Toward Maxwell-Like Equations for Language and Signal Fields: Can a Common Mathematical Structure Describe Flow, Coupling, Invariance, and Transformation Across AI Field Regimes?

A mature physical field theory can consolidate phenomena that once appeared to demand separate explanations. Maxwell's electromagnetic synthesis is a powerful historical illustration, but it does not follow that fields proposed for artificial-intelligence systems obey electromagnetic laws or share a physical ontology. The Large Language Field (LLF) and Large Signal Field (LSF) research trajectory has introduced a mission-relative language for describing how models, evidence, memory, tools, signals, constraints, evaluators, and authorized actions form interacting operational regimes. Its later investigations have raised questions of coupling, invariance, field motion, identification, and the survival of properties under transformation. This paper asks what must be clarified before one can responsibly investigate Maxwell-like equations across such regimes. It develops a pre-equational semantic correspondence framework rather than a mathematical theory: source and conditioning, passage and flow, interaction and coupling, circulation and feedback, change and induction-like dependence, invariance, boundaries and admissibility, and return after consequential action. The framework distinguishes operational counting from semantic preservation, mathematical representability from causal identity, and architectural constraints from natural laws. It places established graph theory, discrete Hodge methods, causal modeling, distributed systems, provenance, multimodal learning, and access-control models in the foreground as strong alternatives, not merely as precursors awaiting replacement. Concrete thought experiments expose where an analogy may survive, where it fails, and which missing definitions would obstruct formalization. No equations, axioms, theorems, experiments, benchmarks, or observed improvements are claimed. The contribution is a bounded semantic research program intended to make a future mathematical investigation sharper, more discriminating, and more open to the possibility that no distinct Field mathematics is required.