The L–E–Ω Model: Geometric Constraint Theory for Cognitive–Affective Dynamics

This document presents the L–E–Ω Model, a structured hypothesis framework for analyzing cognitive–affective dynamics through constrained dynamical systems theory. The model introduces a three-dimensional state space (L, E, A) representing phenomenological cognitive–affective variables and formalizes Ω(t) as a geometric boundary operator that constrains admissible trajectories. The framework does not treat Ω as a latent dynamical variable, control gain, or metaphysical construct; rather, Ω parameterizes the boundary manifold that ensures forward invariance and boundedness under specific nonlinear coupling regimes. The central contribution of the model is conditional: If unconstrained L–E–A dynamics exhibit finite-time divergence or lack a global Lyapunov function for realistic parameter regimes, then boundedness must be structurally justified via explicit constraint geometry. If global boundedness can be constructively demonstrated without constraint, Ω becomes unnecessary and must be removed. The document includes: Logical preconditions for persistence and structural encoding Explicit instability construction for unconstrained nonlinear coupling Lyapunov-based analysis of boundedness Formal definition of Ω as a boundary operator in ℝ³ Projection operator and forward invariance theorems Operational detection criteria and empirical testing protocol Explicit falsification and removal conditions Epistemic Status: Structured hypothesis with model characteristics (estimated epistemic gravity ≈ 0.55). The framework makes no claim to grand unification or ontological primacy. It proposes a specific geometric solution to a specific boundedness problem in coupled cognitive dynamics. Keywords dynamical systems constraint manifolds cognitive-affective dynamics geometric stability boundedness theory nonlinear coupling Lyapunov analysis 🖤