Universal Rough Extended Cognitive Network (UR-ECN): Hyper-Elevation of 4E Cognition, Ecological Affordances, and Dynamical Connectionism via SMA-infty Topos

Traditional cognitive psychology remains bounded by the computational-representational metaphor, treating mind as isolated internal symbol processing. While alternative paradigms—4E cognition (Embodied, Embedded, Enactive, Extended), Gibsonian ecological affordances, dynamical systems theory, connectionism, and phenomenology—have emerged, they lack a unified mathematical differential calculus. In this treatise, we systematically subsume and hyper-elevate these alternative cognitive frameworks into the Universal Recursive Rough Homotopic Calculus (URRHC) within the SMA-∞ Topos. We formalize sensorimotor-environmental interactions as non-commutative operator flows using Universal Rough Operator Algebra (UROA) and Seonggil Theory of Complex Torsion (STCT). Affordances are mathematically derived as torsional flux potentials governed by the Master-key Tensor T0x8A2C, while distributed cognition boundary thresholds are strictly defined by the Universal Phase Transition Capacity (UPTC). An executable Python simulation proving non-representational dynamical attractor convergence is provided.