A Curvature-Deformed Arcsine Coordinate for Cubic Geometric Refinement

This paper develops a curvature-deformed analogue of the arcsine coordinate arising from an intrinsic geometric encode–scale–decode construction on Riemannian surfaces. A normalized equal-radius geodesic side law is shown to define an exact local linearizing coordinate for the associated refinement maps. In the Euclidean limit this coordinate reduces to the ordinary arcsine, while on a sphere it admits a closed-form expression and yields an exact Schröder/Koenigs conjugacy together with a local multiplicative semigroup. For a general Riemannian surface, the variable-curvature expansion is derived analytically through geometric weight five. The resulting coefficient functions have a finite Laurent structure in r=1−z2r=\\sqrt{1-z^2}, which organizes the Taylor coefficients into finite combinations of generalized binomial sequences. At geometric weight six, a high-precision intrinsic torus computation is used to reconstruct a rational candidate side-law formula in eight natural curvature invariants. Independent geometry and angle holdout tests support this reconstruction. The weight-six result is explicitly conditional: it has not yet been independently derived from a degree-eight normal-coordinate expansion or certified by interval arithmetic. Conditional on the reconstructed side law, the subsequent symbolic inversion and Laurent-kernel formulas are exact. The work connects intrinsic geodesic geometry, Schröder–Koenigs linearization, curvature expansions, and cubic geometric refinement within a common local framework.