{"ID":2826877,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.17170","arxiv_id":"2512.17170","title":"Ethic Duality: A Homological Framework for Primal-Dual Problems","abstract":"We develop a homological duality framework based on a contravariant functor $D=\\operatorname{Hom}_E(-,R)$ with dualizing object $R$. A morphism is called ethic when it satisfies the canonical double-dual compatibility $D^2(f)η=ηf$. In the derived setting, the functor $\\mathrm{RHom}_E(-,R)$ produces a graded family of Ext-groups that measure all failures of this compatibility. The first layer $\\operatorname{Ext}^1$ identifies primal-dual gaps, while higher $\\operatorname{Ext}^k$ provide a systematic hierarchy of derived obstructions to exactness. This formulation specializes uniformly across several classical domains. In linear and conic optimization, Farkas- and Slater-type exactness criteria correspond to the vanishing of $\\operatorname{Ext}^1$, and integer duality gaps coincide with torsion Ext-classes. In graph theory, Kirchhoff- and Baker-Norine-type dualities arise as instances of ethic exactness. In dynamical systems, the higher derived layers encode nonvanishing persistence phenomena. Additional examples include social-choice configurations, categorical factorization in scattering formalisms, coding-theoretic duality, and Bellman-type recurrences, all appearing as concrete instances of Ext-controlled exactness. All resulting invariants are stable under derived Morita equivalence and depend only on the dualizing pair $(E,R)$. The framework therefore provides a substrate-independent criterion for primal-dual exactness and a uniform homological description of its obstructions.","short_abstract":"We develop a homological duality framework based on a contravariant functor $D=\\operatorname{Hom}_E(-,R)$ with dualizing object $R$. A morphism is called ethic when it satisfies the canonical double-dual compatibility $D^2(f)η=ηf$. In the derived setting, the functor $\\mathrm{RHom}_E(-,R)$ produces a graded family of E...","url_abs":"https://arxiv.org/abs/2512.17170","url_pdf":"https://arxiv.org/pdf/2512.17170v1","authors":"[\"Dmitry Pasechnyuk-Vilensky\",\"Martin Takáč\"]","published":"2025-12-19T02:15:44Z","proceeding":"math.CT","tasks":"[\"math.CT\",\"math.OC\"]","methods":"[\"Large Language Model\"]","has_code":false}