Central Obstructions and Block-Complement Categories for Schur Covers

We study explicit ways to replace the symmetric-group automorphisms of the category of finite sets and injections by their Schur double covers. If the central involution acts freely on a finite lift fibre of an injection of codimension \(d\), then the fibre cardinality is divisible by \(2^{\lfloor d/4\rfloor+1}\). Thus free two-sheeted morphism lifts, and uniformly polynomial-sized lift fibres, are impossible. We construct an ordinary category using unordered families of internally ordered four-element complement blocks, and a complex superlinear category using two-element blocks. Their composition laws follow from explicit coherent even lifts; the phase needed for pairs is not a group-theoretic section. Finally, a coherent Clifford quotient of the fully framed model has kernel requiring new relations in every rank \(n\geq 3\), with quotient dimension \((n-2)2^{n-2}\). Hence that model is not locally noetherian. Classical Schur, Clifford and spinoriality facts are credited. These results concern specified models; they do not establish a general spin representation-stability theory or settle the broad AIM construction program. This is an unrefereed preprint, with no absolute priority claim.