A Fully Nested 729 x 729 Unique-Prime Magic Square Constructed from Nine Correlated 243 x 243 Prime Magic Blocks

Description of attached documents

This Zenodo record is accompanied by three documentation files describing, contextualising, and auditing the construction of a fully nested 729×729 unique-prime magic square.

The first document, “A Fully Nested 729×729 Unique-Prime Magic Square Constructed from Nine Globally Disjoint, Correlated 243×243 Prime Magic Blocks,” is the main companion note for the dataset. It describes the 729×729 construction, whose 531,441 entries are all distinct prime numbers, with master centre 10,000,000,033 and magic constant 7,290,000,024,057. It records the verified recursive nesting at the 3×3, 9×9, 27×27, 81×81, 243×243, and 729×729 levels, and explains the block-as-entry construction using nine globally disjoint, correlated 243×243 prime magic-square blocks. It also states the scope limits of the result: the dataset is presented as a verified computational construction, not as a proof of an infinite family or a theorem about prime distribution.

The second document, “Prime-Shell Fertility and Recursive Prime Magic Squares,” is the main construction-method report. It formalises the centre-shell prime-sourcing method, prime-shell fertility, coordination capacity, residue-lane filtering, recursive block assembly, global uniqueness pressure, and the distinction between reusable recursive grammar and non-reusable numerical displacement templates. It situates the 729×729 construction within the verified sequence of 81×81, 243×243, and 729×729 unique-prime magic squares, while also clarifying prior-art boundaries and avoiding overclaiming against ordinary recursive magic squares or earlier prime magic-square work.

The third document, “Optional Hyper-Symmetry Addendum for the 729×729 Nested Unique-Prime Magic Square,” records a secondary structural audit. It separates the core verified claims from optional stronger symmetry properties. The addendum confirms the required structure, including prime entries, global uniqueness, 13 mod 30 residue-lane purity, aligned recursive block magic, recursive centre-grid magic, and exact local 3×3 centre-shell orientation. It also reports optional hyper-symmetry tests, including full opposite-pair associativity, exact D4 value invariance, and arithmetic-progression centre-line behaviour, noting which of these stronger properties are not satisfied by the present object and should instead be treated as possible targets for future constructions.