A Fully Nested 81×81 Unique-Prime Magic Square Constructed by Recursive Centre-Shell Expansion

This record provides companion documentation for the dataset “A Fully Nested 81×81 Unique-Prime Magic Square Constructed by Recursive Centre-Shell Expansion” (DOI: https://doi.org/10.5281/zenodo.20005776). The dataset contains an 81×81 magic square with 6561 globally distinct prime entries. The square is fully nested: every aligned 3×3, 9×9, and 27×27 sub-square is itself magic, and the full 81×81 square has constant row, column, and main-diagonal sums. The master centre is 800011 and the global magic constant is 64800891. These companion notes describe the construction method, verification procedure, and mathematical framework behind the dataset. The construction is based on recursive 3×3 centre-shell expansion, where prime centres are expanded into local 3×3 prime magic shells and assembled into a nested higher-order structure. The document includes: 1. A concise note accompanying the 81×81 nested unique-prime magic square.2. A constructional chronicle describing the development of the method and the verified result.3. A discussion of prime-shell fertility, residue admissibility, and the magic-square sieve framework.4. A source reconciliation note explaining how earlier draft materials were consolidated.5. A minimal verification appendix for independently checking the CSV dataset. The notes also clarify the limits of the claim. This work presents a computationally checked construction and does not claim a proof of an infinite family, a theorem about the distribution of primes, or a completed theory of recursive prime magic squares. No independent verification had been performed at the time of publication; readers are invited to run the included verification procedure and report results. The document was produced through human-directed, AI-assisted mathematical exploration. AI tools were used as coding, verification, analysis, and documentation assistants, while the constructional framing, constraint design, interpretation, and publication decision were directed by the author.