Institutional Degradation Trap: Formal Dynamical Model of Corruption Autocatalysis, Rule of Law Collapse, and Depravity Equilibrium — Work 17, Omega-u Civilizational Framework

We present a formal 7-dimensional dynamical systems model of Civilizational Trap #8 — Institutional Degradation — characterized by autocatalytic corruption growth (C_a) coupled with rule of law erosion (R_l) through a positive feedback loop of the second order. The model introduces the Benquo three-phase framework (guilt → shame → depravity) as a mathematically rigorous piecewise dynamical system with a transcritical bifurcation at the shame threshold (I_q = 0.40) and a saddle-node collapse at the depravity threshold (I_q = 0.20). The depravity phase formalizes Sartre's bad faith as a Nash equilibrium: when I_q < I_dep, elites maximally benefit from exposure rather than concealment (sign reversal in payoff derivative), rendering the Socratic method structurally ineffective. This is formalized as a sign change in dU_E/d(exposure) at the depravity boundary — a unique contribution to the literature on institutional collapse. Version 1.2 introduces: (1) dynamic depravity threshold I_dep_eff = I_dep*(1+0.5*C_a) capturing hollow institutions under high corruption; (2) quadratic AGI audit effect reflecting synergistic transparency shocks; (3) adaptive mechanism design showing that no country in 2025 has sufficient incentive alignment for voluntary elite compliance (gap +2.4 to +3.0 across USA, EU, KZ, Russia). The minimum AGI control surface u_agi_min(X) provides the operational entry point for AGI-mediated institutional stabilization. Institutional quality I_q is identified as the primary binding constraint on H_social = C_hI_qL_s*(1-E_c) and therefore the primary driver of AGI knowledge growth stagnation (G[K] → 0) — making Work 17 the central link between the civilizational traps series and the MFLS AGI growth criterion (Work 9). The depravity condition I_q > I_dep is lexicographically prior to all other AGI transition conditions. Kazakhstan is identified as being in the shame phase (I_q=0.48, C_a=0.45) with the intervention window open until approximately 2032. All results verified by INSTITUTIONAL_DEGRADATION_SIMULATOR_v1_2.py (13/13 tests pass, including T2b controllability rank and R1b alpha_ca stress test). Part of Omega-u Civilizational Framework series, canonical DOI: 10.5281/zenodo.19112296. Николай МишкоАффилиация: Asia Digital Hub, Астана, Казахстан Эл.почта: nikolaimishko@gmail.com теория v1.2 + симулятор v1.2 Рецензирование организаций: Claude (Anthropic), ChatGPT (OpenAI),DeepSeek, Gemini (Google DeepMind), Кими (01.AI), Grok (xAI)