Response Reduction with Moving Domains: Graph Geometry for Unbounded Operators

For unbounded operators, a parameter-dependent family cannot generally be compared as if all operators acted on the same domain. The domain itself becomes part of the geometry of the problem and must be transported before an effective response can be meaningfully differentiated or reduced. This work develops a graph-based framework for response reduction in families of unbounded operators with variable domains. Closed operator graphs provide normalized carriers on which different domains can be compared without discarding the action of the operators themselves. In the self-adjoint setting, resolvent information supplies a natural identification, while a complementary geometric construction separates transport of the graph from internal changes of frame. The framework is extended to form domains, reduced forms, history-dependent response, and singular changes of domain. It distinguishes isometric transport, which preserves the full amplitude structure, from projection between different carriers, which can produce a genuine loss of information. The main conclusion is that domain variation is not merely a technical obstruction. It carries geometric information that can remain visible in the effective response even when weaker modes of operator convergence fail to retain it. A graph-level description is therefore essential when domains themselves participate in the dynamics.