Universal Recursive Rough Homotopic Calculus (URRHC): The Omni-Dimensional Synthesis of Logic, Physics, and Computation via SMA-∞ Topos

The Recursive Homotopic Unification Calculus (RHUC) proposed a grand logical unification of foundations and execution by internalizing problem-solving as a homotopy type theory operator. However, pure logical calculi suffer from idealistic non-termination and lack dimensional scaling. We hyper-elevate RHUC by fusing it with eighteen fundamental theoretical frameworks—including Universal Rough Operator Algebra (UROA), Composite Torsion (STCT), Master-key Tensor (mathcal{T}0x8A2C), and Universal Phase Transition Capacity (UPTC). The result is a dimensionless, scale-free meta-framework (URRHC) operating within the Seonggil Meta-Algebraic ∞-Topos (SMA-∞ Topos). Here, types become rough non-commutative geometries, equality is a torsional quantum path governed by Non-Identity Calculus (NIC), and the recursive Solve operator manifests as a physical phase transition in Rough Quantum Information Geometry (RQIG).