Discrete 31-Based Magic Structures: The 31×31 Square, 31×31×31 Cube, and 31⁴ Hypercube

This paper presents a finite-field framework for a family of discrete mathematical structures of order 31: a 31×31 square, a 31×31×31 cube, and a four-dimensional 31⁴ hypercube. Using affine coordinate mappings over ℤ/31ℤ, the construction places consecutive integers bijectively while producing constant sums along coordinate-axis lines under explicitly stated algebraic conditions. Additional conditions establish the toroidal panmagic property for the two-dimensional square, including rows, columns, and broken diagonals in both principal directions. The original 31×31 construction begins with 36 at row 1, column 16. Its exact average, 516, occurs at the geometrical center, row 16, column 16. The paper also examines complementary pairs, line-sum constants, prime factorizations, repeating base-31 digit patterns, and the general arithmetic relationships extending through the cube and 31⁴ hypercube. For an initial value of 36, the calculated coordinate-axis line sums are 15,996 for the square, 462,861 for the cube, and 14,315,676 for the four-dimensional hypercube. The paper clearly distinguishes proved mathematical properties, computationally verifiable properties, proposed interpretations, and unsupported physical claims. These structures are finite abstract mathematical objects; no claim of an additional physical dimension or subatomic phenomenon is made.