Informational Inequivalence of Formally Equivalent Mathematical Frameworks: A Multi-Framework Analysis of the Inverse Square Law

We distinguish between formal equivalence (identical result in the ideal case) and informational equivalence (identical predictive content in all cases). Six independent derivations of the inverse square law are applied to predict gravitational acceleration at six real-world locations. Each framework fails distinctly: spherical harmonics reduce errors by two orders of magnitude, Gauss uniquely handles sub-sea-level data. We propose the principle of informational inequivalence and extend the analysis to higher dimensions and Bertrand's theorem. Supplementary Python code included.