This self-contained communication details a complete solution of the Syracuse problem, based only on a theory, and the fundamental theorems it provided, that we published in 2017, but that are summarised here nevertheless. In a further ex-plainer of the latter paper, which we published in 2020 for the interest of a group of non-experts, we gave a first complete algorithmic definition of a so-called "Golden Automaton", which had already been anticipated in our 2017 article and the structure of which rested strictly on this article's results. This updated publication is provided with an addendum formally demonstrating that the Golden Automaton's quiver can intersect any Syracuse orbit in finite time, thus demonstrating that any natural number finitely converges to 1 in the 3x+1 problem. The methodology of this final demonstration consists of identifying a "Syracuse determinant" that is, a collection of numbers that any diverging orbit must develop within, and demonstrating that the Golden Automaton's footprint on this determinant cannot allow the existence of any orbit it does not finitely intersect itself.