IOL power calculation in cataract surgery: closed-form sensitivities to biometric uncertainties

We derive closed-form analytical sensitivities of pseudophakic refraction with respect to corneal power (K), axial length (L), effective lens position (E), and intraocular (IOL) power (P) from a single implicit paraxial vergence equation. Each partial derivative is expressed in both local-vergence and parameter-only forms, with explicit reference-plane conventions (corneal versus spectacle). The axial-length sensitivity scales as T^-2, predicting increased vulnerability in short eyes. The effective lens position (ELP) sensitivity admits a factorization explaining the empirical heuristic ΔR≈0.0006(P²+2KP)ΔE_mm. The IOL-power adjustment ratio Q=(∂R/∂P)^-1 matches a previously published expression, confirming independence from P and L. Clinical application to uncertainty budgeting reveals that the ELP prediction dominates refractive variance in short eyes, while keratometry dominates in long and post-refractive-surgery eyes.