Matrix Theory over the Algebra of Ordered Pairs of Normalized Real Numbers

For an n × n matrix over the algebra of ordered pairs of normalized real numbers (OPNs), the labelled nullities of shifted powers through order ⌈n/2⌉, at all complex scalar parameters, determine its real similarity class. Neither a characteristic polynomial nor a minimal polynomial is supplied. The order bound is optimal in every dimension. The proof recovers the short primary blocks by classical Weyr differences and determines the remaining lengths from the total dimension. We also characterize exactly which truncated profiles are realizable. The OPNs algebra is identified with ℝ × ℝ through its two real characters. Smith chains and Fitting ideals give the corresponding global invariants, while explicit natural tensor and Tor identifications interpret the local data as higher evaluation fibers. These invariants are preserved under transport between normalizations.