The closed world of the minimal subfactor: an exhaustion theorem

The relations forced by the index-two subfactor form a provably closed world. By Jones's planar algebra theorem, the spherical (non-braided) operations on the fusion category are exactly the planar tangles — tensor, composition, duality, trace, recoupling — and there are no others. Every structure-adding completion of the category requires a choice the category does not make (a braiding, an extension, a module category, a fibre functor), so no completion produces a forced invariant; the forced content is the choice-free content regardless of how completions are classified. The unique operation beyond the closed set is the crossing — the introduction of a braiding — which is the onset of the modular, contingent sector of companion paper II. The laws are finite, complete, and self-bounding: this is the exhaustion theorem, proved given minimality, with no additional hypothesis. The paper also shows the closed world carries a holographic structure: by the Turaev–Virelizier theorem, the spherical laws fully determine a three-dimensional bulk theory whose boundary data is the braided completion — the space of all games the rulebook permits. This is companion paper IV of a quartet.