Carrier: F_31 x Z_29. Encode (x,u) by i = 29*x + u.
Operation:
(x,u) * (y,v) = (3x - 2y mod 31, F_{y-x}(u,v)),
where d = y-x mod 31 and
F_0(u,v) = u + f(v-u) mod 29,
f(0)=0, f(z)=27z for nonzero QR_29 z, and f(z)=3z for NQR_29 z;
F_d(u,v) = v for nonzero QR_31 d;
F_d(u,v) = -u-v mod 29 for NQR_31 d.
This is a QR/NQR fiber extension of the idempotent Type-I base x o y = 3x-2y on F_31. The diagonal fiber is the size-29 two-piece E677 magma; QR cross-fibers are right projections; NQR cross-fibers use the semisymmetric law -u-v.
Properties checked directly from the 899x899 table: satisfies Eq 677 and Eq 255; idempotent; left-cancellative; not right-cancellative; nonmedial. Right-cancellation witness: 87*0 = 261 = 88*0. Medial failure: (0*0)*(17*0) = 10, while (0*17)*(0*0) = 8.
The construction is not a direct product: the fiber law depends on the QR/NQR class of y-x in F_31.
omegaestable · 2026-08-17 03:48:36