<![CDATA[Twisted centralizer codes determined by A = xJₙ + yIₙ are examined over a finite field F_q, with twist a ∈ F_q \ {0,1}. Previous universal claims of one-dimensionality for this family require qualification, particularly in modular characteristic. We establish a casewise dimension classification using the decomposition of the all-ones matrix when char(F_q) does not divide n and a nilpotent-operator argument when char(F_q) divides n. In the semisimple regime, the code dimension is governed by eigenvalue resonance; the one-dimensional full-weight MDS subfamily occurs when y + nx = 0, x ≠ 0, and char(F_q) does not divide n. Additional cases have dimension (n − 1)² or n − 1 with minimum distances 4 or 2n, respectively, under stated conditions. In the modular regime, x ≠ 0 and y = 0 yield dimension n² − 2n + 2; the minimum distance equals 4 for n ≥ 3 and 3 for n = 2, in which case a nontrivial [4,2,3] MDS code occurs. Explicit small-field examples check the derived parameters. The resulting distinctions correct overgeneralized spectral arguments and clarify the difference between extremal distance and information rate.]]>
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