The Brocard–Ramanujan Equation in the Dodecahedral Cubic Field of Discriminant −23

We revisit the Brocard–Ramanujan Diophantine equation n!+1=m2n!+1=m2 from the viewpoint of a specific cubic number field of discriminant −23−23.In this framework, the factorial is not a scalar but an orbit in Z3Z3 generated by explicit 3×33×3 matrices with determinants k3−k+1k3−k+1, arising from the norm form of the field.The associated Koecher reduction theory produces a dodecahedral polyhedron with symmetry group A5A5, on which the factorial orbit projects to a finite set of vertices and inscribed tetrahedra.We show that the three known solutions (n,m)=(4,5),(5,11),(7,71)(n,m)=(4,5),(5,11),(7,71) are precisely the vertices where the cubic norm, the dodecahedral A5A5–symmetry, and a distinguished elliptic curve y2=x3−x+1y2=x3−x+1 intersect, and explain why the orbit cannot hit any further vertex compatible with n!+1=m2n!+1=m2.This yields a geometric saturation picture: the finiteness of Brocard–Ramanujan solutions is interpreted as a packing limit of factorial torsion in the dodecahedral cubic field.