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]
dwrensha · 2026-04-29 16:57:20
Size-29 sporadic magma. Prime size (29), but unlike sizes 5, 7, 11, 13, 19, 31, 37, 41, ... there is NO F_29-linear magma satisfying equation 677 — the system "ab(b²+1)=1 and a+a²b²+b³=0 (mod p)" has no solution for p=29. So this is a genuinely non-linear 677 magma at prime size. It is a quasigroup (LC + RC), all 29 elements are idempotent (x ◇ x = x), and it has NO non-trivial sub-magmas — every pair of distinct elements generates the whole magma. Both L_x and R_x are permutations with the same uniform cycle type (1 + 14 + 14): the unique fixed point is x itself (since x is idempotent), and the other 28 elements split into two 14-cycles. The magma is highly homogeneous with no obvious block decomposition.
dwrensha · 2026-05-15 13:11:33
dwrensha · 2026-04-29 16:57:20