A Five-Dimensional Nonassociative Algebra: elliptic integrability and bracketing-dependent dynamics

Abstract

We introduce a five-dimensional nonassociative real algebra $\mathbb{P}$ whose one-generated subalgebras are complex while its left, middle, and right nuclei are all $\mathbb{R}$. This contrast permits an unambiguous one-element functional calculus but makes polynomial ordinary differential equations sensitive to bracketing. We classify the homogeneous quadratic words containing one fixed coefficient $B$ and two copies of $p$ that preserve $\operatorname{span}_{\mathbb{R}}\{1,B,p\}$. For a fixed-coefficient family of Pentarnion ODEs interpolating between $B(pp)$ and $(Bp)p$, changing only the bracketing preserves the linearisation but changes the observed global dynamics. Exact balance identities isolate the associator contribution, while numerical diagnostics support bounded chaos at one endpoint and an attracting periodic orbit at the other. The flexibility defect $p(Bp)-(pB)p=-[p,B,p]$ also produces a rank jump: after polarisation, one-generated subalgebras have dimension three on a nonempty Zariski-open set, rather than at most two. On a canonical invariant three-dimensional space the resulting flow has two first integrals and an explicit periodic orbit on an elliptic curve. We also construct a dense five-dimensional quadratic flow with a full-affine-span Jacobi-elliptic solution. Its projective closure is an elliptic normal sextic, and representation by two ordered products selects the modulus $m=1/2$.