Title : Minimal local antisymmetric dynamics on simplicial complexes
Abstract:
A central question in dynamical systems is how much structure is required to represent conservative, oscillatory behavior. Standard formulations rely on continuous settings, differential operators, or complex-valued formalisms, resulting in representations that are often nonlocal, implicit, or structurally opaque. We show how known properties of conservative oscillatory systems can be constructed from a single local rule, offering a simplified and structurally transparent alternative for representing such dynamics. We construct a minimal local discrete system on simplicial complexes that realizes this class of dynamics directly. A purely local antisymmetric update rule induces a global skew-symmetric operator ???? satisfying ???? = −???????? , without assuming any global operator a priori. This local-to-global structure emerges solely from consistency constraints of the rule. The resulting system guarantees both conservation laws and oscillatory behavior. In particular, a linear invariant ∑ ???? and a quadratic invariant ∑ ????2 arise naturally, and oscillatory dynamics follow directly from antisymmetry, without requiring explicit spectral construction, wave equations, or complex-valued representations. This provides a compressed representation of conservative oscillatory dynamics: the behavior is specified entirely by a local combinatorial rule, with no need for differential operators, global coupling definitions, or auxiliary algebraic structure. The system is fully explicit, local, and computable, and exposes the underlying mechanism generating oscillation as a direct consequence of antisymmetry. Additionally, local consistency constraints impose a global topological structure, effectively enforcing a Möbius-like doubling. This yields a constraint-to-topology emergence phenomenon within the same minimal framework. The contribution is not a new class of dynamics, but a minimal constructive realization that serves as a compact representation framework.
