From Gravity Tunnels to Complex 6_Sphere and Fermat's Last Theorem

A. J. Wiles 1985 obtained a (long) indirect implicit proof on the FLT.

After proving semi-simple almost complex structure on extended triple (G, h, k_B) here Newton G-gravity constant variety, G₀=2/3; h- reduced Planck constant variety, h=2π√3; k_B=8√3 - Boltzmann constant variety or extended Boltzmann constant variety k_B^N= N2√3 (N=1,2,3,... ∞), I obtain an important result on an extended Fano variety k_B^N /h as the "key" component (the so-called Kuznetsov component K(Y)) in following semi-orthogonal decomposition. Then I can give a generic stability condition σ=Stab^†(Ku(Y)).

The exact Weierstrass form is:

E: y² = x³ + a₄x + a_6, where two constants are uniquely determined by my nonlinear Quantum Vortex Field Equation (QVFE)

∂u/∂t = ∇D(u)·△u.

I presented all exact solution of QVFE at the APS 2021 March Meeting. Then, in fact, I already known a unique exact stable Elliptic curve and its exact analytical description. This paper provides a key transition: from  the discriminant △= -16(4 a₄³ + 27 a₆²)  in a classical 4-d Parallel Space-Time into a real torsion space-time called as a 1-d Unitary Space-Time ,then the reduced △ transfers an explicit automorphism modular form. As a result, I prove that there exist three fundamental complex domains:

1- Cohomology domain; 2- Homology domain; 3- Most important Torsion domain. In the Torsion domain, natural gravity potential = natural chemistry potential. I found 4 flex critical points, which determine singular torsion intersection.

Then I recover an exact stable quantum gravity orbit in a maximal complex torus- Fu-Xi holography.

As two co-products, I here give out a direct explicit proof on the existent complex 6-sphere and the Fermat's Last Theorem. A key result is that in CW- Complex manifolds:

a³ + b³ = 0 (n= 3, 5,7,...) or by a critic character representation in complex tori: β³ + ω³ = 0, which means that the Cup product of gravity potential and chemistry potential =0 and S⁶- Sphere is integrable Complex Structure as S²-Sphere.