterion in the ungated zero-noise limit (G = 1) ; Dₜhreshold is the stochastic operational threshold incorporating noise amplification through the logistic gating function G (Xc) = 1/2. Dₜhreshold is interpreted as a phenomenological scaling ansatz — a rigorous Fokker-Planck derivation under non-conservative forcing is identified as future work. Theorem 2 (Buffer-First Principle): Buffer depth increases are monotonically stabilizing within the coarse-grained model class for all Δb > 0: d (ΔV) /d (Δb) > 0. Buffer investment unconditionally deepens the potential well and expands the safe intervention window, making it categorically superior to direct state forcing near criticality. Recommended Three-Phase Governance Protocol (Corollary of Theorems 1–2): (1) maximize buffer depth b first, (2) apply bounded direct intervention ||U|| ≤ Dₜhreshold, (3) transition to structural steering if ξ persists. The rationale follows directly from Theorems 1 and 2; the protocol is not formally proved optimal. Key scaling relations near criticality (A ~ |ξ|^1/2, mean-field): Barrier height: ΔV ~ |ξ| → 0 Well minimum: Xc ~ |ξ|^1/4 → 0 Geometric threshold: Dgeom ~ |ξ|^3/4 → 0 Stochastic threshold: Dₜhreshold ~ |ξ|^1/4 → 0 (Dₜhreshold > σ_η². Near criticality (ΔV → 0), the system enters a crossover regime where classical Kramers asymptotics break down; the analysis is therefore a scaling argument indicating that intervention-induced barrier reduction accelerates transitions, not an exact escape-rate formula. The qualitative conclusion Dₜhreshold → 0 as ξ → 0 is robust to this limitation. The effective potential Vₑff (X) = V (X) − G (X) ·U·X is a first-order approximation near the well minimum; U is not a conservative force. The Buffer-First unconditional claim holds within the model class only. The order parameter X is related to the critical coordinate ξ of Paper I through a smooth reparameterization X = f (ξ) with f (0) = 0, representing the normal-form reduction of the coordination dynamics near the bifurcation point and formalizing the connection between the GGT critical coordinate and the double-well potential structure. This paper is Series Paper III of the Deficit-Fractal Governance (DFG) Framework. Together with Paper I (self-tuned criticality) and Paper II (cascade universality), it completes the trilogy: critical coordination systems simultaneously generate universal power-law cascades, restrict safe intervention magnitude to zero at criticality, and require logarithmic hierarchy depth — three structural consequences of the same coordination balance Γ ≈ Γc.