Topological Spacetime Separation via Two-Stage Non-Hermitian Renormalization of 16-Dimensional Lövheim Sedenion Algebra and Trivalent Complex Analytic Continuation: Exact Identification of Planck's Constant and Geometric Emergence of Three Particle Generations

An analytical framework is established to derive the four-dimensional physical spacetime manifold (M4) from a sixteen-dimensional sedenion algebraic space (S16) under a velocity-normalized reference system (c \equiv 1). The dimensional reduction is formulated via an intermediate projection onto an eight-dimensional octonion subspace (O8), which is isomorphic to the co-variant phase space fulfilling the mass-shell constraints. The supremum bound of the non-associative associator generated during this algebraic reduction is identically mapped to the cosmological quantum scale, identifying it precisely as Planck’s constant (\hbar). Furthermore, a trivalent complex analytic continuation centered at the primordial singularity (z = 0) topologically isolates three independent spacetime sheets characterized by positive, negative, and imaginary temporal dimensions. This geometric architecture ensures that localized pair-productions within planckian volume elements (\Delta VPlanck) conserve the global energy balance while inducing the macroscopically observed accelerated cosmic expansion. Concurrently, the topology of the overlapping sheets provides a deterministic geometric derivation for the exact existence of three generations of elementary particles, establishing a strict boundary condition governing the multi-sheeted manifold.