To start from X³ + Y³ - Z³ = k , k a Integer wich is valid for every n, and to say we are at the conjecture point Z=t*z, then to force T³=T³*x³ + T³*y³ - T³*z³, to Z³=U³ + V³ - Z³*z³ and X³ + Y³ -(U³ + V³ - Z³*z³) = k ,If X³ + Y³ - Z³ = 0 , then (-1)(U³ + V³ - Z³*(z³-1)) = k : k have a Unique SOLUTION k=1. I hope it will save the people work in this direction, but maybe will need also more work from the youngest matheamaticians to be understood, because seem strange, and then I wll be coming with a primari suplimentary explanation.
I WILL SAY:´NOT MODULARITY IS A PROOF FOR, BUT INSTEAD Fermat - Murgu Quadruplets with theirs conjecture to UNITY.´
WITHOUT TO ASSURE I WAS IMPSSING DOUBLE FALSE REDUNDANCY OF TRUTH(which we impassed into Fermat - Murgu Impossible Equations) I will try a small Demonstration:

Fermat - Murgu Quadruplets are ∞ a problem of future , but we have already a lot:
x _i³ + y _i³ - z _i³ = 1
To supose by absurd we have a X³ + Y³ - Z³ = 0
to wrote those 2 as:
x _i³ + y _i³ = z _i³ + 1
X³ + Y³ = Z³
and are in a rational raport k, which can be k³ or expanded to, then:
X³ + Y³ = k(x _i³ + y _i³)
Z³ = k(z _i³ + 1) and
Z³ = k(x _i³ + y _i³) and
If pay attention , this Equation already solved our problem, because k forced to be INTEGER and of form k=p³ , and Z at power 3 to admit 2 diferent sum of powers.
But already we get more then 28 FERMAT - MURGU QUADRUPLETS, AND connecting our Fermat Equation to every one, every connection will bring each, another 2.
Z³ = l(z _k³ + 1) and
Z³ = l(x _k³ + y _k³) and so on..
IMPOSSIBLE
I DON'T LIKE THIS KIND OF SOLUTIONS, USING APPARENT EVIDENT THEOREMS BUT, LEAVING BACK DOUBTS!

FERMAT MURGU IMPOSSIBLE EQATIONS
SOLVED, CERTIFY AND SENT IN FUNDAMENTAL
Fermat's Last Theorem
USING ALL MATHEMATICS AND SCIENCE ACCURACY AND RIGOR.

Fermat-Murgu Impossible Equations - IS A ANALYTIC MATHEMATICS METHOD - which not force via any unproved postulate , Fermat's Last Theorem IMPOSSIBLE, but instead by all rigor calculus REVEAL an ABSOLUTE Conditional, brought By Fermat-Murgu Second Grade Impossible Equations.
Maybe, for future Research into we will need in the future also 3-th, 4-th , i-th Grade Impossible Equations where is the case but For Fermat's Last Theorem , Second Grade Sent it in FUNDAMENTAL for Math and Science.