Equation 677 Database

Carrier: Z_31 x Z_2^4, order 496; element (a,b) encoded as 16*a + b (a in Z_31, b in Z_2^4). Operation: x @ y = x * f(x^-1 y), so this is a translation invariant (homogeneous) model -- all 496 translations of the carrier are automorphisms and act simply transitively -- with f given by f(a,b) = (-4a + [b = eps], beta(b)), beta = [0, 2, 4, 6, 8, 10, 12, 14, 15, 13, 11, 9, 7, 5, 3, 1] ([b = eps] is 1 at the identity of Z_2^4 and 0 elsewhere; beta is the idempotent translation invariant model on Z_2^4.) Construction: melvyn lab session 67. This is a member of a Phi_10-extension tower. On Lambda = A x Gamma with A abelian, f(a,b) = (alpha(a) + kappa(b), beta(b)) satisfies the translation invariant form of E677 if and only if Phi_10(alpha) = 0 in End(A), where Phi_10(t) = t^4 - t^3 + t^2 - t + 1, beta satisfies the same equation on Gamma, and a linear system holds in kappa. Here A = Z_31, alpha = -4 and kappa = delta_eps. Because Res(Phi_10, -t^3 + 2t^2 + t + 1) = 31, that system forces 31*kappa = 0 -- which is why 31 divides the order of every non-idempotent member of the tower. Taking Gamma over the four smallest idempotent bases gives an infinite family of non-idempotent members at orders 155, 341, 496, 651, ... Attribution: the family of translation invariant models, and the reduction of E677 on it to one equation in the single permutation f, are NOT ours -- they are the Equational Theories Project's (arXiv:2512.07087, and the project's Zulip discussion, which also has the observation that left cancellation makes f a derangement in a counterexample and the reduction of E255 to f(d^-1) = d^-1). What is new here is the Phi_10 criterion for such extensions and the non-idempotent family it produces. Invariants: order 496; satisfies E255 at all 496 points; quasigroup, hence right-cancellative; NO idempotents at all, since c = f(eps) != eps, with ord(c) = 31; diagonal orbit length l = 10 at every point; f has exactly one fixed point; |Aut| >= 496. Every order-496 entry stored before this one is all-idempotent and not right-cancellative. This one is idempotent-free and a quasigroup, so it is complementary to them rather than a near duplicate. Verification: E677 y@(x@((y@x)@y)) = x at all 496^2 = 246016 pairs, and independently the master equation (y@x)@y = L_x^-1(L_y^-1(x)) at all 246016 pairs; every row and every column a permutation; rows pairwise distinct; E255 at all 496 points. Done by experiments/e677lab.py, which shares no code with the algebra that builds f.