Projection Algebra TOE Ω — V12.24-R6.0

Projection Algebra TOE Ω — V12.24-R6.0 (Paper I) This publication presents a pregeometric mathematical framework in which observable spacetime and physical interactions are modeled as projections of a common algebraic source. The construction begins with signed Cayley–Dickson recursion and develops explicit finite-dimensional operators, nonassociative and nonalternative algebraic structures, zero-divisor geometry, Leech-lattice harmonic sectors, projection crystals, and source-to-observable mappings. The manuscript develops a rank-based account of emergent spacetime geometry, a coupled quantum–geometric variational framework, spectral-gap constructions, Schur-complement descriptions of hidden-sector responses, and reprojection dynamics. A central addition in version R6.0 is the algebraic reconstruction of π through the complex Cayley–Dickson subalgebra, ternary lattice characters, and recursive phase refinement. The resulting source-defined phase normalization is incorporated into the framework's calculations of the fine-structure constant and Higgs-sector observables. The paper also investigates Yang–Mills and strong-sector structures, connections between quantum mechanics and gravitational geometry, and the role of OR closure and P5 reprojection. Exact finite-algebra identities and computationally reproducible constructions are distinguished from proposed physical identifications and questions requiring further validation. Files included: Complete English and Czech editions of Paper I, version V12.24-R6.0. Status: Independent theoretical research preprint. The proposed physical interpretations have not been established by independent experimental validation.