Mathematical Foundations of the Single Monad Model of the Cosmos and Duality of Time Theory: Emergence of Lorentzian Geometry from Unicity to Multiplicity via Admissible Histories, Quadratic Carriers, and Hilbert–Algebraic Structures

This paper develops the mathematical foundations of the Single Monad Model of the Cosmos and the Duality of Time Theory, with a central focus on the emergence of Lorentzian geometry from a unified generative origin. Starting from a dual-time architecture—comprising generative inner time, completed outer time, and a completion–projection interface—the framework constructs a precise pipeline from unicity to observable multiplicity. At the generative level, admissible histories form a monadic structure acting on states, while completion and observation induce a quotient-based descent to irreversible observable dynamics. Quadratic carrier algebras provide the minimal algebraic setting, separating a compact circular branch associated with recurrence and phase from a split hyperbolic branch associated with causal readout. A key representation-theoretic result shows that compact inner-time symmetry enforces canonical complex Hilbert structure on irreducible sectors, while invariant-form rigidity excludes Lorentzian signatures from these phase sectors. The central new contribution is a stabilization interface that maps completed histories to probability measures on candidate event structures. From these stabilized statistics, the paper reconstructs effective causal order and volume, and proves that, in a manifoldlike regime, these data determine a Lorentzian geometry up to coarse equivalence. This establishes a theorem-bearing bridge from generative structure to spacetime geometry. The resulting framework organizes complex Hilbert structure, observable irreversibility, and Lorentzian geometry within a single constrained emergence chain: compact recurrence yields phase and complex structure; completion yields irreversible observables; stabilization yields persistent event statistics; and reconstructed order plus volume yields spacetime geometry. The analysis is structural and constraint-driven, showing that phase and causal geometry arise at distinct levels and are necessarily carried by different quadratic branches.