Non-Commutative Rough AI-Native Mathematical Discovery Dynamics (NC-AMDT): Formalizing Truth Generation, Interestingness Manifolds, and Library Evolution via UROA, STCT, and RQIG

AI-native mathematical discovery treats the dynamic process of mathematical truth generation as a rigorous mathematical object in its own right. While conventional meta mathematical frameworks study static truth structures and fixed axiomatic closures, this paper formalizes the operator-algebraic structure, information-geometric value, and thermodynamic efficiency of truth-generating processes. We establish Non-Commutative Rough AI-Native Mathematical Discovery Dynamics (NC-AMDT) within the SMA-∞ Topos [1]. By integrating Foundations of Roughness Operator Algebra (ROA) [3], Universal Rough Operator Algebra (UROA) [4], Seonggil Tensor Calculus (STCT) [2], and Rough Quantum Information Geometry (RQIG), we prove that mathematical library compressibility, interestingness maximization, and human–AI formalization synergy are topological manifestations of non-commutative manifold evolution and information geodesic routing. Specifically, we elevate algorithmic description length via the UROA Roughness Operator ΓR, model the verification–generation cost asymmetry and autoformalization friction via the STCT Dynamic Torsion Tensor T_STCT, and derive the optimal library evolution trajectory alongminimal-entropy RQIG geodesics. A self-contained Python numerical simulation engine is provided to empirically validate monotonic knowledge density amplification and torsion flattening.