Global Perelman-Ricci-Poincare-Inspired Inequality Diagnostics: Curvature, Entropy and Graph-Topological Modelling of Macro-Regional Pressure and Smoothing Capacity

This study develops a Global Perelman-Ricci-Poincaré-inspired economic inequality diagnostic framework for measuring the world economy as a macro-regional pressure-smoothing manifold rather than as a collection of isolated inequality indicators. The framework does not claim to prove, test or replicate Perelman's proof, the classical Ricci-flow theorem or the Poincaré conjecture. Instead, it translates their structural modelling logic into a bounded economic analogue in which inequality pressure, smoothing capacity, curvature-like deformation, entropy-like complexity and graph connectedness are jointly measured. The empirical architecture uses ten macro-regional blocks over 2010-2024 and sixteen stress-coordinate variables, yielding a balanced diagnostic panel of N = 150 region-year observations. Raw macro-social indicators are transformed onto a common 0-100 pressure-oriented scale and then used to construct a correlation-based metric tensor, Ricci-style component curvature, normalised Ricci-flow summaries, world-weighted scalar curvature, Perelman-type F- and W-functionals, graph Laplacian connectedness diagnostics, Betti numbers, Ollivier/Forman network-curvature proxies, Ricci-surgery rankings, counterfactual sensitivity tests and econometric validation layers. The corrected results identify 2020 as the strongest crisis-deformation year: global pressure rises to 38.079, scalar curvature falls to 6.216, and the W-functional reaches 508.023. By 2024, pressure declines to 30.314, and curvature rises to 10.264, indicating partial recovery, while elevated W-functional complexity and persistent regional dispersion prevent a full-convergence interpretation. Component results show that curvature energy is concentrated mainly in investment, productivity, technology access, unemployment and fiscal-pressure channels. Graph diagnostics show a connected, dense and looped inequality system with beta_0 = 1, algebraic connectivity of 0.994 and beta_1 = 72. The contribution is methodological and empirical: a transparent mathematical-econometric architecture for global inequality diagnostics, bounded validation and claim-disciplined interpretation.