Negative-Entropy Construction Theory I

This paper establishes a methodological foundation for Negative-Entropy Construction Theory by distinguishing logical domains from the formal systems used to reason within them. Its central concern is not the construction of barriers against external interference, but the identification of domains of discourse, the conditions under which their internal reasoning is valid, and the boundaries beyond which their descriptive and inferential capacities cannot automatically be extended. We begin with a source contrast that establishes an initial distinction within a broader universe of consideration. This distinction provides a basis for defining a local logical domain. Within such a domain, a formal system can be constructed from a formal language, a set of axioms, and a collection of inference rules. The resulting system may possess internal consistency, semantic completeness relative to an appropriate class of models, or deductive closure. None of these properties, however, independently establishes that the system provides a complete description of a broader domain. The paper then develops a precise account of logical-domain boundaries and structural residuals. We distinguish expressive residuals, representational collapse, undecided sentences, and mapping residuals. These distinctions prevent the different limitations of a formal system from being collapsed into a single undifferentiated notion of incompleteness. We prove a factorisation criterion for representational adequacy and construct a counterexample showing that a consistent, syntactically complete, and deductively closed theory may still fail to recover specified target features.