Yauni's Algebra: Admissibility Constrained Partial Composition and Identity-Violating Operations

Classical algebraic systems presuppose total composition or encode failure through internal error elements such as nulls or zeros. This assumption breaks down in systems where certain operations are not merely invalid but structurally inadmissible, and where some transformations terminate identity persistence itself. This work introduces Yauni’s Algebra, a foundational framework for admissibility-constrained partial composition. In this setting, admissibility is treated as a primitive structural property rather than a derived logical predicate, and undefined composition is interpreted as genuine non-existence rather than as a representable algebraic value. Identity-violating operations are shown to be terminal and non-extendable within any faithful algebraic completion. A minimal axiom system is presented, followed by explicit finite models demonstrating realizability. The theory establishes existence and boundary theorems for maximal admissible regimes, proves the impossibility of faithful total completion, and characterizes invariant structure via reachability classes and a regime-dependent entropy measure. Attractor existence and mixing behavior are analyzed in the finite case. Yauni’s Algebra is explicitly falsifiable through concrete counterexamples and delineates a sharp boundary at which algebraic reasoning ceases to apply. The framework is intended as a foundations level contribution, clarifying the limits of algebraic composition in identity-bearing systems and distinguishing structural non-existence from value coded failure. Name dedication to my heart. My niece; for breaking generational curses that may never multiply.