Mathematical Existence Proof of Global Non-Closure ℵ-Centumduoctonion Algebras over Completed Artinian Local Rings and the Duplicate Conjugate Coset Partition Theorem inside Copied Covariant Modules

This study introduces the 128-dimensional ℵ-Centumduoctonion algebra (Cℵ) to address algebraic non-closure and identity breakdowns at the eighth Cayley–Dickson stage through transfinite formal power series and an Artinian local ring structure. To ensure complete algebraic isolation and prevent error propagation within transfinite ordinal boundary treatments established in broader set-theoretic formulations, it is necessary to construct an independent proof for the existence of the ℵ-Centumduoctonion algebra (Cℵ) as demonstrated in foundational frameworks. By enforcing the 4th-order Artinian filtration (m4 ≡ 0, ϵ4 = 0) on the sheaf of minimal left ideals, structural and pathway non-closure anomalies are uniquely decomposed and bounded via the Duplicate Conjugate Coset Partition Theorem, establishing robust bi-regularity independent of analytic limits.