Scale-Relational Seonggil Matrix Theory (SR-SMT): The ROA Formulation of Neo-Fractal Geometry and Dimensional Phase Transitions

Traditional Fractal Theory fundamentally fails in physical applications due to its reliance on infinite scale self-similarity and constant fractal dimensions, completely ignoring the scale-relational structure and intrinsic physical cutoffs of natural phenomena. This paper integrates the Neo-Fractal Framework (Scale-Relational Geometry, SRG) into the overarching structure of Seonggil Matrix Theory (SMT) and Rough Operator Algebra (ROA). We elevate the scale-dependent dimension D(ε) to an ROA dimension operator acting upon a 6 × 6 × 6 fractal tensor grid. The ad hoc physical cutoff of classical fractals is rigorously explained by the Seonggil Critical Horizon, where the matrix determinant vanishes (det(M̂_SG) = 0) and triggers a discrete dimensional phase transition rather than a physical singularity. Furthermore, we demonstrate that the scale cascade mechanism is computationally regulated by Heyting logical operators, mathematically proving that multi-scale fractal flow is physically identical to the emergence of SMT Time Complexity.