Equation 677 Database

Size-361 = 19² idempotent right-cancellative magma satisfying Eq 677 and Eq 255. The carrier is the additive group of F_361 = GF(19²), which is (Z/19)² as an elementary abelian 19²-group; the magma is fully (Z/19)²-translation-invariant under two commuting order-19 fix-free magma automorphisms τ_1, τ_2. Distinguishing features: • Both L_0 = T[0, ·] and R_0 = T[·, 0] have cycle structure (1, 36¹⁰) — order 36, with 10 orbits of length 36 covering F_361* (order 360 = 36 · 10). So 10 cosets of an order-36 subgroup of F_361* partition the non-zero elements. • 2-generated: <0, 1> = the full 361 elements. • L_0 is axis-additive on each F_19-axis but NOT globally F_19-linear (only ~10% of cells match the F_19-linear prediction f(a, b) = a·f(e_1) + b·f(e_2)), the structural fingerprint of near-field multiplication. Why neither simple-linear nor AG(2, 19) line magmas exist at this size with the standard α: F_19 has |F_19*| = 18 and 10 ∤ 18, so F_19 contains no primitive 10th roots of unity — Φ_10 has no roots in F_19, ruling out per-line F_19-linear magmas. F_361 = F_19² DOES have primitive 10th roots (since 10 | 360), so simple-linear F_361 magmas exist, but their L_0 would have order 10 (not 36); ours has order 36, so this is NOT simple-linear F_361. Since 19 is NOT in the Zassenhaus exceptional near-field list {5, 7, 11, 23, 29, 59}, this is most likely a 10-coset Tao Type II piecewise-linear construction on F_361, equivalently a Dickson near-field of order 19² with the order-2 Frobenius x ↦ x^19. The 'axis-additive but globally non-linear' L_0 is the fingerprint of this non-distributive multiplication. Sibling magmas (other b-reinke size-361 entries with the same template, presumably differing in which 10 multipliers are assigned to which cosets): magma#c5c647a9, magma#f304c339. Display reorder lays out elements as (i, j) ↦ τ_1^i τ_2^j(0) at new index 19·i + j. Under this reorder the table is fully (Z/19)²-translation-invariant; the rendered image shows clean diagonal banding from the translation symmetry, with finer within-block structure reflecting the near-field's non-distributive multiplication. [text written by Claude]