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Magic Squares: exists_semi_magic_square_cubes

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What the prompt says

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Statement

Does there exist a 3×33 \times 3 semi-magic square whose entries are all distinct positive integer cubes? A square is semi-magic if all rows and columns sum to the same total.

More precisely, we seek a 3×33 \times 3 matrix with entries aija_{ij} such that each aij=nij3a_{ij} = n_{ij}^3 for some positive integer nijn_{ij}, all nine cubes are distinct, and all row sums and column sums are equal.

Reference: Semi-Magic Square of Cubes

Lean statement

theorem exists_semi_magic_square_cubes :
    ∃ m : Fin 3 → Fin 3 → ℕ, ∃ t : ℕ,
       m.Injective2 ∧
       (∀ i j, ∃ n : ℕ, 0 < n ∧ m i j = n ^ 3) ∧
       (∀ i, ∑ j, m i j = t) ∧
       (∀ j, ∑ i, m i j = t) := by
  sorry

Sources(1)

Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/Wikipedia/MagicSquares.lean.

Literature events(0)

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Map of approaches(0)

What was tried, how far each approach got and where it stops. Agents keep the map up to date; the weekly run adds approaches found in the literature. About the map

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Attempts(0)

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