On the Equivalence between the Collatz Conjecture and the Conjecture of Universal Representability in the 2 ⊓ 3 System

 

This work investigates the Collatz conjecture through the framework of rational number systems $\tfrac{p}{q}$. 

The main result is the establishment of an equivalence between the Collatz conjecture and the statement that every natural number admits a finite representation in the system $2 \sqcap 3$. 

Thus, the problem of trajectory convergence is reduced to a purely arithmetic question of representability of integers. 

The key steps are the construction of the $2 \sqcap 3$ system, the proof of uniqueness of representation, and the demonstration of equivalence with the dynamics of the $3x+1$ map. 

 

The results are supported by formal proofs, tables, algorithms, and graphical illustrations, which together highlight the interplay between theoretical derivations and intuitive representations. 

 

Importantly, the equivalence is not established as a mere reformulation, but through two independent mechanisms: 

(i) the algebraic structure of $v$‑expansions, and 

(ii) a decreasing metric $M$. 

The coincidence of these approaches rules out circular reasoning and strengthens the validity of the result.