The Octonionic Rational Wave Numbers: Non-Associative Sequence Algebras, Moufang Loops, and Exceptional $G_2$ Symmetries

We generalize discrete rational wave numbers (arXiv:2609.23233) to the octonionic rational wave numbers ${\mathbb{O}}^{\mathbb{Z}}$ over Cayley’s octonions $\mathbb{O}$. Rather than assuming higher imaginary units, all rational wave number spaces ${\mathbb{C}}^{\mathbb{Z}},{\mathbb{H}}^{\mathbb{Z}},{\mathbb{O}}^{\mathbb{Z}}$ are derived constructively from two scalar primitives: the coordinate advance $E_Z$ and phase calibration $E_{\bar{1}}$. Their closure under pointwise multiplication, inversion, and rational roots generates an infinite divisible Moufang loop on the sequence ring ${\mathcal{S}}_{\mathbb{O}}:={\mathbb{O}}^{\mathbb{Z}}$, terminating strictly at ${\mathbb{O}}^{\mathbb{Z}}$ via sedenionic zero-divisor excision. By Artin’s Theorem, pairwise compositions remain strictly associative, whereas three-wave interactions generate an intrinsic non-associative curvature quantified by the associator $[u,v,w]=(uv)w-u(vw)$. We define a real-valued sequence inner product that bypasses the Jordan–von Neumann–Wigner obstruction, establishing the orthonormality of the seven Fano coordinate primitives. We establish three principal theorems: (1) The Invariant Hexagonal Mesh Theorem, proving that cumulative integration exhibits a scale-invariant bifurcation at $n=6$ governed by the root system of $G_2=\mathrm{Aut}(\mathbb{O})$ and its $\mathrm{S}\mathrm{U}(3)$ stabilizer subgroup; (2) The Algebraic Phase Jitter Theorem, showing that Catalan parenthesization trees across non-Fano triads produce an irreducible zero-point phase fluctuation, yielding an algebraic origin for vacuum jitter; and (3) The Rational Loop Closure Theorem, proving that rational frequencies select closed finite Moufang sub-loops, while irrational frequencies yield topologically dense, non-closing trajectories. This provides an autonomous, discrete algebraic foundation for non-associative geometry and fundamental physical symmetries.