Non-Commutative Rough Quantum-Cryptographic Dynamics (NC-RQCD) Subsuming Fault-Tolerant QEC and Lattice-Based Post-Quantum Cryptography Limits via UROA, STCT, and RQIG

Abstract

Fault-tolerant quantum error correction (QEC) overhead and lattice-based post-quantum cryptography (PQC) infrastructure migration are inextricably coupled, two-sided limits of the quantum era. This paper supersedes independent subsystem views by establishing Non-Commutative Rough Quantum-Cryptographic Dynamics (NC-RQCD) within the SMA-∞ Topos. By integrating Universal Rough Operator Algebra (UROA), the Seonggil Theory of Composite Torsion (STCT), and Rough Quantum Information Geometry (RQIG), we prove that physical qubit explosion and cryptographic transition delays are macroscopic topological manifestations of non-commutative boundary noise and informational shear stress. Furthermore, qLDPC codes and hybrid PQC migration are formally derived as geodesic optimization sub-loops protecting the quantum-classical manifold.