The Kodaira dimension and minimal models of A₆

We prove that the moduli space A₆ of principally polarized abelian sixfolds has Kodaira dimension zero, determining the remaining case in the Kodaira classification of the spaces Ag. More precisely, every nonzero scalar Siegel modular form of genus six, weight k and positive boundary order b satisfies k≥ 7b, with equality precisely for powers of the weight-fourteen harmonic theta series of Dittmann, Salvati Manni and Scheithauer. The canonical ring is therefore a polynomial ring on one generator of degree two. We also prove numerical dimension zero and construct the unique projective Q-factorial terminal minimal model. Its Picard number is one and its canonical divisor has order two. The perfect cone compactification is a Mori dream space, and the birational automorphism group of A₆ is trivial. The proof combines normal vanishing on the Prym locus, polar intersections on abelian boundary fibers, and an exact theta calculation. A discrepancy estimate recovers the modular open subset from the minimal model.