Topological Phase Transitions and Hodge Theory: Algebraic Foundations of Sphere-Torus Dynamics. Unified Lagrangian for Heaviside Formalism and Algebraic Switching

We establish a profound connection between topological phase transitions (sphere ↔ torus) and algebrai structures through Hodge theory. Demonstrating that toroidal topology encodes Clifford algebra operations generating fermionic statistics and Dirac equations, while spherical topology manifests exterior algebra structures governing bosonic fields and Yang-Mills theories. The Hodge decomposition provides the fundamentalbridge between topological invariants and operator algebras. Experimental verification via quantum simulation confirms the emergence of spin-statistics relations at critical points, resolving long-standing questions about geometric origins of particle physics. We construct a unified Lagrangian for topological phase transitions (sphere ↔ torus) incorporating discontinuous algebraic switching via generalized Heaviside formalism. The topological order parameter Θ triggers Clifford algebra emergence in toroidal phase (Θ = 1) and exterior algebra in spherical phase (Θ = 0). Gauge-covariant derivatives maintain consistency across transitions, with junction terms preserving Bianchi identities. Quantum simulations confirm discontinuity-induced Berry phase accumulation at criticality.