A Normal Ground State at a Momentum Minimum: A reciprocal lattice field with a complex native source

This note derives a normalized ground state at some global momentum minimum for a specified reciprocal lattice field with an arbitrary bounded Hermitian finite native source, including the complete 512-state single-source operator at its declared coupling. All photon modes and oscillator occupations are retained. Exact finite-mode compressions, global minimization of hopping phases and an explicit soft-photon bound yield strong compactness and a ground in the original transverse Fock space. Significance within Recognition Science. This removes an existence obstacle for the declared interacting source candidate: its global spectral bottom is attained by a normal dressed state, rather than being only a lower energy bound. It provides an actual state on which to study source visibility and energy response. The minimum location, overlap with a prescribed bare source, positive mass curvature, physical species and generation assignments, and selection of this candidate as the physical source law remain open. Significance for broader science. Soft photons can prevent a spectral infimum from being represented by a normal Fock state. The result adapts established current-response and compactness methods to a nonlinear reciprocal lattice Hamiltonian with a complex finite source. Its useful addition is a global phase-envelope bound that replaces a native positive-cone assumption at the moving compression minima, together with the required factor-two correction for complex current coefficients. This is a model-specific mathematical result; it is not a new general solution of the infrared problem or an empirical particle-spectrum prediction. Related continuum methods are credited to Hasler and Siebert, Journal of Functional Analysis 284 (2023), 109725, doi:10.1016/j.jfa.2022.109725. Evidence. The PDF and source archive include the analytical derivation, eight supporting scalar Lean results, eighteen exact algebra checks and their receipts. The full infinite-dimensional existence argument is analytical, not formalized wholesale. Common energy-unit covariance and a quantitative soft-photon tail are included. AI assistance. Codex assisted with the derivation, source comparison, scalar formalization, algebra checks and preparation. Jonathan Washburn is the named author and retains responsibility for the claims. This technical research note does not claim external peer review or independent human verification.