[CD-PMS-05] Universal Quadratic Fisher Scaling in Finite-State Parametrized Markov Generators: A General n-State Theorem

We establish a universal quadratic scaling law for Fisher information in finite-state continuous-time Markov generators with single-row logistic-gate perturbations. For the n-state uniform master-seed family $Q_\kappa = Q_\infty + p(\kappa)\Delta Q$, we derive the closed-form stationary distribution and obtain an explicit expression for Fisher information along the parameter axis. We prove that for every finite $n \ge 2$, $$g(\kappa) \asymp d_\Pi(\kappa)^2,$$ with limiting constant $C(n) = (n-1)a^2/n^2 \in (0,\infty)$. The Fisher–Rao distance to the degenerate infinite-friction limit is finite for all $n$. These results generalize the two-state analysis of companion works to arbitrary finite state-space dimension within the uniform master-seed class. Results are derived for the uniform master-seed structure with single-row logistic perturbation. Extension to non-uniform rate matrices and multi-row perturbations remains future work.