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This paper formalizes the limitations of using scarce resources to prevent Sybil attacks in permissionless systems. It proves that if a resource is divisible, additive, temporally reusable, and identity-transferable, then influence can be concentrated at sublinear cost, regardless of the protocol. Conversely, throughput-bounded, non-transferable, window-local resources enforce linear cost, demonstrating an asymptotic separation between resource classes.
No matter how cleverly you design your protocol, if your resource is divisible and reusable, Sybil attackers can always concentrate influence at sublinear cost.
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.