Proposition 2, p. 24 — bond voting is robustly v-implementable

Let the voting bond rate be $r > 0$ and the commitment horizon $T > 0$. Then bond voting is robustly v-implementable:

1. the bond vote function $v(q, t) = q e^{rt}$ ($t > 0$), $v(q, 0) = 0$, values commitment (Definition 2);
2. it has non-increasing influence of tokens (Definition 3);
3. for every admissible agent type $(\theta, u, r_i)$ — that is, $|u|\theta > 1$ and $r_i > r$ — the bond-voting payoff

is Sybil resistant (Definition 6): for no budget $q > 0$ is there a split $q_1, \dots, q_M \ge 0$, $\sum_m q_m \le q$, with commitments $t_1, \dots, t_M \in (0, T]$ such that $\sum_m \pi_i(q_m, t_m) > \pi_i(q, t)$ for every single-entity commitment $t \in [0, T]$.

This is the main result of the paper: time-weighted voting through bonds rewards commitment, does not reward concentrated token holdings more than linearly, and gives no agent a reason to create multiple identities.

Formalization Note. The hypothesis $r > 0$ is added (the paper does not state the sign of $r$; Definition 2 fails for $r \le 0$). Sybil resistance compares an attack with every commitment the agent could choose as a single entity, not with a fixed one or with the formula $t^*_i$; this keeps the statement meaningful for agent types with $r_i \ge \theta|u| r$, for which equation (6) gives no maximizer.