More Calls, Not Necessarily Better Coordination: A Budgeted Study of Multi-Principal Reasoning

When one language-model service chooses for several principals, plausible text can conceal infeasibility, unequal compromise, or avoidable inefficiency. We introduce CoordCap, a deterministic finite-catalogue protocol whose local solver—not model-authored claims—scores hard constraints, principal regret, and Pareto efficiency. A fixed-budget main study contains 640 attempts over 80 tasks, two API models, and four prompt organizations. The observed Gemini-minus-GPT route difference is +17.50 pp in joint success (95% CI [+7.81 pp, +27.50 pp]) and +15.31 pp in Pareto efficiency ([+5.31 pp, +25.00 pp]), while no organization is a uniform winner. On a prospectively selected 24-task paired panel, one blind self-revision pass changes no scored outcome vector; a two-call blind audit-and-revision condition adds tokens, latency, and cost without reliable coordination gain and worsens pooled hard-constraint violations by +14.58 pp ([+2.08 pp, +27.08 pp]) and mean regret by +0.1338 ([+0.0292, +0.2541]). The ablation’s frozen 98% validity gate formally fails (96.53%), so we retain all 144 records, disclose complete-case sensitivity, and derive solver-based adversarial bounds. Those bounds preserve the pooled HCVR and regret directions but can change secondary model-conditional and Pareto conclusions. Thus our negative result is qualified: additional blind model-generated verification does not yield reliable gains in this controlled setting; it says nothing universal about tool-grounded or externally verified repair.