Scarcity Is Not Enough: An Impossibility Result for Linear Sybil Cost Under Parallelizable Resources

Permissionless systems resist Sybil attacks by binding influence to scarce resources. We show that scarcity alone is insufficient: the structural properties of the resource determine whether influence can be concentrated at sublinear cost through identity replication, delegation, or pooling. We model this through the adversarial cost C(s,T): the minimum expenditure required to achieve influence proportional to s independent participation units over T windows. We prove that any resource satisfying divisibility, additivity of influence, temporal reusability, and identity transferability admits influence amortization: C(s,T)=o(sT), regardless of protocol design. This is an impossibility result: no protocol rule can enforce linear cost of influence concentration over a structurally parallelizable resource. We further prove that throughput-bounded, non-transferable, window-local resources enforce C(s,T)=Omega(sT): each additional unit of sustained influence incurs marginal cost Delta(s,T)=Omega(T), growing with the time horizon. The two resource classes are asymptotically separated. As a direct design consequence, any mechanism targeting linear cost of influence concentration must ground participation in a resource that violates at least one parallelizability property.