Since Mautner in 1946 the invariance programme has demarcated logical notions as those preserved by a group acting on the domain of discourse, a strategy refined through Tarski, Sher, McGee, Feferman, Casanovas, and Bonnay into the modern invariance criteria for logicality. The programme's finiteness underwriting comes from classical invariant theory: for a linearly reductive group acting linearly on a finite-dimensional space the invariant ring is finitely generated, and for the classical groups its generators are named explicitly. This paper identifies a register at which that machinery cannot run, and shows the failure is structural rather than technical. The invariance criterion succeeds on the domain register because permutations of a domain form a group by construction. It fails on the register of semantic decomposition, where the operative transformation is not permutation but abstraction, the discarding of a constituent, which is idempotent and carries no inverse. The transformation set is therefore a monoid once the empty abstraction is adjoined, group-hood fails before reductivity is ever tested, and neither the Hilbert finiteness theorem nor the Weyl explicit generators apply. We give the obstruction in exact form, an operator with singular values one, one, zero whose distance from left-invertibility is bounded below by exactly one in spectral norm and attained there. Admissibility, that every constituent of a claim is abstractable, then forces a stronger conclusion than the failure of a finiteness theorem. Invariance under each of the three coordinate abstractions forces every invariant function to be constant on the whole space in three substitutions, equivalently because the semigroup they generate contains an annihilator: finite generation holds vacuously, generated by the constant one, while the separating power collapses absolutely, no invariant distinguishing any two points whatever. The criterion at this register does not lack its underwriting theorem. It demarcates nothing. We show the consequence for any pipeline composing a semantic decomposition with a decidable arithmetic verdict, namely that the non-effectiveness Church and Tarski require of a truth-tracking procedure must reside on the decomposition side since the remaining stages are finite and decidable, and we state the one hypothesis on which the collapse turns, which is not closure under composition, that being automatic, but admissibility: that every constituent is abstractable.