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

A health-pool member who pays every year and never claims should, by insurance logic, leave first. She often stays, held by dual benefit: her own return is worth more as the collective's output grows. The coupling is the cross-partial δ ≡ ∂²W/∂G∂M between collective output G and personal benefit M. Under separability, the lower tail exits; where collective channels are closed, non-claimants leave first: Akerlof's unraveling. A stable positive membership exists if and only if δ exceeds a threshold δ*, where participation folds in a saddle-node bifurcation. Where δ > 0 the collective's value differs across members by δ·G·ΔM, and the best uniform transfer falls short by half that gap: the coupling cannot be bought out. In Indian panel data, illness does not predict who joins; delivered benefit predicts who stays. A Nepal renewal panel shows no claim × tenure interaction once calendar years and household frailty are controlled.

Version note. Version 112 (3 October 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. Changes from version 103: the abstract is revised; the Nepal renewal evidence in Section 7 is now counted by household (7,758 households, 16,550 household-years, 12,725 renewal decisions; households that filed no claim in the year renewed at 51.8 per cent and claimants at 79.4 per cent), with the de-identified household records and the script that reproduces every Banke figure deposited separately at https://doi.org/10.5281/zenodo.23114484; the funding, data and generative-AI statements are rewritten; the reference list is updated. 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. The record holds one archive, DBP_Preprint_v112.zip, which contains: (1) DBP_Canonical_v112.pdf and .docx — main text, nine sections and Appendix A. (2) DBP_Online_Supplement_v10.pdf — full proof of Theorem 1 (Section 3.3), the worked numerical example (Table 2, referred to from Section 3.4) and the axioms behind Assumption S (Section 6.2). (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 7; each line prints the manuscript value beside the recomputed value. (4) MANIFEST.md5 and README.txt — checksums of the four files and a guide to the archive.