Emergence of Fractional Anyons from a Closed Quartic Variational Structure

This work presents a complete derivation of fractional anyons from a closed quartic variational structure in 2+1 dimensions. No topological term, fractional parameter, or phenomenological field theory is introduced externally. The construction begins from a single integral quartic functional and proceeds through explicit static variation, full Hessian analysis, and global stability selection via a decision operator. Finite energy vortex solutions arise naturally under dimensional restriction. Spectral analysis demonstrates that the first stable non trivial minimum corresponds to a Z3 internal structure. From this structure emerge, without additional assumptions, fractional statistical phase, fractional charge equal to one third of the elementary charge, and Hall conductivity equal to one third of the quantum unit. Integration of massive modes generates an effective topological term of Chern Simons type with coefficient fixed by the internal winding. No parameter is inserted by hand. Extension to a degenerate internal sector produces pairing and a reduced spectral gap, yielding a non Abelian braid representation with explicit two by two unitary exchange matrices. The energetic hierarchy follows directly from the Hessian spectrum: the one third sector is the first robust gapped minimum, while paired sectors possess parametrically smaller gaps. The Z2 sector is shown to be marginal and gapless. The entire structure — fractional statistics, fractional charge, effective topological response, non Abelian pairing, and filling hierarchy — emerges from the same closed variational origin. The framework is internally consistent, spectrally stable, and requires no phenomenological input.