Neutrino Mass and Mixing: Exact Analytical solutions from the Generalized Newton's Laws Theory - Courant Algebroid on neutrino mixing will be rendered as Dirac Operator- its differential form as spinors for the metric bundle T+T^*

The origin of fermion mass hierarchies and mixing is one the unresolved and most difficult open problem in high-energy physics. Using the Generalized Newton's Laws, I will close this open problem. Main result includes three base theorems in the Topological Field Theory (TFT):

Theorem -1. The mass hierarchies of matter can be represented by a sine function in maximum complex torus, such as for photon mass: m_γ = sin(1/(2π)) = 0.1584838866... , photon speed: v_γ = 4(1 - m_γ²) = 4(1 - sin²(1/(2))) = 3.8995314308... > light speed c = 3.0 x ... km/s, which is a transitive state or quasi-stable state speed, rather than a limit in vacua.

Theorem -2. The unitary Space-Time U = x²/(2t) ↔ x/(√2t) induces the base relation between energy E and mass in quantum gravity action : Gmv = Gm r_coh² = Id =1, there G- Newton's Constant, reduced momentum r_coh = √(- E) = 2cos(φ) or v r_coh² = 4cos²(φ), where φ is phase angle, and there exists m(1-m²) =m(1+m)(1-m) = 1/(4G) = (8/3)^-1 = (2.666 666...)^-1.

Theorem -3. The neutrino mass and speed are critically governed on the Generalized Newton's Law with quantum gravity: m₁₂ = sin(θ₁₂) = 1/√3 = 0.577350..., then v₁₂ = 4(1- 1/3)

=4×2/3 = 8/3 = 2.666 666; m₂₃ = sin(θ₂₃) = 1/√2 = 0.70710..., then v₂₃= 4×(1-1/2) = 2.0...; m₁₃ = sin(θ₁₃) ≡ λ_C ~ 0.2 (Cabibbo angle), v₁₃= 4(1 - 0.2²) = 3.84 ≈ photon speed v_γ= 3.8995.... These fact shows that all neutrinos mass, energy and mixing have a unitary topological structure, m₁₃= sin(8.5^o) = 0.1478094111, then v₁₃= 3.91260... more near to photon mass v_γ = 3.8995, which means that neutrino with reactor angle θ₁₃≈ 8.5 is just like a photon.

In oder to deepen the research on neutrino mixing, one can extend three flavor mass m_i to three flavor speed V_i (i=1,2,3), i.e., including two sheaves: neutrino momentum r_coh = c_ij ≡ cos(θ_ij) and neutrino mass s_ij ≡ sin(θ_i,j), then from V_ij=4(1- sin²(θ_ij)) ⇒ sin²(θ_ij) = 1- V_ij/4. It also may embed in tangent and cotengent t_ij = tan(θ_ij) = cot^-1(θ_ij) to show deformation retract.