The Dual-Benefit Threshold: Why No Transfer Replaces a Collective

A member of a community health pool pays every year and never claims. By insurance logic, she should be the first to leave. She often stays, and dual benefit is what holds her: her own return is worth more as the collective’s output grows, a coupling measured by the cross-partial δ ≡ ∂²W/∂G∂M between collective output G and personal benefit M, zero when the benefit separates. The coupling changes who exits: under separability the non-claimant leaves first; where the collective channels are legible and clear the premium, she renews, so the lemons selection is switched off. Two results carry the paper. A pool retains a stable, positive membership if and only if δ exceeds a critical value δ*, at which the participation dynamics fold in a saddle-node bifurcation; where the collective channels are closed to the member, the non-claimants leave first, an unraveling like Akerlof’s. And where δ > 0, what the collective adds differs across members by δ·G·ΔM, so no single cash transfer fits all, and the best one is out by half that gap: the coupling cannot be bought out. The testable content is about staying, not joining. Indian panel data show illness does not predict who joins, and delivered benefit predicts who stays; a five-year Nepal renewal panel shows the claimant–non-claimant gap widening with tenure on the log-odds scale, the sign the coupling predicts. Version note. Version 103 (30 September 2026) of the canonical statement of the dual-benefit factor DB of the Trinity Law S = T × C × DB. Dual benefit is defined as the complementarity between a member's private return and the collective output of a voluntary collective organization, the cross-partial δ = ∂²W/∂G∂M of the money-metric benefit, with DB = δ/δ_max. This deposit supersedes the conjunctive formulation of dual benefit in the authors' earlier deposits (Zenodo 10.5281/zenodo.17582549 and 10.5281/zenodo.17726291). Files. (1) DBP_Canonical_v103.pdf and .docx — main text, nine sections and Appendix A. (2) DBP_Online_Supplement_v5.pdf — full proof of Theorem 1, the worked example (Table 2) and supporting characterizations. (3) dbp_audit_checks.R — replication script in base R (no packages; run with Rscript dbp_audit_checks.R) that recomputes the worked example of the supplement (Table 2), the wedge of Theorem 4, the fold constant of Theorem 3, the constant of Lemma A.4.1 and the two-scale renewal interaction of Section 8.2; each line prints the manuscript value beside the recomputed value. Keywords: dual benefit; voluntary collective organizations; strategic complementarities; community-based health insurance; complementarity; institutional design Declaration of generative AI use. During the preparation of this work, the authors used a generative AI system (Claude, Anthropic) as a drafting and editing aid, under the authors’ direction and review throughout. All theorems, proofs, definitions, and interpretations of results are the authors' own; the tool was used for prose drafting, consistency checking, and formatting, and the authors verified every substantive claim before inclusion. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.