Directional Curvature Expansion of the Baker Geometric Refinement

This mathematical preprint investigates the directional curvature expansion of the Baker proportional-subtended geometric refinement construction on smooth Riemannian surfaces. The analysis extends the Euclidean cubic refinement-error coefficient to curved geometry through local geodesic side-law expansions. The resulting formulas identify corrections involving Gaussian curvature, its directional derivative, and higher-order geometric invariants. For the Baker construction with proportional parameter q = 4/3, the Euclidean limiting cubic coefficient is −7/384. Curvature-dependent corrections describe how this coefficient changes with the local geometry and direction of measurement. The manuscript develops higher-order expansions, examines their algebraic structure, and presents numerical verification on a torus. Its analytical derivations and numerical experiments are explicitly distinguished, with the scope of the reproduced and externally reported checks documented. The reported torus calculations concern the specified surface and meridional directions. They are not presented as universal numerical certification for arbitrary Riemannian surfaces. Version 2.5 incorporates mathematical and editorial corrections, clarifies regularity assumptions and verification provenance, and states the limitations of the numerical evidence. Research status: Independently authored mathematical preprint with AI-assisted mathematical and numerical review. Independent journal peer review is not claimed. Supporting software/repository archive: https://doi.org/10.5281/zenodo.23806184 GitHub repository: https://github.com/CWBaker360/curvature-deformed-arcsine