σ-Equivariant Magmas on AG(2,3): finite landscape, intrinsic selection, and linearization.

We study a finite family of σ-equivariant magmas on M=F3×F3 with left-row output and a fixed same-row Steiner rule. The resulting landscape has size 3²1 (10 460 353 203 different magmas). The paper develops three layers. First, we establish finite landscape results, including the exact associativity atlas and a global hidden-continuation frontier. Second, we prove a finite intrinsic selection theorem isolating a unique magma from the landscape by four internal conditions: zero hidden-column access, zero diagonal entropy, diagonal idempotence, and pure C/J directed-edge geometry. Third, we analyze the selected magma through its operator structure, finite composition envelope, binary term algebra, matrix-carrier bridge, and companion Lie/symplectic structures. The organizing implication is: finite landscape → intrinsic finite selection → linearization of the selected magma. The work combines finite universal algebra, information-theoretic selection principles, and computer-assisted finite certification.