Equation 677 Database

Size-63 non-idempotent right-cancellative magma satisfying Eq 677 and Eq 255. A genuinely sporadic structure — neither a fiber-bundle / direct product, nor a translation-invariant magma in any abelian group sense. Aut(M) has order 48 and acts on the 63 elements with 4 orbits of sizes (1, 6, 8, 48): • Orbit {62}: a single fixed element — the unique idempotent of M (verified directly). • Orbit {48, 49, 50, 51, 52, 53}: six elements. Together with 62, they form a unique 7-element sub-magma isomorphic to magma#7981e2df (the linear F_7 magma x ◇ y = 4x + y mod 7). • Orbit {54, 55, 56, 57, 58, 59, 60, 61}: eight elements. Together with 62, they form a unique 9-element sub-magma isomorphic to magma#2925dc18 (the linear Z/9Z magma x ◇ y = x + 3y mod 9). • Orbit {0, 1, …, 47}: 48 elements (the 'free' / off-axis points), acted on REGULARLY by Aut(M) — orbit size equals |Aut|, so the action there is a faithful regular representation. Sub-magma counts: there are EXACTLY one size-7 sub-magma and EXACTLY one size-9 sub-magma (besides the trivial size-1 sub-magma {62}). Every other pair (out of 1896 of the C(63, 2) = 1953 pairs) generates the full 63-element magma — i.e. the magma is 'almost everywhere 2-generated'. Why this is NOT an F_7 × F_9 direct product: a direct product would have 7 + 9 = 16 axes (7 rows + 9 columns) as sub-magmas of size 9 and 7 respectively. We observe only 1 + 1 = 2 such sub-magmas, so this is not a product structure. R_0 = T[·, 0] has uniform cycle structure (3²¹) — order 3, with 21 orbits of length 3. The R_0 cycles decompose by Aut-orbit composition as follows: 8 cycles entirely in the 48-orbit; 6 mixed (F_9 element + 2 free); 4 mixed (F_7 element + 2 free); 2 mixed (F_7 + F_9 + free) — these are the 'bridge' cycles linking the two special sub-magmas; 1 mixed (2 free + idempotent 62). Total 8+6+4+2+1 = 21 ✓. So R_0 mixes elements across all four Aut-orbits. L_0 = T[0, ·] has mixed cycle structure (1, 6, 8, 24, 24) — orders 1, 6, 8, 24 mixed. Compare to R_0's uniform order 3: the magma has highly asymmetric left vs right behaviour. S map (x ↦ x ◇ x): not a permutation. Only one S-fixed point (the unique idempotent 62). The existing display_reorder organizes elements by the 21 R_0 cycles (with the 7 'S-anchored' cycles — those whose elements eventually map to 62 under iterated S — placed together). [text written by Claude]