Declarations

and materials. This is a theoretical work with no empirical dataset. All definitions, calculations, tables, proofs, and schematic TikZ figures are contained in this source.

citations. Cheung manuscripts are cited by registered identifier and supplied version as unpublished documentary comparisons. Their relevant mathematical statements are proved locally when used. They are not treated as independent verification of this manuscript. No publication date or journal status is assigned to those manuscripts.

and submission checks. The manuscript gives explicit symbolic calculations of the branch bounds, particle intersections, cell coverage, edge lengths, projected and induced areas, adapted charts, affine translation cancellation, primitive winding, intrinsic angle defects, and rejection residuals.

No successful pdflatex execution, executable exact-arithmetic verification, proof-assistant check, independent numerical execution, independent referee review, or journal acceptance is asserted. Author information and a clean local build with resolved references remain necessary submission checks.