THE PMNS MATRIX, NEUTRINO MASSES, AND THE b → sµ+µ− ANOMALY FROM GEOMETRIC HYPERSENSITIVITY ON THE NONCOMMUTATIVE SUPER-TORUS

We derive the PMNS matrix, the neutrino mass spectrum, and a quantitative explanation of the b → sµ+µ− anomaly from the geometric hypersensitivity offermionic torsion points on the noncommutative super-torus T 2 θ .The central mechanism is the proximity of lepton torsion points to the zeros of the Jacobi theta function ϑ(z|τ) on the elliptic curve Eτ. We prove four principal results:(1) Large PMNS angles from hypersensitivity: The neutrino torsion points z (g) ν lie exponentially close to the theta zero, making the Yukawa eigenvalues hypersensitive to the anisotropy δz of the Nash embedding. This translates a small geometric misalignment into O(1) mixing angles, yielding θ12 ≈ 31.8◦, θ23 ≈ 47.2◦, and θ13 ≈ 7.9◦, in agreement with experiment.(2) Neutrino masses: Dirac and Majorana scenarios: The framework admits both Dirac and Majorana neutrino masses. The Dirac mass arises from the standard Berezin integral; the Majorana mass arises from the odd subspace A1 of the superalgebra, which identifies zf with −zf via the Berezinian. The theory predicts normal ordering with P mν ≈ 0.061 eV.(3) The b → sµ+µ− anomaly from neutrino loops: Dirac neutrino masses generated by the Berezin integral contribute to the Wilson coefficient Cµ9 throughloop diagrams. The contribution is proportional to the hypersensitivity factor κµ ∼ 1/|zνµ − zzero|, which is maximal for the second generation. This yields∆Cµ9 ≈ −1.0, ∆Ce9 ≈ 0, and the prediction ∆Cτ9 ≈ −4 to −9.(4) Integer quantisation of ∆Cµ9 : The value ∆Cµ9 = −1 arises from the topological index of the Berezin integral at the 2-torsion point zνµ = (1/2, 1/2). The integer nature of the shift is a consequence of the discrete geometry of the super-torus, analogous to the quantisation of the Hall conductivity in the quantum Hall effect.The theory contains no free parameters. All predictions are determined by fixed τ and θ = ℓ^2P.