FCU–002: Entropy as Relational Variance – Toward a Structural Definition of Entropy in Finite, Non-Equilibrium Systems

This working paper extends the Finite Cyclical Universe (FCU) framework by proposing a structural reinterpretation of entropy. Rather than defining entropy primarily as disorder or as a count of microstates, entropy is described here as the distribution of relational variance across a structured system. Let Vi denote the relational variance at node i, and let Lij encode the connectivity between elements of the system. The evolution of the system follows a non-equilibrium dynamical equation combining diffusion, nonlinear growth, and stochastic fluctuation. Within this framework, entropy is not a terminal quantity, but an evolving functional of variance distribution. A central proposal of this work is the entropy functional: SV = Σi log( σi² ), with σi² = Σj Lij (Vi − Vj)² In the continuous field limit, this becomes: SV = ∫ log( V(x) · (−∇²V(x)) ) dx This formulation captures entropy as a logarithmic measure of the structure of variance across relations or fields. It unifies discrete (network-based) and continuous (field-based) perspectives within a single conceptual framework. Three fundamental processes define the evolution of entropy in this system: → Diffusion redistributes variance and promotes global coherence→ Nonlinear growth localizes variance, enabling structure formation→ Fluctuation maintains dynamical openness and prevents equilibrium collapse This leads to a reinterpretation of entropy as a dynamic balance between smoothing, structuring, and renewal processes. The framework suggests that stability in physical systems is not characterized by the absence of change, but by the regulated coexistence of global coherence and local instability. In this view, entropy is not merely a measure of disorder, but the evolving architecture of variance across relational structures. Potential implications of this approach include: → A bridge between thermodynamics and network/field dynamics→ A structural perspective on non-equilibrium systems→ New interpretations of pattern formation, coherence, and cyclicity in cosmology→ A possible extension of entropy concepts beyond equilibrium statistical mechanics Entropy, in this framework, is not the measure of disorder, but the measure of how variance is structured across relations. This work remains exploratory and is intended as a conceptual and mathematical proposal for further investigation.