Social Instability Trap: Formal Dynamical Model of Elite Extraction, Middle Class Erosion, and Civilizational Legitimacy Collapse — Work 15, Omega-u Civilizational Framework

We present a formal 7-dimensional dynamical systems model of Civilizational Trap #3 — Social Instability — characterized by dual autocatalytic instability channels: inequality growth (Gini coefficient Gₛ) and elite institutional capture (Ec). The model introduces a 7D state space: Gini coefficient, middle class income share, elite capture index, social legitimacy, human cognitive capital, non-productive allocation share (IEP master variable u), and institutional quality. Two independent positive eigenvalues (g1 = 0. 0176, e1 = 0. 0112, lambdaₘaxₙumeric = +0. 018) are proven via spectral continuity from decoupled subsystems and confirmed numerically (QR algorithm). The uncontrolled system converges to one of three boundary attractors: oligarchy, failed state, or AGIFROZEN — the latter directly linking social collapse to AGI knowledge growth stagnation via the MFLS criterion (Work 9). Trap #3 is formalized as the post-1991 Nash equilibrium of democratic capitalism without an external competitive mirror (socialism), where alpha >> beta makes elite extraction the dominant rational strategy. A dynamic Nash extension shows that collapse penalty k > 1 on projected future legitimacy shifts the equilibrium toward moderated extraction. An 8-lever proportional feedback controller U* (X) is specified with Lyapunov analysis and empirical calibration from Piketty-Saez (2014), Gilens-Page (2014), V-Dem Institute, and Pew Research (2021). Bifurcation structure includes saddle-node legitimacy collapse (normal form derived), transcritical class replacement, and pitchfork media capture. A 5-level Early Warning System (Dₜrap metric) and AGI-feedback loop (deltaₐgiₑffective threshold model) complete the framework. All results verified by SOCIALINSTABILITYSIMULATORᵥ1₄. py (13/13 tests pass, including elite sabotage T9 and closed-loop diagnostic T8). Part of the Omega-u Civilizational Framework series, canonical DOI: 10. 5281/zenodo. 19112296. Николай МишкоАффилиация: Asia Digital Hub, Астана, Казахстан Эл. почта: nikolaimishko@gmail. com Рецензирование организаций: Клод (Anthropic), ChatGPT (OpenAI), DeepSeek, GLM, Кими (01. AI), Grok (xAI)