Equation 677 Database

Size-76 idempotent right-cancellative magma satisfying Eq 677 and Eq 255. The structure is a **'pencil' of 5 size-16 sub-magmas through a common pivot element**, with the 75 non-pivot elements glued together by a **transversal design TD(5, 15)** of size-5 sub-magmas. Concretely: • There is one distinguished element (the 'pivot', canonical label 62) that lies in 5 sub-magmas of size 16 and 0 sub-magmas of size 5. • The 5 size-16 sub-magmas (the 'petals') all contain the pivot and partition the other 75 elements into 5 groups of 15 each. Every petal is isomorphic to magma#6fa95655 (the size-16 idempotent linear magma over Z/16Z with x ◇ y = 9x + 8y mod 16) — verified by submitting each petal's 16×16 sub-Cayley table to the canonicalization API. • There are 225 = 15² size-5 sub-magmas, each isomorphic to the unique size-5 Eq 677 magma over F_5 (the affine sharply-2-transitive one). Every such size-5 sub-magma is a **transversal of the 5 petal groups**: it has exactly one element in each petal-group (excluding the pivot). The 225 transversals form a TD(5, 15) (transversal design), equivalently 3 mutually orthogonal Latin squares of order 15. • Every pair of distinct elements (x, y) generates: - the full petal (size 16) if x, y are in the same petal (= same group plus possibly the pivot); - a unique transversal (size 5) if x, y are non-pivot elements in different petal-groups; • Each non-pivot element sits in exactly 1 petal and 15 transversals; the pivot sits in all 5 petals and no transversals. This is a 'near-pencil' / Tao-style design construction: 5 copies of a size-16 building-block Eq 677 magma, glued at one shared point, with cross-petal multiplication coordinated by the TD(5, 15) so the whole structure satisfies Eq 677. Display reorder places the pivot at index 0 and then lays out the 5 petals in consecutive 15-blocks (indices 1-15, 16-30, 31-45, 46-60, 61-75). Under this reorder the Cayley table shows a clear 5×5 grid of 15×15 blocks plus a pivot row/column: the 5 diagonal blocks correspond to within-petal operations (each block is the action of one F_16-petal on itself), and the 20 off-diagonal blocks reflect the cross-petal TD(5, 15) structure. [text written by Claude]