Mathematical Existence Proof of Non-Alternative and Non-Flexible ℵ-Hexacontatetrany Algebras over Completed Artinian Local Rings and Regularity Identities of Vector, Matrix, and Tensor Spaces

The 64-dimensional hypercomplex domain, designated as Hexacontatetrany (H), which constitutes the seventh stage of the Cayley–Dickson construction, inherently suffers from the total dissolution of commutativity, associativity, the alternative identity, and the flexibility identity (a(ba) = (ab)a). This structural collapse induces an uncontrollable spontaneous generation of zero divisors and nilpotent elements, severely destabilizing multilinear algebraic operations. This study establishes a rigorous, mathematically self-contained existence proof for the ℵ-Hexacontatetrany algebra (Hℵ) as an autonomous topological barrier isolated from general hyper-hierarchical number frameworks. By formulating arbitrary elements within Hℵ as countably transfinite formal power series over a completed Artinian local ring, and by strictly enforcing the Artinian descending chain condition (m³ ≡ 0) along with the crucial vanishing axiom (ϵ³ = 0) onto the sheaf of minimal left ideals, the non-alternative and non-flexible algebraic anomalies are identically annihilated. Furthermore, we explicitly demonstrate the bi-regularity and well-definedness of vector components (modules), matrices (square operators), and tensors (multilinear contraction systems) within this domain without relying on analytic limits (lim), thereby realizing a pure, stably well-defined algebraic infrastructure for non-Archimedean hyperspace modeling.