Recursive Unique-Prime Magic Squares by Centre-Shell Expansion - 2187 x 2187 Solution

A Fully Nested 2187 × 2187 Unique-Prime Magic Square with Verification Package This record contains a completed 2187×2187 recursively nested unique-prime magic square, together with supporting verification materials and methodological documentation. The square contains: 2187 Sqaured=4,782,969 entries;all entries are distinct prime numbers;centre value: 80,999,999,953;magic constant: 177,146,999,897,211;verified aligned recursive levels: 3,9,27,81,243,729, and 2187. The construction extends the recursive unique-prime magic-square sequence previously documented at 81×81, 243×243, and 729×729. The square is not merely a large prime-valued magic square: it is recursively nested, meaning that aligned sub-squares at each claimed power-of-three level satisfy their own magic-square conditions. The record also includes an independent verification package, prime_magic_square_verifier.py, designed to audit the released dataset. The verifier checks file shape, primality, global uniqueness, full row and column sums, both principal diagonals, recursive aligned subblock structure, optional residue conditions, SHA-256 hashes, and JSON/TXT audit reports. This release is presented as a finite verified computational construction. It does not claim an infinite-family theorem for all orders 3n, nor does it claim a universal law of prime distribution. The purpose of the record is to provide the dataset, verification tools, audit materials, and accompanying papers needed for independent inspection and reproducibility. The work was produced through human-directed, AI-assisted computational exploration. All mathematical claims are subordinate to explicit verification.