This work presents the complete invariant algebra formulation of the Information Invariance Model (IIM) closure auditor within the fixed dependency chain MOF → PFC → Alpha Locking → IIM. Finite-resolution closure produces a multiplicity-free invariant scalar algebra of dimension two, fully determining closure-admissible invariant scalar readouts. Within the primary closure class, the invariant algebra uniquely yields two independent scalar functionals corresponding to the proton–electron and muon–electron mass ratios. We further demonstrate that the existence of an additional physical scalar observable, the tau–electron mass ratio, cannot arise within the same invariant algebra. Instead, it emerges uniquely as the minimal admissible element of a higher closure assembly governed by an independent hierarchical admissibility predicate. This construction does not extend the invariant algebra but introduces a distinct closure class whose admissible assembly is uniquely determined by discrete structural constraints and minimality. All results follow from closure-determined structural constants without continuous fitting parameters. The paper includes a complete invariant algebra derivation, admissibility predicate formalization, uniqueness proofs, and a fully reproducible reference implementation.