A convergence proposal for the Collatz sequence: the personalised conjecture

An alternative conjecture to the classic formulation of the Collatz sequence is proposed based on the analysis of the ratio between even and odd steps. It is demonstrated that, except for exceptional cases, in each iteration (understood as a set of odd and even steps dependent on the initial number of the same iteration), the total number of even steps ( b ) exceeds that of the odd steps ( a ) by at least one or two units, leading after a certain number of iterations to the inequality 3𝑎⋅𝑁2𝑏<1 , which implies the convergence of the sequence towards 1. In fact, below, my personalised Collatz conjecture will be applied in a faster way, considering only natural numbers N modulo 4 and that the sequence of these numbers (after an initial number that can also be even mod. 4) is reduced to the sequence of only odd numbers ≡ 1 mod. 4 and ≡ 3 mod. 4. This approach leads to the same results as the canonical Collatz sequence.