Finite σ-Equivariant Row-Graded Magmas on AG(2, 3): Parameter Space, Intrinsic Selection, and Linearization

We study the finite parameter space of σ-equivariant row-graded magmas on M = S × S, where S = 𝔽₃ (equivalently, the point set of the affine plane AG(2,3)), subject to left-row output and a fixed same-row off-diagonal Steiner rule. The parameter space admits a normal form Ω′ ≅ S¹⁸ × S³ ≅ S²¹ of cardinality 3²¹ = 10 460 353 203. Two finite enumeration theorems completely map the ambient space. The associativity spectrum satisfies 63 ≤ Assoc ≤ 597, with exact endpoint loci and the full partition function. The path-resolved Hamming defect satisfies min H_tot = −2268, max H_tot = 7302, and maxN₋=0 H_tot = 7020. Inside this parameter space, the four internal finite conditions 𝒽_acc = 0, H_diag = 0, (r,r)² = (r,r), StabMaps_R = {C,J} select a unique point — the right-row projection magma RP. After selection, RP admits a complete finite linearization profile: rank-three left translations, a 51-element left-translation semigroup, an explicit endomorphism and intertwiner classification, the binary part of the term clone (size 630), a row-sum contraction matrix realization, a Hamming-defect operator identity, associated compact Lie groups (SU(3), SU(2), U(1)), and a signed symplectic lift of the C/J right-stabilizer geometry. The theorem package is organized as finite parameter space ⟹ intrinsic finite selection ⟹ linearization of RP. All finite claims are supported by exhaustive certificate files available in the public GitHub repository (see “Finite certificate protocol” in the paper).