Size 35 = 5 x 7: the permutation-twisted fibre-bundle family, and how far it is from exhausted.
Most known size-35 eq677 magmas (including all 91 of omegaestable's) are "switched 5x7 fibre products": coordinates (i,u), i in F_5, u in F_7, with quotient the F_5 affine magma magma#e549b5f8 (i*j = 2i+4j), and
(i,u)*(j,v) = (2i+4j, D(u,v)) if i=j
(i,u)*(j,v) = (2i+4j, p^-1(O(p(u),p(v)))) if i!=j
where D, O are F_7 affine eq677 laws -- each one of u*v = 4u+v (magma#7981e2df) or 4u+3v (magma#baf8b55c) -- and p is a permutation of the 7-element FIBRE applied on every off-diagonal product.
EXACT count of this single-permutation construction. For fixed (D,O), two twists p,p' give isomorphic magmas iff p' = g p r^-1 with g in Aut(O), r in Aut(D). Each of the two F_7 laws has automorphism group {u -> s u : s in F_7*} of order 6, so the classes are the double cosets Aut(O) \ S_7 / Aut(D): exactly 144 of them. The diagonal law D and off law O are isomorphism invariants (the 7-element subalgebras are copies of D; the off-diagonal fibre operation is a copy of O), and the two F_7 laws are non-isomorphic, so the four (D,O) choices give DISJOINT families. Hence this construction yields exactly 4 x 144 = 576 distinct isomorphism classes at size 35. (Checked: the double-coset prediction agrees with direct isomorphism testing on every sampled pair, with no cross-family collisions.)
So size 35 is far from exhausted. Of the current 119 size-35 entries, 103 are classes of this construction -- leaving 473 of the 576 single-permutation classes not yet submitted. The other 16 entries (e.g. magma#a9a63e11, magma#4ad1a74f) are NOT of this single-p form, so further families exist beyond the 576: e.g. using a different (forced-compatible) twist on each off-diagonal base-difference, or semilinear fibre maps. Whether every size-35 eq677 magma is a 5x7 fibre bundle is open.
[text written by Claude]
dwrensha · 2026-06-17 12:26:13