Optimal unrestricted containment of one regular polygon in another

Croft asked for the largest regular polygon that can be translated, rotated, and scaled inside another. For coprime side counts at least five, we prove the conjecture of Dilworth and Mane that an optimum occurs at one of two endpoint configurations. The endpoint values never tie, and every interior orientation in the fundamental interval is strictly suboptimal. Together with the published cases, this determines the exact optimum for every ordered pair of regular polygons. In the coprime case, the optimal placement is unique modulo the natural dihedral symmetries of the two polygons and has exactly four contacts between inner vertices and the relative interiors of outer sides. These form two reflected pairs. They lie in a common open half-plane whose boundary passes through the outer center exactly for (m, n) = (2k + 1, 2k), k ≥ 3; otherwise one lies in each open quadrant determined by the common symmetry axis and its perpendicular. The proof uses support constraints, balanced three-constraint certificates, and centered modular residues. Supplementary computations check identities and finite ranges without serving as proof premises. License scope: original scholarly text, figures, and project-generated mathematical data are licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). Identified original software is licensed under the Apache License, Version 2.0. The included file-scope maps and third-party notices specify exceptions; historical input archives retain their existing terms.