18.1. Finite‑Dimensional Geometric Reduction of Kink Dynamics with Boost‑Closure Diagnostics and Leakage Functional

We present a self‑contained theory of finite‑dimensional geometric reduction for relativistic scalar field theories admitting kinks. The central object is the leakage functional, which measures the failure of a truncated submanifold to be invariant under the Hamiltonian flow. We show that exact finite‑dimensional symplectic geometry can exist even when dynamical closure fails, and we separate the symplectic, Hamiltonian, dynamical, covariant, and quantum layers of the reduction. For localized fluctuation eigenmodes we prove exact overlap identities and a boost sum rule. We apply the formalism to the sine‑Gordon kink (continuum only) and to the \(φ^4\) kink (discrete shape mode). For the latter, we compute the frozen‑embedding leakage, the perturbative Darboux correction, and the Lorentz‑adapted ansatz, showing that a fixed finite‑dimensional submanifold cannot be exactly invariant for a moving discrete mode because the linearised field equations have time‑dependent coefficients in the laboratory frame. Numerical benchmarks and complete Python code are provided.