The Geometry of Flavor: Computational Visualization of the Albert Algebra and the Octonionic Blade of Stability

The origin of the three generations of fermions (Flavor Problem) remains one of the open questionsof the Standard Model. Within the Axiomatic Physical Homeostasis (APH) framework, we proposethat particle generations arise from the intrinsic triality of the Albert Algebra (J3(O)). We presenta high-performance computational visualization of the zero-determinant surface of J3(O) projectedinto 3-space. Our simulations reveal that stable geometric structures only emerge within a narrowband of vacuum stiffness (β ≈ 1.91), which we term the “Blade of Stability.” Outside this blade,the geometry dissolves into a Sedenion bulk, visually confirming the phase transition between theassociative Standard Model and the non-associative pre-geometric foam.