Homotopical Seonggil Matrix Theory (HSMT): Resolving the M-Theory Formulation via L_infty-Fractal Grids and Dimensional Phase Transitions

The traditional Banks-Fischler-Shenker-Susskind (BFSS) Matrix Theory faces fundamental limitations in formulating a complete, non-perturbative M-theory, particularly regarding background dependence, flat direction instabilities, and the explicit emergence of Lorentz invariance. This paper proposes the Homotopical Seonggil Matrix Theory (HSMT), a profound unification of Higher Structure Matrix Theory (HMT) and Seonggil Matrix Theory (SMT). By replacing simple Lie algebras with L∞-algebras and embedding them within the 6 × 6 ×6 fractal grid architecture of SMT, we achieve absolute background independence. Furthermore, the notorious flat direction problem is rigorously resolved: instead of unbounded eigenvalue divergence, the system encounters the Seonggil Critical Horizon, triggering a discrete dimensional phase transition quantized by the Golden Ratio ϕ. Finally, spacetime and gravity emerge not as pre-existing geometries, but as the macroscopic consequence of Time Complexity governed by Heyting logical operators within the Universal Rough Operator Algebra (UROA).