A proof of the Collatz conjecture and connection of intervals based on ``tetrada''.

This paper proposes be reveal the mechanism underlying systems of numbers, which suggests the existence of the Collatz conjecture (transformation) and as result a proof of the Collatz conjecture, also known as the 3x+1 problem.

To determine the underlying mechanism of the Collatz conjecture, I convert the recursive-conditional Collatz transformation to a system of recursive linear transformations. Our study reveals a hidden “mechanism'' in numerical systems based on the fractional bases  \( \frac{2^a}{3}\)and \( \frac{4}{3}\). It should be noted that the method of numeral systems of rational bases appeared as a result of the transition from “recursive destruction'' to “recursive conservation'' while modifying the Collatz transformation.

In this paper, we study the distribution of a number, as a potential, on selected levels (scales) of numerical systems based on the fractional bases:

• we obtain and proof sufficient conditions for the Collatz conjecture;
• we obtain and prove the necessary conditions for the Collatz conjecture by the procedure, that it is impossible not to obtain sufficient conditions for any natural number. During the recursive Collatz transformation, the ``length of the significance part'' of the number can remain unchanged, or can decrease by at least half-digit. The ``length of the significance part'' reduces when the oddness of the form 4k + 1 appears, and is kept when keeping the oddness of the form 4k+3. As the oddness of the form 4k+3 cannot be kept indefinitely, the ``length of the significance part'' periodically decreases. Therefore, it is inevitable that after an appropriate number of iterations, the number transforms into the form  \( \frac{2^{p(q)}}{3^q}\)(number with unit length)
• we also provide a proof, that all natural numbers get their expansion in a set of numbers with a fractional bases \( \frac{2^a}{3}\) .