A Unified Variational-Constraint and Feedback Framework for Cosmological Low Entropy, Gravitational Order, and the Emergence of Time

Abstract The origin of the universe's low-entropy initial condition remains one of the most fundamental open problems in cosmology. While standard physics describes entropy evolution forward in time, it provides no dynamical explanation for why the initial value was so exceptionally small. This work proposes a unified variational and constrained-systems framework in which the universe is modelled as a dynamical system evolving on a constrained state manifold. The constraint set C(t) is rigorously defined as a measurable subset of phase space, and its evolution is derived from an explicit feedback functional via a gradient-flow equation. Entropy is reinterpreted as the logarithm of the Liouville measure of the accessible region under constraints. We prove a monotonic entropy-production theorem, derive an explicit Lyapunov function establishing stability of the dynamic equilibrium, and show that the thermodynamic arrow of time emerges from irreversible constraint relaxation. Extensions include gravitational entropy via the Weyl curvature hypothesis, black hole thermodynamics, and a generalized Universal Balance Principle connecting constraint reduction to entropy increase.