Equation 677 Database

Asymmetric "mixed-pencil" magma at size 69 = 3 * 23. Size 69, exactly 1 idempotent (element 68), right-cancellative. |Aut(M)| = 1 (only the identity automorphism); every element is a singleton Aut-orbit. So the magma is maximally rigid. Sub-magma structure: there are exactly 6 proper non-singleton sub-magmas, all containing the unique idempotent 68 (= "the origin"): - 1 size-9 sub-magma: {60, 61, 62, 63, 64, 65, 66, 67, 68} — iso class magma#2925dc18 (a size-9 Eq 677 magma with exactly 1 idempotent; NOT the fully-idempotent F_9 linear magma). - 5 size-13 sub-magmas, ALL of the same iso class magma#babb8d44 (a size-13 Eq 677 magma with 1 idempotent): {0, 1, 2, 3, 4, 5, 6, 16, 17, 45, 46, 47, 68} {7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 40, 41, 68} {19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 42, 68} {30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 43, 44, 68} {48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 68} The 6 sub-magmas all pass through 68 (the only shared element). Outside of 68, they partition the remaining 68 elements as: 8 (= size-9 minus 68) + 5*12 (= 5 * size-13 minus 68) = 68 ✓. This is a "pencil of 6 lines through a common point" structure with TWO different line sizes (9 and 13) - asymmetric in a way that no AG(2, q) or PG(2, q) geometry produces. 69 = 3 * 23 is composite, and there is no Eq 677 magma at size 3 (see size-3 page commentary), so the construction isn't a Z_3 x Z_23 product either. This is genuinely sporadic. Suggested reorder lays out 7 consecutive blocks: positions 0..7: size-9 sub-magma minus 68 (8 elements) positions 8..19: size-13 line 1 minus 68 (12 elements) positions 20..31: size-13 line 2 minus 68 (12 elements) positions 32..43: size-13 line 3 minus 68 (12 elements) positions 44..55: size-13 line 4 minus 68 (12 elements) positions 56..67: size-13 line 5 minus 68 (12 elements) position 68: the idempotent 68 Within each size-13 line, the 12 non-idempotent elements are reordered via the line's iso to the CANONICAL form of magma#babb8d44 (the size-13 sub-magma iso class). Consequence: all 5 diagonal 12x12 blocks of the Cayley table show the SAME within-block pattern (the canonical babb8d44 layout, with the global idempotent 68 represented in the last row/column). The off-diagonal blocks remain chaotic because |Aut| = 1 - there is no group symmetry to align cross-line operations. L_0 cycle 1 + 12 + 28*2 and R_0 cycle 1 + 4*3 + 14*4 reflect the global asymmetry. Comparable structures: magma#1b32837d (size 49): 8 size-7 lines through a common idempotent, in 2 iso classes (7+1) - a "twisted AG(2, 7) pencil". This magma: 6 lines through a common idempotent, with MIXED line sizes (5 size-13 + 1 size-9) - distinct from any AG(2, q) or PG(2, q) interpretation. [text written by Claude]