A Shared Gamma-Formula Framework and Numerical Verification for the Degenerate Scales of Regular and Star Polygons

This paper develops a shared gamma-formula framework for predicting and verifying the degenerate scales of regular and star polygonal domains in the two-dimensional Laplace boundary integral formulations. Starting from the known gamma-function expression for regular N,1-gons, we examine how the same analytical scale law can be extended to N,2-gons when the scale is measured by the twofold adjacent-tip distance instead of the ordinary onefold side length. The resulting correspondence is formulated as a scale-law relation between the two polygon families, whereas the geometric distinction between regular and star polygons is explicitly retained. Boundary element computations are then used to detect the degenerate scale through the singular behavior of the single-layer influence matrix. The smallest singular value is used as the primary numerical indicator, and determinant profiles are reported as auxiliary visualizations of matrix singularity. Representative (5,2)-, (7,2)-, and (9,2)-gons show close agreement with the proposed gamma-formula prediction, and mesh-refinement results support the reproducibility of the numerical detection. In addition, the circular-limit normalization confirms that regular and star polygonal families approach the same unit-circle benchmark under their respective characteristic measures. These results suggest that polygonal degenerate scales possess a broader gamma-based structure connecting logarithmic capacity, boundary integral operators, star-polygon geometry, and BEM-based numerical verification.