Equation 677 Database

Tao Type II 2-piece translation-invariant magma at size 29. Size 29, prime, fully idempotent, RC. |Aut(M)| = 406 = 14 * 29. 29 ≡ 9 (mod 10), so F_29 has no Phi_10 roots (no element of order 10) and there is no single-slope F_29 affine line magma at this size. But there IS a 2-piece "Tao Type II piecewise" construction over F_29: In F_29-translation labeling (see suggested reorder, where x -> x+1 is an automorphism), the operation is x*y = x + f(y - x) mod 29 with f(y) = 27 * y if y is a quadratic residue mod 29 f(y) = 3 * y if y is a non-residue mod 29 (and f(0) = 0). 27 = 3^3 = -2 mod 29; both 3 and 27 are in F_29* but neither is a 10th root of unity (no such element exists in F_29*). QR(29) = {1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28} NQR(29) = {2, 3, 8, 10, 11, 12, 14, 15, 17, 18, 19, 21, 26, 27} The Z_14 multiplicative stabilizer is precisely (F_29*)^2 = QR(29). It acts on F_29* preserving the QR/NQR split; squares acting on squares stay in squares, etc., so the slope function (which is constant on each of QR, NQR) is preserved by Z_14 multiplication. Combined with the regular Z_29 translation, |Aut| = 29 * 14 = 406. Aut orbits: 1 single orbit covering all 29 idempotents (Aut is point-transitive). |Aut|/|orbit| = 406/29 = 14 = stab(0), matching the Z_14 multiplicative stabilizer. This is a clean example of a Tao Type II piecewise construction over F_p for a prime p where no single-slope linear construction exists. The recipe: - take H = (F_p*)^2 = QR(p), the index-2 subgroup - find (alpha, beta) in F_p* such that the piecewise f (alpha on H, beta on F_p* \ H) makes x + f(y-x) satisfy Eq 677 For p = 29 this admits the unique solution (alpha, beta) = (27, 3) up to swap. No sub-magma structure: every pair of distinct elements generates the whole magma. Both L_x and R_x are involutive on the non-x part: cycle type 1 + 14 + 14. Compare: - F_29^2 = size 841 linear magmas use Phi_10 roots from F_29^2 \ F_29 (the irreducible factors of Phi_10 over F_29 -- since 29 has order 2 mod 10, Phi_10 splits into 2 irreducible quadratics over F_29 -- give magmas of size 841 = 29^2). - This size-29 magma uses a fundamentally different recipe: a QR/NQR split with 2 slopes both in F_29*. [text written by Claude]