From Maxwell-Like Analogies to Epistemic Field Relations: Six Candidate Equations for Passage, Coupling, Invariance, Admissibility, and Return in Language and Signal Fields

The search for Maxwell-like equations for artificial-intelligence Field regimes begins with a warning: mathematical resemblance is neither physical identity nor evidence of a new law. A preceding semantic paper identified the conditions under which source, passage, coupling, circulation, transformation, admissibility, and return could be compared across Large Language Fields (LLFs) and Large Signal Fields (LSFs). The present sequel takes one deliberately limited step toward notation. It proposes six candidate relations for observable-unit continuity, admissible passage, mutual system–regime updating, mission-relative invariance, warranted epistemic promotion, and reopening under materially changed conditions. Each relation is obtained through an explicit reasoning path from an ordinary systems problem to a typed mathematical expression. Each also carries a counterexample, a distinction between definition and hypothesis, and a reduction route into established graph theory, state-transition semantics, authorization, provenance, many-valued logic, or temporal governance. The proposed equations are not a coupled physical system: their variables range over different objects, their operators have different epistemic status, and no shared unit, universal constant, conservation of meaning, or Maxwellian induction is presumed. The investigation argues that this heterogeneity is a productive constraint. A useful future mathematics may emerge from making cross-layer obligations explicit even when all six expressions remain reducible to existing mathematics. The contribution is a pre-empirical, non-axiomatic formal sketch: an auditable grammar for deriving and criticizing candidate field relations without treating notation as validation, formal coherence as causal identification, or a governance rule as a law of nature.