Symmetria: Trinary Symmetry, Trions, and the Geometry of Neutrality and Polarity

This paper introduces Symmetria, a new mathematical framework for analyzing systems composed of two opposing components mediated by a neutral element. Such triadic structures appear in opinion dynamics, physics, classification theory, and information geometry, yet they lack a unified representation. We formalize these systems as Trions, ordered triples (L, N, R), and define their invariants: mass, commit, and lean. We prove that all admissible Trions project into a Transition Cone, a convex polyhedral domain characterized by normalization, edge, and mirror symmetries. Within this setting, we derive structural identities linking invariants, including the quadratic coexistence law and purity-balance relation, and introduce alternative coordinate systems that reveal neutrality as a contraction operator. Geometrically, we establish convexity, boundary characterizations, and a rectangular representation of the state space. Dynamically, we analyze trajectories confined to the cone, oscillatory states described by spectral ellipses, and the role of the Symmetria Laplacian in harmonic analysis on cones. Case studies demonstrate applications to political polarization, statistical classification with abstention, two-phase physics with disorder, and noisy communication channels. Together, these results show that Symmetria is elementary in definition yet fertile in consequence: a coherent analytic framework uniting neutrality, polarity, and symmetry.