A K-R defect framework for classical inequalities: curvature recovery, uniqueness, stability, and applications to nonlinear boundary value problems

We develop a K--R K – R defect framework for classical inequalities centred on the normalised defect functional Φ f(x,y;K,R) = Df/(KR(x-y)²) Φ f ( x , y ; K , R ) = D f / ( K R ( x − y ) 2 ) , where Df(x,y;K,R) = Kf(x)+Rf(y)-f(Kx+Ry) D f ( x , y ; K , R ) = K f ( x ) + R f ( y ) − f ( K x + R y ) for an admissible pair satisfying K≥ 0 K ≥ 0 , R≥ 0 R ≥ 0 , $K+R=1$ K + R = 1 . The framework establishes a complete forward-inverse curvature theory. The Curvature Recovery Theorem proves that Φ f Φ f equals 1/2f''(ξ ) 1 2 f ″ ( ξ ) for some interior point ξ , making Φ f Φ 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 C² 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 C² C 2 functions with constant normalised defect. The Reconstruction Formula recovers f fro