A Formal Viability Framework for Living and Self-Reproducing Systems: A Categorically Minimal Axiomatic Theory Integrating Autopoiesis, Dissipative Thermodynamics, Holland Tag Automata, and the Topological Fold

We present a mathematically closed, axiomatically minimal formalization of living and self-reproducing systems by synthesizing five foundational pillars: (1) Maturana–Varela autopoietic boundary closure, (2) Prigogine far-from-equilibrium dissipative thermodynamics, (3) Holland's tag-mediated addressing and viability-grounded arbitration, (4) Von Neumann's dual-modality tape architecture, and (5) Barbieri's biosemiotic organic codes. By lifting internal state space to a smooth Riemannian Viability Manifold (M_V, g) governed by Aubin's Tangency Condition and modeling the translation apparatus as an active recursive constructor, we eliminate ad-hoc teleological reward engineering. We resolve the infinite meta-computational regress of arbitration through the Topological Fold—a Lawvere metric closure inducing non-trivial Riemann curvature and an egocentric phenomenal present (hic et nunc). Furthermore, we establish the Sensory Restriction Principle, proving that omnidirectional perception causes fatal dynamical paralysis via dense saddle networks and Poincaré dithering, whereas directional Line-of-Sight truncation guarantees autopoietic persistence. Finally, we provide an Ehresmann categorical sketch proof demonstrating that the five generating morphisms are strictly irreducible, independent, and initial.