Non-crystallographic systems of integers over composition algebras

Abstract

In this work we revisit classical systems of integers inside the real normed division algebras from the point of view of finite norm shells and root systems. Building on the icosian framework of Moody--Patera and on the integral root-system viewpoint of Chen--Moody--Patera and of Johnson, we isolate the precise axiomatic ingredients of the non-crystallographic analogue: an order over the golden ring \(\Zphi\) together with a distinguished finite root shell whose Cartan coefficients lie in \(\Zphi\). We show that the usual Gaussian, Eisenstein, Hamilton, Hurwitz and Coxeter--Dickson examples are recovered by separating the order, its units, and its distinguished finite shells; once the lattice requirement is replaced by a finite root-shell requirement, the golden integer ring becomes the natural coefficient ring for the non-crystallographic cases \(H_2\) and \(H_4\). We then construct a weak golden octonion order by Cayley--Dickson doubling of the icosian ring; the resulting free rank-\(8\) \(\Zphi\)-order has a \(240\)-element finite shell of type \(H_4\oplus H_4\) and its multiplication is genuinely octonionic. Finally, we prove (i) that this weak double is self-dual with respect to the polar norm pairing, hence has no strict norm-integral overorder, and (ii) that the first trace-integral discriminant tower over it contains no octonion-stable nonzero isotropic gluing.