Generative Polarity and Structural Emergence: A Common Algebraic Architecture in Foundational Physical Formalisms

Several foundational formalisms of theoretical physics — the Hamiltonian formalism, the Schrödinger equation, the Klein–Gordon scalar field, the Dirac spinor structure, and the Maxwell propagation equations — share an algebraic architecture that has not been identified as a common structure in the foundations of physics literature. Each of these formalisms describes a system through the interplay of two complementary domains and two transition operators governing the passage between them. This paper identifies and formalizes this common architecture as the G→X→Q→N cycle, where G designates the centripetal generative domain, Q the centrifugal manifest domain, X the torsional transition operator, and N the return operator. The operators X and N are defined rigorously as endomorphisms on a graded vector space V = VG ⊕ VQ, with three algebraically derived properties: anti-diagonal action, involutive torsion (X² = −IdV), and projection-contraction return (N = πG − ε·πQ). The self-composition law (N∘X) ·A·U = A²·U is derived as a theorem. The Möbius topological representation of the cycle is established as a direct algebraic consequence of X² = −IdV. The structural isomorphisms with the physical formalisms are discovered — they follow from independently derived algebraic properties — not constructed. The paper examines implications for the problem of time (time as generated by the X transition, consistent with the Page-Wootters mechanism) and the measurement problem (reformulated as N incompleteness). The paper does not propose a new physical theory; it identifies a formal structure common to existing foundational formalisms and examines its epistemological implications for the foundations of physics.