Scarcity Is Not Enough: Structural Limits of Linear Sybil Cost Under Parallelizable Resources

Abstract

Permissionless systems resist Sybil attacks by binding influence to scarce resources. Yet influence concentration persists across systems built on computation, capital, and other reusable resources despite substantial differences in protocol design. This raises a fundamental question: is concentration primarily a consequence of protocol rules, or of the structural properties of the underlying resource itself? We address this through the adversarial cost function $C(s,T)$, the minimum expenditure to sustain influence equivalent to controlling $s$ independent participants over a horizon of length $T$. We develop an axiomatic resource taxonomy connecting resource structure to adversarial cost scaling. We show that resource-mechanism pairs satisfying divisibility, additivity of influence, temporal reusability, and identity transferability admit influence amortization, yielding $C(s,T)=o(sT)$. Conversely, throughput-bounded, non-transferable, window-local resources enforce $C(s,T)=Ω(sT)$, with marginal cost $Δ(s,T)=Ω(T)$ increasing with time. These results reveal a fundamental asymptotic separation between resource classes that admit amortization and those that enforce linear cost. These results shift the focus of decentralization from protocol design to resource design. If concentration is a structural consequence of the properties that make resources parallelizable, redesigning consensus rules alone cannot eliminate it. The same properties enable concentrated control to be projected across many nominally distinct participants through delegation, pooling, or identity replication, obscuring the relationship between visible identities and underlying control. The search for stronger decentralization must therefore begin with the choice of the underlying resource.