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This paper introduces TOENS-Q—a revolutionary numerical representation framework that ad- dresses fundamental limitations of traditional computational systems in high-dimensional data pro- cessing and uncertainty quantification. By integrating quaternion vector algebra with an intensity parameter system, we define the quaternion TOENS number Q = (q,⋆, s), where q ∈ VH is a quaternion vector, ⋆ ∈ {· , ∗ , ∼ , ?} identifies the value type, and s ∈ [0, 4095] controls the error bound ε = 2- s . Numerical simulations demonstrate unprecedented performance: quantum computation achieves ultra-low errors of 10-1234, structural monitoring false alarm rates drop to 5%, and chaotic prediction efficiency improves by 1.9x. The core innovation lies in establishing a complete operational system (including addition, multiplication, exponentiation, logarithm, matrix operations, and inte- gration) with rigorous error propagation models. Theoretical analysis proves that TOENS-Q forms a non-associative but distributive algebra over R, enabling geometrically consistent uncertainty prop- agation in high-dimensional spaces.