Equation 677 Database

Size-49 "pivot + 7 petals" magma — 1 size-7 sub-magma + 7 size-6 Aut-orbits. Size 49, exactly 1 idempotent (element 48), right-cancellative. |Aut(M)| = 6 with 14 Aut-orbits: - 7 singletons: {30}, {31}, {32}, {33}, {34}, {35}, {48} — the elements of the unique size-7 sub-magma - 7 size-6 orbits: {0..5}, {6..11}, {12..17}, {18..23}, {24..29}, {36..41}, {42..47} — the 7 "petals" Sub-magma structure: exactly ONE proper non-singleton sub-magma exists, the size-7 set {30, 31, 32, 33, 34, 35, 48}, isomorphic to magma#7981e2df (the size-7 Eq 677 magma with exactly 1 idempotent). Every pair of distinct elements (x, y) with at least one element outside the pivot generates the full magma — i.e., no size-7 sub-magma is "shared" between petals, unlike the AG(2, 7) pencil magma#1b32837d which has 8 size-7 sub-magmas through a common pivot. The 7 petals exactly coincide with the size-6 Aut-orbits. The pivot's 7 elements are exactly the L_48-fixed points (where T(48, y) = y), and they form the size-7 sub-magma containing the unique idempotent 48. So 49 = 7 (pivot) + 7 × 6 (petals) = 7 × 7 in the multiplicative sense, but NOT factored through any AG(2, 7) line structure — this is a different kind of "size-49 quotient by F_7-acting-on-something" construction. The Aut-orbit structure (7 fixed singletons + 7 transitively permuted size-6 sets) suggests a cyclic Z_6 action on each petal, with the 7 petals collectively permuted as a single Aut-orbit at a higher level — though |Aut| = 6 (not 7 × 6 = 42), so the petals themselves are NOT exchanged by Aut. Each petal is fixed setwise, with Aut acting on the 6 elements within each petal. In the suggested reorder, positions 0..41 contain the 7 petals (each 6 consecutive cells, in canonical-label order), and positions 42..48 contain the pivot (with idempotent 48 last). Diagonal 6×6 blocks at positions 5k..5k+5 reveal within-petal structure; the last 7×7 sub-table is the pivot sub-magma; off-diagonal blocks capture cross-petal cross-pivot operations. Compare: magma#1b32837d (size 49, AG(2,7) pencil): 8 size-7 lines through a common pivot, |Aut| = 252 magma#fa885ee6 (size 49 AG(2,7)-like): 8 size-7 sub-magmas, non-RC this magma: ONLY 1 size-7 sub-magma + 7 petals of 6, |Aut| = 6, RC [text written by Claude]