{
  "abstract": "We develop a K – R $K\\text{--}R$ defect framework for classical inequalities centred on the normalised defect functional Φ f ( x , y ; K , R ) = D f / ( K R ( x − y ) 2 ) $\\Phi _{f}(x,y;K,R) = D_{f}/(KR(x-y)^{2})$ , where D f ( x , y ; K , R ) = K f ( x ) + R f ( y ) − f ( K x + R y ) $D_{f}(x,y;K,R) = Kf(x)+Rf(y)-f(Kx+Ry)$ for an admissible pair satisfying K ≥ 0 $K\\ge 0$ , R ≥ 0 $R\\ge 0$ , K + R = 1 $K+R=1$ . The framework establishes a complete forward-inverse curvature theory. The Curvature Recovery Theorem proves that Φ f $\\Phi _{f}$ equals 1 2 f ″ ( ξ ) $\\tfrac{1}{2}f''(\\xi )$ for some interior point ξ, making Φ f $\\Phi _{f}$ a derivative-free curvature estimator. The Local Limit Theorem establishes exact pointwise curvature recovery in the limit, with an explicit error bound. The Uniqueness Principle proves that two C 2 $C^{2}$ functions with identical normalised defect fields differ by at most an affine function. The Constant Defect Characterisation identifies quadratic polynomials as precisely the C 2 $C^{2}$ functions with constant normalised defect. The Reconstruction Formula recovers f from its defect measurements by double integration. The K – R $K\\text{--}R$ Stability Theorem proves that a C 2 $C^{2}$ function with small defect magnitude satisfying homogeneous boundary conditions is uniformly close to zero, with an explicit constant. A K – R $K\\text{--}R$ deficit theory for the Hölder inequality is established, including a new K – R $K\\text{--}R$ Isoperimetric Deficit Ratio, with equality conditions strictly stronger than the classical case. Applications to the nonlinear boundary value problem − u ″ = f ( u ) $-u''=f(u)$ include derivative-free source term curvature bounds, a polynomial reconstruction algorithm with a proved positive-definite Hessian guarantee, and a derivative-free convexity diagnostic for solution profiles. A multivariable extension connects Φ f $\\Phi _{f}$ to the Rayleigh quotient of the Hessian, motivating a programme for full Hessian recovery from defect measurements.",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "RamaKrishna Pasupuleti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2pbq595wms81v.bib",
  "cited_by": 0,
  "doi": "10.1186/s13661-026-02333-6",
  "fulltext_url": "https://doi.org/10.1186/s13661-026-02333-6",
  "id": "2pbq595wms81v",
  "issns": [
    "1687-2770",
    "1687-2762"
  ],
  "landing_url": "https://doi.org/10.1186/s13661-026-02333-6",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "lcc",
      "QA299.6-433"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7170150110",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-01",
  "publisher": "SpringerOpen",
  "ris": "https://research.draconitic.com/paper/2pbq595wms81v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T13:46:10Z",
      "native_id": "753e66d72d404524922d107deedef2ba",
      "source": "doaj",
      "url": "https://doi.org/10.1186/s13661-026-02333-6"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2604|Mathematical Inequalities and Applications",
      "subfield": "general"
    }
  ],
  "title": "A K-R defect framework for classical inequalities: curvature recovery, uniqueness, stability, and applications to nonlinear boundary value problems",
  "title_text": "A K-R defect framework for classical inequalities: curvature recovery, uniqueness, stability, and applications to nonlinear boundary value problems",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T13:46:10Z",
  "url": "https://research.draconitic.com/paper/2pbq595wms81v",
  "venue": "Boundary Value Problems",
  "versions": []
}