The Unitary Symmetry Series: Mathematical Formalization of Multiplicative Equilibrium and Reflexive Scaling Topologies

This paper presents the formalization of the Unitary Symmetry Series (USS), a purely multiplicative geometric sequence anchored to a Unity baseline $U_0$ = 1.0. While classical additive models often suffer from uncontrolled divergence, the USS framework utilizes exact reciprocal fractional operators—$\alpha$ = 1.25 (Expansion) and $\beta = 0.8$ (Contraction)—to maintain systemic equilibrium. Key Mathematical Insights: The Unity Baseline: Establishes the invariant anchor $U_0$ = 1.0 for all topological transformations. Unitary Additive Identity: A rigorous proof demonstrating that the ratio of linear expansion to contraction is locked to the expansion constant: $\frac{\Delta_{exp}}{\Delta_{con}} = \alpha$. Theorem of Reflexive Power Scaling: Formalizes the identity $\frac{\alpha^n}{\beta^m}$ = $\alpha^{n+m}$, enabling seamless dimensional scaling without informational loss. Topological Archetypes & Applications: The framework is mapped onto four fundamental physical archetypes: Radial Vortex Dynamics: Conserving spatial density and angular momentum via reciprocal radial vectors. Harmonic Oscillation: Modeling homeostatic rhythms (e.g., cardiovascular cycles) through the pulse function P(t) = $\alpha^{\sin(\omega t)}$. Recursive Branching: Optimizing fluidic distribution and structural growth according to Murray’s Law. Wave Interference: Establishing stable informational nodes on a spacetime grid through frequency saturation. Grounded in the theoretical constraints of the Anadihilo unified space-grid, this research provides a standardized mathematical vocabulary for spatial conservation and structural stability in complex dynamic systems.