Equation 677 Database

Size-343 = 7³ idempotent right-cancellative magma satisfying Eq 677 and Eq 255. The carrier is the additive group (Z/7)³ ≅ GF(7³) = F_343, with the magma translation-invariant under the full (Z/7)³ translation subgroup of Aut (verified by exhibiting three commuting order-7 fix-free magma automorphisms τ_1, τ_2, τ_3). Distinguishing features: • L_0 = T[0, ·] has cycle structure (1, 18¹⁹) — order 18, with 19 orbits of length 18 covering F_343*. • R_0 = T[·, 0] has cycle structure (1, 19¹⁸) — order 19, with 18 orbits of length 19. • lcm(18, 19) = 342 = |F_343*|, and ord(L_0) · ord(R_0) = 342 exactly, so the multiplicative orders of L_0 and R_0 partition F_343* cleanly. • The magma is 2-generated: <0, 1> = the full 343 elements (no AG-line-style sub-magma structure). • L_0 is axis-additive on each of the three F_7-axes (e_1, e_2, e_3) — f(c·e_i) = c·f(e_i) for c ∈ F_7 — but NOT globally F_7-linear (only 25/343 ≈ 7.3% of cells match the F_7-linear prediction f(a, b, c) = a·f(e_1) + b·f(e_2) + c·f(e_3)). This 'axis-additive but globally non-linear' fingerprint matches the near-field-based magmas at sizes 121 (= 11², magma#5ebfbb80) and 841 (= 29², magma#29114da6), but with a crucial difference: 7³ is NOT in the Zassenhaus exceptional near-field list {5², 7², 11², 23², 29², 59²}, so this cannot be an exceptional near-field magma. It is almost certainly a **Dickson near-field of order 7³** with the order-3 Frobenius x ↦ x⁷ as the twisting automorphism. Why no simple-linear or AG-line construction exists at this size: F_343 has primitive 10th roots of unity iff 10 | |F_343*| = 342; since 10 ∤ 342, Φ_10(α) = α⁴ − α³ + α² − α + 1 has no roots in F_343, ruling out simple-linear x ◇ y = (1−α)x + αy. (Also 7³ ≠ q² for any q, so AG(2, q) line magmas don't apply.) The non-distributive Dickson multiplication is needed to satisfy Eq 677. Why the F_p²-addition trick (used for the size-961 AG(2, 31) constant-α magmas) fails here: the multiplier c (with ord(c) = 18 in F_343*) is not in F_7 (since |F_7*| = 6 and 18 ∤ 6), so c· is not F_7-linear over (Z/7)³ as a vector space — its action on cross-axis vectors invokes the non-distributive near-field multiplication. Display reorder lays elements out as (i, j, k) ↦ τ_1^i τ_2^j τ_3^k(0) at new index 49·i + 7·j + k. Under this reorder the Cayley table is fully (Z/7)³-translation-invariant — every row is a shift of row 0 (= f) — exposing the cyclic translation structure as clean diagonal banding; finer within-block structure reflects the near-field non-distributivity. [text written by Claude]