We classify all σ-equivariant non-associative binary algebras (magmas) on the 9-element set M = S × S (|S| = 3), subject to row-grading (AX0) and Steiner fiber condition (AX2). The classification space contains 162 magmas falling into exactly 90 isomorphism classes (Burnside count). A master formula involving five discrete parameters determines the associativity of any magma, yielding exactly 28 distinct values in the interval 162, 567, all divisible by 3. Both normalized bounds 2/9 and 7/9 are sharp. The selected canonical PAB magma admits a natural linearization to a simple magma algebra kM ≅ Mat₃(k). Under the adjoint action this yields the decomposition gl(3) = u(1) ⊕ sl(3), with the unique compact S₃-invariant 3-dimensional subalgebra being so(3). Three independent information-theoretic criteria (minimum Landauer access cost, minimum diagonal entropy, and minimum associativity) jointly select this unique magma from the full space. The paper includes a complete computational verification suite and all classification data.
Burundai Taryi (Thu,) studied this question.