The M_R Family: Gaussian Regularization of the Prime Alpha Family, Geometry of Zeros, Arithmetic Structure, and Thermal Laws

We introduce the M_R family of logarithmically weighted prime series as a Gaussian regularization of the prime Alpha family. The paper develops its joint holomorphy, exact differential hierarchy, vertical heat equation, Gaussian semigroup, Mellin and Laplace transforms, positivity properties, moment inequalities, almost periodicity, annular value geometry, zero-free regions, thermal asymptotics, and spectral reconstruction of the primes. A central result associates asymptotic vertical zero chains with consecutive prime pairs and proves that the slope and spacing of each chain recover the pair that generates it. The paper also develops a sequential limiting measure of zeros, zero–cumulant identities, additive and multiplicative arithmetic companions, operator-trace and Fredholm-determinant representations, and a multiscale thermal reflection framework. A selected algebraic, geometric, differential, and local two-term core was independently formalized and verified in Lean 4.12.0 with Mathlib. The formalization is partial and does not yet cover the infinite prime tail, the Rouché transfer argument, the complete asymptotic localization of the zeros, or the planar limiting-measure theorem. The verified Lean package is available as related software at DOI 10.5281/zenodo.21581767. The manuscript does not claim to prove the Riemann hypothesis, Goldbach’s conjecture, or any other classical open problem. It presents the definitions, proofs, computational evidence, limitations, and open questions required for independent mathematical examination.