arXiv:2611.06268v1 Announce Type: new Abstract: Sybil defenses price the creation of an identity, yet influence goes, again and again, to active identities. We study the cost of maintaining influence. A defense has a bilinear Sybil cost with ratio $A$ if keeping $s$ identities active for $T$ periods costs at least $sT/A$ honest units: the average cost per identity-window is at least $1/A$ times an honest participant's cost. Recurrence alone does not guarantee this cost. The cost can be carried over from earlier periods, shared among many identities, or be negligible when fast hardware does the work of thousands of honest devices. We give four sufficient conditions for a defense that checks every identity each period to achieve bilinear Sybil cost: identity binding, freshness, a shared deadline, and sequential, non-batchable work. They yield two guarantees: a bilinear total cost, with $A$ the speed advantage of the adversary's fastest hardware, and, because the work must be fresh and finished by a common deadline, a bound on the identities active in a period by the processors running at that time. We prove that none of the four follows from the other three. A verifiable delay function on an identity-bound input meets all four without payments, with $A$ of 2 to 8 for CPUs and about 28 for a hypothetical ASIC, against $10^3$ to $10^7$ for hash puzzles. We measure one active identity per core per period against about 80,000 under a signed heartbeat, no gain from batching up to 1,000 identities, and rejection of all tested attacks.