Tsuidoku: Where Sudoku Meets MOLS

We show that imposing a second orthogonal Sudoku constraint on a 9×9 grid — turning a Sudoku into a Tsuidoku (Suirodoku), i.e. a Sudoku Graeco-Latin square — relaxes rather than tightens three foundational features of the Sudoku problem. First, the value 81, algebraically forbidden as an automorphism group order of any classical Sudoku grid, is realized as the structural stabilizer of a Tsuidoku orbit over the Sudoku-162 base. Second, uniquely-solvable puzzles exist with 13 clues, well below the Sudoku minimum of 17 (McGuire, Tugemann and Civario, 2014). Third, rigidity under paratopism is a property of the algebra of the base (Cayley table), not of the constraint count: over the back-circulant (isotope of Z_9) only 0.8% of orbits merge under paratopism, while over the Sudoku-162 (isotope of Z_3 x Z_3) 87.4% do. The paper connects two literatures previously kept separate: the Sudoku constraint-programming tradition (AllDifferent filtering, Cardinality (0,1)-Matrix propagation) and the Latin-square equivalence machinery (isotopism, paratopism, species). The 13-clue result is obtained via a native C solver implementing Régin and Gomes' Cardinality Matrix constraint. Full enumeration of the back-circulant slice yields 3.62 billion Tsuidoku, organized into 740 structural orbits and 734 paratopism species, with the Crystal Grid as a singleton invariant under all 1,296 symmetries of its stabilizer.