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.