We develop a $K\text{--}R$ defect framework for classical inequalities centred on the normalised defect functional $\Phi _{f}(x,y;K,R) = D_{f}/(KR(x-y)^{2})$, where $D_{f}(x,y;K,R) = Kf(x)+Rf(y)-f(Kx+Ry)$ for an admissible pair satisfying $K\ge 0$, $R\ge 0$, $K+R=1$. The framework establishes a complete forward-inverse curvature theory. The Curvature Recovery Theorem proves that $\Phi _{f}$ equals $\tfrac{1}{2}f''(\xi )$ for some interior point ξ, making $\Phi _{f}$ 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 $C^{2}$ functions with identical normalised defect fields differ by at most an affine function. The Constant Defect Characterisation identifies quadratic polynomials as precisely the $C^{2}$ functions with constant normalised defect. The Reconstruction Formula recovers f from its defect measurements by double integration. The $K\text{--}R$ Stability Theorem proves that a $C^{2}$ function with small defect magnitude satisfying homogeneous boundary conditions is uniformly close to zero, with an explicit constant. A $K\text{--}R$ deficit theory for the Hölder inequality is established, including a new $K\text{--}R$ Isoperimetric Deficit Ratio, with equality conditions strictly stronger than the classical case. Applications to the nonlinear boundary value problem $-u''=f(u)$ 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 $\Phi _{f}$ to the Rayleigh quotient of the Hessian, motivating a programme for full Hessian recovery from defect measurements.
