A Spherical N-Series Area Law for Regular Refinement Families

This preprint investigates multiscale extrapolation of spherical area approximations computed from geodesic boundary polygons and chordal surface triangulations. The principal mathematical contribution is an analysis of the even-power asymptotic error structure underlying the observed second-, fourth-, sixth-, and eighth-order convergence sequence in regular spherical refinement families. Analytic results establish even-power asymptotic expansions for regular geodesic polygon approximations of spherical caps and uniformly triangulated latitude–longitude cells away from the poles. Other spherical geometries, including irregular polygon refinements, are examined through numerical evidence rather than a universal convergence theorem. The manuscript develops Richardson–Romberg extrapolation operators for cancelling successive even-power error terms and derives generalized multiscale weights for prescribed combinations of even and odd error exponents. A conditional perturbation analysis shows how mesh displacements proportional to the refinement scale can introduce a cubic error term that survives conventional even-power extrapolation. Exact residual factors and potential numerical validation tests are presented. The proposed mechanism is conditional on the stated smooth-coefficient expansion assumptions. Version 2.3 clarifies the mathematical scope, distinguishes analytic proofs from numerical observations, and excludes exploratory numerical diagnostics lacking complete provenance. No general even-power convergence theorem for arbitrary spherical meshes is claimed. Research status: Independently authored mathematical preprint with AI-assisted mathematical and numerical review. Independent journal peer review is not claimed. Supporting software archive: https://doi.org/10.5281/zenodo.23806064 GitHub repository: https://github.com/CWBaker360/spherical-nseries-area-law