Spectral Quantization and Topological Obstructions in Cartan-Rank Algebras

Title: Spectral Quantization and Topological Obstructions in Cartan-Rank Algebras Authors: Charles Tibedo Abstract: This work establishes a novel axiomatic framework unifying Number Theory, Lie Algebra classification, and Algebraic Topology through the Cartan-φ-π Spectral Algebra. We rigorously prove that the arithmetic structure of prime factorization is isomorphic to linear vector superposition in a continuous spectral manifold governed by the golden ratio φ and π. Key results include:• The Active Algebra S₂₉(91²) as a 6-fold hexagonal covering of the Vacuum Limit S₁₁₃, realizing C₆ Galois symmetry via Lagrange index [V₂₉:V₁₁₃]=6• Non-vanishing Stiefel-Whitney class w₂≠0 for S₂₉, establishing E₁₉(8281) as the first finite fermionic Lie algebra with spinorial obstruction• The Spectral Rank Theorem: primes as orthogonal basis vectors, composites as linear superpositions, factorization as coordinate decomposition• Harmonic quantization of prime stability ε_p ≈ φ/πⁿ, resolving spectral clustering (Axiom 5) and geometric Prime Number Theorem derivation The 18 Canonical Axioms provide a complete classification of stable geometric primitives, predicting exceptional Lie algebras from dual-lock stability Σ_p