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.

翻译：去中心化系统通过将影响力与稀缺资源绑定来抵御女巫攻击。我们证明稀缺性本身是不够的：资源的结构特性决定了能否通过身份复制、委托或聚合来以亚线性成本集中影响力。我们通过对抗成本C(s,T)建模：在T个时间窗口内，为获得相当于s个独立参与单位的影响力所需的最小支出。我们证明任何满足可分性、影响力可加性、时间可复用性和身份可转移性的资源都会允许影响力摊销：C(s,T)=o(sT)，与协议设计无关。这是一个不可行性结果：对于结构上可并行的资源，没有任何协议规则能强制影响力集中的线性成本。我们进一步证明，吞吐量受限、不可转移、窗口局部的资源能强制执行C(s,T)=Omega(sT)：每增加一单位持续影响力将产生边际成本Delta(s,T)=Omega(T)，该成本随时间跨度增长。这两类资源在渐近意义上被严格分离。作为直接设计推论，任何以线性影响力集中成本为目标的机制，必须将参与建立在至少违反一个可并行化属性的资源之上。