Mathematical Structural Critical Phenomena: Bridging Abstract Mathematics and Statistical Physics

This paper introduces the theoretical framework of Mathematical Structural Critical Phenomena (MSCP), a novel paradigm that extends the concepts of phase transitions and critical phenomena from statistical physics into the domain of pure abstract mathematics. When a mathematical system, parameterized by a control variable denoted as lambda, approaches a critical threshold denoted as lambda_c, the underlying system undergoes profound structural transformations, giving rise to emergent properties that are absent in non-critical regimes. We formalize the foundational concepts of MSCP, define abstract critical exponents, construct an analogue of the renormalization group for mathematical spaces, and analyze topological transitions near critical boundaries. By establishing a rigorous dictionary between physical criticality and mathematical structural shifts, MSCP offers a unifying perspective on phenomena observed across algebra, topology, graph theory, and number theory.