Graph-Local History Identifiability under Nuisance Quotients: Topological Obstructions and Perturbative Delay

We study the local identifiability of continuous parameters describing intermediate history interventions in a graph-local quantum system from a single final-time measurement architecture in the presence of nuisance parameters. For a fixed-axis commuting Hamiltonian, we prove that back-propagated single-site observables obey an exact one-hop support closure and derive a topology-dependent ceiling on the rank of the history Jacobian. Under A3, which provides one single-site scalar readout per site, a chain geometry confines a five-dimensional history space to at most four history-sensitive observation channels, yielding exact local non-identifiability. We then introduce weak noncommuting perturbations. Although the deficient raw history direction opens at first order in the perturbation strength ε, under the fixed A4 row partition and the baseline exact structural zeros of A5, graph-local support structure and nuisance-space placement force that first-order correction to remain inside the baseline joint history–nuisance observation space, J_h,1v₀ ∈ col 𝓙₀. Hence no new independent nuisance-quotient history direction can open at first order. Combining rigorously interval-certified nonzero second-order coefficients with real-analytic genericity, we obtain, outside a proper analytic exceptional set within a fixed right-analytic structural component, σ_raw=Θ(ε) and σ_quot=Θ(ε²). Unknown-input observability, nuisance projection, graph distance–relative degree, graph-local perturbative suppression, second-order identification, and singular-value perturbation are established prior results. To our knowledge, within the prior art we examined, the narrow contribution not directly anticipated is the sufficient-condition chain by which graph-local support placement structurally traps an already first-order-visible raw history tangent inside the baseline joint history–nuisance observation space, delaying only the first new quotient-identifiable direction to second order, together with the resulting strict raw/quotient order split.