High Influence Is Not New Influence: A No-Arbitrage Analysis of Multi-Capability Data Selection

When is one influence score per example enough to select data for several target capabilities? At rank one with a positive direction, fixed prices, task-wise max, round-robin, capability-level BIDS and greedy gap maximization select the same data, and near rank one a fixed linear price keeps a guaranteed share of every capability's best gain, tight up to a constant for two capabilities. On real payoffs, far from this regime, a fractional coverage ceiling bounds what one score loses. Beyond rank one, high influence is not new influence: to first order, a candidate is non-redundant only if no cost-matched mixture of selected data matches it on every capability. The replication program's dual prices capabilities by scarcity; with an inactive cap, the gap is the candidate's one-step, price-weighted advantage over every such mixture. On image benchmarks, the rules coincide on payoffs deformed to rank one; the one-step identity holds for small steps but stops predicting within 3% of the parameter distance training travels; and scalar influence falls below random sampling in all 32 targeted comparisons, a failure that first-order coverage flags, while gradient distance to the targets also ranks the other rules. Gradient matching is the most accurate rule that does not know the targets; our greedy rule is a probe, and we do not fine-tune language models.