This repository contains FRC 100.007 — The Λ-Field: A Scalar Completion of Coherence Dynamics, a core theoretical paper in the Fractal Resonance Cognition (FRC) framework. The work introduces the Λ-field, a real scalar field defined on spacetime, as the minimal mathematical structure required to render coherence locally defined, transportable, and dynamically active. The Λ-field provides a unifying physical basis for results established in earlier FRC papers, including entropy–coherence reciprocity, coherence-driven state evolution, and the emergence of the Born rule as a dynamical equilibrium rather than a postulate. Key contributions include: A logarithmic definition of coherence as a scalar field, ensuring additivity, thermodynamic consistency, and information-theoretic grounding. An explicit Lorentz-invariant action and field equation for Λ, yielding a massive Klein–Gordon–type dynamics in the absence of sources. A CPTP-preserving coupling between Λ-gradients and quantum density-matrix evolution, consistent with standard open quantum system theory. A demonstration that standard quantum mechanics is recovered exactly in the zero-coupling limit. Clear scope constraints excluding macroscopic forces, gravity, cosmology, consciousness models, and symbolic interpretations. This paper closes the FRC 100-series at the level of mathematical field definition and serves as a foundational reference for subsequent computational (FRC 100.008) and cosmological or vacuum-selection extensions (FRC 821-series). The framework is presented as falsifiable, extensible, and open to refinement through both analytical and experimental investigation.