Equation 677 Database

Translation-only Steiner-line magma at size 61, "half-Phi-tribe" slope pattern. Size 61, fully idempotent, RC. 183 size-5 sub-magmas (each = the F_5 affine line magma#e549b5f8). |Aut(M)| = 61 (only Z_61 translation). In the F_61 translation labeling, x*y = x + f(y - x). Slope alpha(y) = f(y)/y takes 12 distinct values: 4 Phi_10 root slopes {3, 27, 41, 52}, each on a size-7 subset (28 elements total) 8 non-Phi_10 slopes {7, 10, 21, 30, 32, 35, 55, 59}, each on a size-4 subset (32 elements total) The 8 non-Phi slopes form 4 inverse pairs in F_61*: {7,35}, {10,55}, {21,32}, {30,59}. This (Phi-sum=28) fingerprint is shared by exactly 2 magmas in the DB: this one and magma#2426bbe0. Both belong to the same "Phi+inverse-pair tribe" as magma#0bcf3cca (Phi-sum=60), magma#e87f6bb9 (Phi-sum=44). All four magmas use the same 12-element slope value set {3, 7, 10, 21, 27, 30, 32, 35, 41, 52, 55, 59} = (4 Phi_10 roots) union (8 non-Phi inverse-paired extras). They differ in how those 12 values are assigned to the 60 elements of F_61*. The other ~26 translation-only Steiner-line magmas at size 61 use larger and more chaotic slope-value sets. [text written by Claude]