Equation 677 Database

Size 343 = 7³ currently has 5 magmas in the DB, falling into two distinct families. Family A: Dickson-near-field-style construction on F_343 = GF(7³) (1 magma, right-cancellative, idempotent). magma#ff552011. Translation-invariant on (Z/7)³ — the additive group of GF(7³) — with three commuting order-7 fix-free magma automorphisms exhibiting the (Z/7)³ structure. L_0 has cycle structure (1, 18¹⁹) and R_0 has (1, 19¹⁸), with ord(L_0) · ord(R_0) = 18·19 = 342 = |F_343*|. L_0 is axis-additive on each F_7-axis but NOT globally F_7-linear (only 7.3% of cells match the F_7-linear formula) — the structural fingerprint of near-field multiplication. Since 7³ is NOT in the Zassenhaus exceptional list {5², 7², 11², 23², 29², 59²}, this is presumably a Dickson near-field construction with the order-3 Frobenius x ↦ x⁷. Why no simple-linear or AG-line magma exists at this size: F_343 has primitive 10th roots of unity iff 10 | 342; since 342 = 2·3²·19 is not divisible by 10, Φ_10(α) has no roots in F_343. (Also 7³ ≠ q² so AG(2, q) line magmas don't apply.) The non-distributive Dickson multiplication is required to satisfy Eq 677. Family B: Direct products of a non-right-cancellative size-49 factor with a linear F_7 factor (4 magmas, all non-right-cancellative, non-idempotent except for one idempotent fixed point, all satisfying Eq 255). These are all dwrensha submissions of the form (nRC-49) × F_7(α, ω): • magma#d25da012 — (nRC-49 b91b0b41) × F_7(4, 1) • magma#102a77e6 — (nRC-49 b91b0b41) × F_7(4, 3) • magma#1ed733cf — (nRC-49 d50b3565) × F_7(4, 1) • magma#97279551 — (nRC-49 d50b3565) × F_7(4, 3) The two non-RC size-49 factors (magma#b91b0b41 and magma#d50b3565) sit in the DB at size 49 (Zassenhaus exceptional near-field size 7² territory). Each is then multiplied by one of two non-isomorphic linear F_7(α=4) magmas (distinguished by their parameter ω = 1 or 3). All 5 size-343 magmas satisfy Eq 255. Notably, the right-cancellative entry is the only idempotent one (Family A), while the non-RC entries (Family B) have exactly one idempotent element each. [text written by Claude]