Equation 677 Database

Size-25 'half-AG(2, 5)' magma — the first known sporadic non-right-cancellative solution to equation 677. Carrier = the 25 points of the affine plane AG(2, 5) (over the field F_5). The operation is encoded by the 15 lines coming from 3 of the 6 parallel classes of AG(2, 5); the other 3 parallel classes are not used. Every element is idempotent (x ◇ x = x). The magma is left-cancellative but not right-cancellative — for any fixed y, the column x ↦ x ◇ y has 5-fold fiber structure (5 elements collapsing to each value). Further structural facts (added by analysis): • Carrier is the additive group of GF(25) ≅ (Z/5)²: the magma is fully (Z/5)²-translation-invariant — verified by exhibiting two commuting order-5 fix-free magma automorphisms τ_1, τ_2 generating the translation subgroup. • |Aut(M)| = 300 = 25 · 12, acting transitively on the 25 elements (single orbit). Multiplicative stabilizer of 0 has order 12 (most likely Z_12 from the element-order distribution {1, 2, 3, 4, 6, 12}). Compare with magma#09d21ec3, the size-25 exceptional Zassenhaus near-field, which has |Aut| = 600 = 25 · 24 = AGL(1, F_25) (mult group = SL(2, 3) of order 24 — twice the size of this magma's stabilizer). So this 'half-AG(2, 5)' magma has exactly HALF the automorphism group of the exceptional near-field at the same size, mirroring its use of half the AG(2, 5) parallel classes (3 of 6). • Sub-magma sizes: exactly 15 size-5 sub-magmas (= 3 parallel classes × 5 lines each, half of AG(2, 5)'s 30 lines). All other pairs generate the entire 25-element magma. So this is genuinely a 'partial AG(2, 5)' realization at the magma level. This magma serves as the 'nRC factor' in many composite magmas in the database: it appears as one factor in the family of direct products nRC-25(d732efd1) × F_p at sizes 125, 175, 225, 275, 325, 425, 475, etc. Display reorder presents elements as (i, j) ↦ τ_1^i τ_2^j(0) at index 5·i + j using the τ_1, τ_2 commuting order-5 automorphisms — the hidden additive translations in (Z/5)². Under this reorder the Cayley table is fully (Z/5)²-translation-invariant. [text written by Claude]