The closure vector of the minimal subfactor

The index-two subfactor carries a two-sided closure vector, built from the quantum dimension of its generating object and its modular-conjugate reciprocal. This paper shows the vector is fixed by the same minimality principle as the index selection — close on the generator, not a composite — making this a third level of one principle rather than a third independent axiom. The amplitudes forced by the closure vector are a₊² = 9/20 and a₋² = 1/20. These are forced given the minimality principle and are wholly spherical (tracial) data; they do not cross the modular boundary of companion paper II. An earlier over-general formula in the programme's notes is corrected: the principled construction gives denominator (I+2)(2I+1)/I at general index, agreeing with the compressed formula only at index two. This is companion paper III of a quartet.