Algebraic Collapse of Sedenion Singularity and Emergent Gravity: A Unified Master Action via Rough Operator Algebra and Composite Torsion

Abstract

This paper presents a complete mathematical framework for a Theory of Everything(TOE) by unifying quantum mechanics and general relativity through the algebraic collapse of a 16-dimensional sedenion (S) space[cite: 3]. We introduce Rough Operator Algebra (ROA) and the Seonggil Theory of Composite Torsion (STCT) to demonstrate that the continuum spacetime is a macroscopic illusion emerging from a non-associative, noncommutative fractal geometry[cite: 3]. First, we prove that the initial universe under goes a zero-divisor collapse at a critical symmetry-breaking index (J_ROA → Λ^4_crit)[cite:3]. This topological tearing projects the observable universe into an octonion (O_1) space governed by G_2 triality, while the decoupled space condenses into a massive topological defect (R_defect)[cite: 3]. We rigorously derive the three-generation fermion mass hierarchy(electron, muon, tau) as a spontaneous geometric consequence of energy closed-loop conditions and Eisenstein Complex Multiplication (CM) phase-locking, yielding the exact ratio of 1 : 206.7 : 3477[cite: 3]. Furthermore, we establish that gravity is an emergent holographic phenomenon driven by the entanglement entropy of the defect[cite: 3]. Finally, we show that the non-associativity intrinsic to O_1 space generates STCT torsion acting as an ultraviolet (UV) cut-off, rigorously resolving the Higgs hierarchy problem and rendering supersymmetry(SUSY) mathematically obsolete, while yielding explicit falsifiable predictions in stochastic gravitational waves, dark matter halo cores, and ultra-high-energy tau neutrino anomalies.