Hexad Computation: From Non-Abelian Emergence to a General Framework of Computation

                                                                                                 Abstract

    Non-Abelian Emergence proved that any coupled system satisfying Axioms 1–5 necessarily gives rise to a gauge group structure. On this basis, this paper answers a deeper question: how do these group structures automatically perform computations internally? We propose a general framework of Hexad computation: taking non-Abelian emergence as the "computational stage", using isoperimetric optimality to drive iterative rule composition, and forming a convergent evolutionary dynamics through the gradient of information density.

Core contributions:

1. Hexad computational structure: Hsystem = ⟨M, A0, g, D, I, T⟩ is precisely defined as a universal computational framework; the rule composition cycle ⊗ combines the information density gradient and the structural method, driving the system toward a stable attractor.
2. Bootstrap completeness theorem: For any computational problem placed within the Hexad framework, its information density is monotonically increasing and bounded, hence it necessarily converges to a steady state.
3. Five general laws of computation: information irreducibility, phase closure, isoperimetric optimality, rotating vector encoding, and bootstrap completeness—automatically derived from the Hexad computational structure, constituting general principles of computation across domains.
4. Complete verification by number-theoretic instances: primality testing, the prime number theorem, perfect number generation, and the Riemann critical line—all emerge automatically from Hexad evolution, with zero dependence on external algorithms.
5. Hierarchical relation with the L-system: Non-Abelian Emergence (L3) establishes the computational stage; this paper (L3 deepening) performs the computational drama on that stage.

Keywords: Hexad computation; non-Abelian emergence; rule composition; attractor; bootstrap completeness; isoperimetric optimality; rotating vector; cross-domain unification