Minimum Deviation Law (MDL): A Unified Geometric Structure in Physics, Number Theory, and Computational Complexity through Galois Covering of Modular Curves

This record contains the preprint “Minimum Deviation Law (MDL): A Unified Geometric Structure in Physics, Number Theory, and Computational Complexity through Galois Covering of Modular Curves” and the accompanying Lean 4 formalization files. The paper proposes the Minimum Deviation Law (MDL) as a unified geometric framework that connects three a priori different domains: three-dimensional physics, analytic number theory, and computational complexity. The central variational principle fixes the projection rate and effective dimension as lambda = 1/nD(n) = n - 1/n which leads to two discrete critical layers n = 3 (icosahedral physics) and n = 2 (modular number theory). Within this framework, the icosahedral group A5 (a subgroup of SO(3)) and the modular group PSL(2,Z) are linked by the unique Galois covering of modular curves X(5) -> X(1), and the resulting information–dimension gap Delta D = 7/6 is interpreted as a universal source of critical fluctuations. On the number-theoretic side, MDL provides a unified geometric interpretation of major prime-related conjectures (Riemann Hypothesis, ABC conjecture, twin primes, and Goldbach), viewing them as different “shadows” of the same icosahedral–modular projection. On the complexity side, the framework formulates a geometric version of the P != NP barrier via an “A5 obstruction” arising from the non-solvable symmetry of the n = 3 layer. All core axioms (F1–F3), the main distribution theorem T1, and the conditional statements relating MDL to RH, ABC, and P != NP are implemented and mechanically checked in Lean 4. The included Lean project provides the formally verified axiom system and key theorems that constitute the mathematical backbone of the MDL framework.