math-014-disprove: Disprove Sums of Three Cubes

[IndieMath:math-014:disprove] Disprove the Sums of Three Cubes conjecture through an IndieMath AI research attempt. Canonical statement: Every integer k with k ≢ ±4 (mod 9) is expressible as x³ + y³ + z³ for some integers x, y, and z. Contributions fund model usage and related compute for this direction.