The Operator Algebra of Physical Relations

Recent work introduced a dynamic operator framework extending the symbolic language of physics through four relational operators designed to describe state-dependent evolution: the Fluctuating Equivalence Operator (\( \fleq \)), the Cyclic Equivalence Operator (\( \ceq \)), the Phase Nexter Operator (\( \pnex \)), and the Phase Reverser Operator (\( \prev \)). These operators provide compact symbolic representations of fluctuation, cyclic reset behavior, forward phase skipping, and reverse evolution in physical systems. While the original formulation focused on conceptual definitions, the present work develops an algebraic structure governing their interactions. We introduce an operator algebra that describes how these dynamic relational operators combine, cascade, transform, or compete within a physical system. Physical evolution is thereby represented as sequences of relational transformations rather than isolated static equalities. The algebra captures key transitions: fluctuations accumulating into cyclic resets, cyclic dynamics generating forward phase advancements, and unstable phases triggering reversals. These relational cascades enable compact description of complex behaviors such as nonlinear oscillations, cosmological phase changes, instability-driven reversals, and threshold-mediated evolution. The proposed algebra provides a formal mathematical foundation for dynamic physical relations and supports applications in emergent spacetime models, nonlinear dynamics, and quantum gravitational frameworks.