We prove a closed-form formula for the cross-ratio of any four consecutive convergents of a simple continued fraction. The result depends only on the product of two partial quotients and takes a remarkably clean form. As a corollary, the Fibonacci ratios — the convergents of the golden ratio — have constant cross-ratio equal to one half, which is the minimum possible value among all quadratic irrationals. This gives a new projective-geometric characterisation of the golden ratio as the most irrational number, alongside the classical characterisations via Hurwitz's theorem and the theory of best rational approximations. The result connects two subjects — continued fraction theory and projective geometry — that have been studied since antiquity but appear not to have been combined in this way before. A complete formal proof verified by the Lean 4 proof assistant (using the Mathlib library) accompanies this paper.
Landon Willow (Wed,) studied this question.