Exact Second-Order Survivors in Decohering Quantum Matrix Algebras

We present a finite-dimensional exact model of second-order survival under decoherence. Working in the matrix algebra M_N(C), we consider a pure dephasing semigroup whose fixed-point algebra is the commutative diagonal algebra. The associated stable memory projection erases first-order off-diagonal matrix units, but preserves diagonal traceless components. For each local nilpotent pair Λ_ij = E_ij and Ω_ji = E_ji, the ordering defect is computed exactly as Hol_ε(Λ_ij, Ω_ji) = −ε²(E_ii − E_jj), with no higher-order remainder. Although the first-order off-diagonal generators are erased by the stable memory projection, this second-order ordering defect survives exactly as a diagonal traceless direction. Under the identification of diagonal density matrices with probability vectors, these survivors correspond to tangent vectors e_i − e_j of the classical statistical simplex, and their span gives the tangent space on which the Fisher–Rao metric is defined. The result does not claim a derivation of spacetime geometry or of the Fisher–Rao metric itself. Rather, it isolates an exact local mechanism by which nilpotent noncommutative ordering defects survive decoherence as statistical tangent directions.