On the Equation A³+B³=Cⁿ: Elementary Methods, Modular Obstructions, and a Brauer-Manin Research Program

We study primitive integer solutions to A³+B³=Cⁿ with gcd(A,B,C)=1 and n≥3. Novel results include: a p-adic valuation closure of Case 2 (gcd(A+B, A²-AB+B²)=3) for all n≥3 via a Ljunggren-type obstruction; explicit vector composition in Z[ω] formalizing composite exponent reduction; a synthesis closing all split Cartan obstruction primes (n≡1 mod 3, n>13) via Bilu-Parent-Rebolledo combined with an explicit j-invariant argument; closure of non-split Cartan primes above 1.4×10⁷ via Le Fourn-Lemos and the observation that j(EA,B) is never an integer for primitive solutions. The remaining gap — approximately 169,000 non-split Cartan primes between 13 and 1.4×10⁷ — is precisely identified. We propose a Brauer-Manin research program on the fiber product Z_n = X₀(3) ×{X(1)} X{ns}^+(n) as a path to closing it, and establish a conditional proof assuming ABC. Repository disclaimer: verify all Galois-modular hypotheses against primary literature before citation as settled mathematics.