Slack and Budget Breaking in Threshold Team Production

A threshold system completes a public task only after $\kappa$ verifiable shares are publicly committed. If the honest schedule creates $\text{Nstar}=\kappa +\mathrm{\Delta }$ share opportunities by deadline $t^{\star }$, then $\mathrm{\Delta }$ shares are slack such that a coalition delays completion if and only if it withholds at least $\mathrm{\Delta }+1$ shares. The incentive problem is therefore to price the cheapest sabotage set. Agents receive a direct fee $f$ per committed share. A delaying coalition may also obtain delay value at most $L$, and may earn additional fee revenue during recovery after the deadline. Let $R_1^{+}$ be a pathwise upper bound on the coalition’s incremental fee revenue in a recovery slot that completes the task, including any same-slot overshoot. The principal can post a nonnegative completion bounty that depends only on committed shares, uses no deposits or punishments, and expires if completion is late. The optimal rule is uniform, as if completion occurs by $t^{\star }$, every admissible horizon share receives $B\mathrm{/}\text{Nstar}$, otherwise no bounty is paid. Full participation is ex-post strongly delay proof exactly when $(\mathrm{\Delta }+1)f+\frac{\mathrm{\Delta }+1}{\text{Nstar}}B\ge L+R_1^{+}\mathrm{.}$ Equivalently, the exact worst-case budget is $B^{\star }=\frac{\text{Nstar}}{\mathrm{\Delta }+1}(L+R_1^{+}-(\mathrm{\Delta }+1)f{)}^{+}\mathrm{.}$ The bound is tight for every nonnegative completion measurable bounty, among the $\text{Nstar}$ horizon shares, some $\mathrm{\Delta }+1$ receive total bounty at most $(\mathrm{\Delta }+1)B\mathrm{/}\text{Nstar}$, and withholding precisely those shares delays completion. The result applies to threshold signatures, data availability certification, coded dissemination, and generic $k$-of-$n$ completion tasks. We also isolate a separate limit, no transfer rule based only on completed shares can remove a final slot race in which a coalition has already observed enough pre-completion shares to act.