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

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