TERM™: Leakage Minimization on Unitary Orbits: Variational Structure and Stability of Finite-Resolution Operator Systems

We study the variational structure of finite-resolution operator systems generated by the leakage functional introduced in previous TERM investigations. Given a finite-resolution sector decomposition of a Hilbert space, leakage quantifies the magnitude of operator components coupling distinct sectors. Rather than analyzing leakage as a passive diagnostic quantity, the present work investigates its geometric and variational properties on unitary orbits of self-adjoint operators with fixed spectrum. A leakage-minimizing configuration is characterized as a stationary point of the leakage functional under infinitesimal unitary variations. We derive the corresponding first-order stationarity conditions and construct the associated Hessian governing local stability. This naturally defines a stability landscape on the space of admissible operators. Particular attention is given to the role of spectral constraints and to the emergence of isolated stable configurations under non-commuting projector families. The resulting framework provides a mathematically consistent description of finite-resolution incompatibility without introducing additional dynamical assumptions. The objective of the paper is not to establish a physical theory, but rather to develop the mathematical structure under leakage minimization.