Icosahedral axis components and the solar mixing angle: a complete enumeration, and a gap

A recurring hope in flavour model building is that the solar mixing angle sits at a value the icosahedral group hands you for free. The group has 6 five-fold axes, 10 three-fold and 15 two-fold, and the squared components of a unit vector along any of them, taken in any icosahedral basis, form a small finite set. If sin^2θ12 were one of those numbers, it would be forced rather than fitted. We enumerate the set completely. There are exactly eight distinct values, and we give them in closed form. We then compare them with the measured solar mixing angle across three independent reductions. None of the eight lands inside any error bar. The nearest candidate differs by reduction — 1/3 at 2.77σ on one, 0.276393 at 4.25σ on another, 1/3 at 3.66σ on NuFIT 6.1 — and every candidate is excluded at better than 2σ everywhere. The enumeration is complete, so this is not a failure to search harder: no bare icosahedral axis component reproduces the solar angle, and any model that gets there must do so through a mechanism that is not a single axis projection. We state the result as a closed gap rather than a near miss, and include the script that asserts the axis counts, the completeness of the eight-value set, and the exclusion of every candidate.