Differential–Integral Symmetry in φ(x−Ct) A Mathematical Basis for Directionality in Space-Time

This work presents a mathematical formulation of directional symmetry in space-time through the function φ(x − Ct), representing a unidirectional propagating wave. By applying the chain rule, we derive a general relation between time and space derivatives of φ for positive integers n and extend this relation to negative indices using iterated Riemann integrals. The resulting symmetry between differentiation and integration provides a dual structure that reflects temporal asymmetry and memory accumulation. This theoretical framework may serve as a foundation for closed-surface physical models and irreversible space-time evolution.

Key words : Directional wave propagation , Closed surface theory , Space-time asymmetry , Fractional calculus , Integral operators , Irreversibility  , Mathematical physics , Memory in field theory