The Rigidity of the G2 Closure: Why the Algebraic Match Is a Constraint, Not a Coincidence

The foundation of the framework is the claim that fourteen operator values, taken from an independently fixed source, satisfy the Jacobi identity and close on the exceptional Lie algebra G2. A natural objection is that any sufficiently large set of numbers might be arranged to close on some algebra, making the match unremarkable. We test this directly. We construct the G2 root system explicitly, identify the sixty bracket relations fixed by its geometry, and show that the structure is rigid: only an overall rescaling preserves it, while a five-percent perturbation of a single root breaks twelve of the sixty relations, destroying closure. Generating G2 therefore requires reproducing its exact root geometry up to scale — a strong constraint, not a generic outcome. The closure achieved by the operator values is thus a non-trivial fact: the values must encode the precise geometry of the algebra, and small departures fail. We present the computation and discuss what it does and does not establish, distinguishing the demonstrated rigidity of the closure from the separate question of whether the specific values are the unique solution.