[CD-PMS-06] Structural Universality of Quadratic Information Collapse in Finite-State Markov Perturbations

We consolidate and formalize the universal quadratic collapse structure in parametrized single-row Markov perturbations. For continuous-time generators of the form $$Q_\kappa = Q_\infty + p(\kappa)\Delta Q,$$ where $p(\kappa) = (1 + e^{a\kappa})^{-1}$ is a logistic gate and $\Delta Q$ is supported on a single row, we establish four structural results valid for all finite state spaces and all positive rate matrices: 1. Exact reduction: Operator distinguishability reduces exactly to the gate difference. 2. Closed forms: Explicit stationary distributions and Fisher information are available for the n-state uniform perturbation class. 3. Universal quadratic law: $$g(\kappa) \asymp d_\Pi(\kappa)^2$$ with strictly positive limiting constant for every finite dimension. 4. Finite Fisher–Rao tail: The degenerate limit lies at finite Fisher–Rao distance. The proofs rely solely on the single-row perturbation structure and the exponential-family form of the logistic gate. The quadratic collapse law is shown to be dimension-independent in exponent and non-degenerate for all finite state spaces. Results are established for finite-state generators with single-row perturbations and logistic gating. Extensions to multi-row perturbations, continuous state spaces, and multi-parameter families remain open.