Invariant-Core Expressive Realization Theory (ICERT): A Formal Framework for Core-Preserving Expression Generation, Cardinality, Structural Diversity and Functional Invariance

This paper specifies the Invariant-Core Expressive Realization Theory (ICERT), a formal framework for investigating how many distinct expressions can represent a specified semantic or functional core while preserving designated properties. Its motivating case contrasts the positive-attitude expression "I like the subject" with the negated negative-attitude expression "I do not dislike the subject." The two forms may be compatible in a given context, but they do not generally express identical commitments.
ICERT separates the specification of an invariant core from candidate generation, admissibility testing, equivalence classification, expression counting, structural-diversity measurement and empirical validation. Its principal counting object is a constrained realization set, whose cardinality is defined relative to a declared grammar, vocabulary, context, feature selection and admissibility policy. The framework distinguishes the number of admissible surface forms from the number of semantic equivalence classes and distinguishes exact core preservation from approximate similarity.
Under explicit assumptions, the paper establishes results concerning equivalence relations, cardinality bounds, monotonicity under nested constraints, finite-state counting, normalized discrepancy scores, entropy bounds, transformation composition and the correctness of finite enumeration algorithms. It also identifies the conditions under which these results do not imply a universal count or guarantee accurate interpretation in unrestricted natural language.
The resulting architecture connects formal semantics, logic, set theory, formal-language theory, computational linguistics, information theory and system-theoretic modelling without treating their distinct notions of equivalence as interchangeable. The paper is a formal theoretical specification and proof dossier rather than a report of completed empirical experiments. A research protocol for implementation, human annotation, metric validation, falsification, reproducibility and subsequent publication is provided.
**Keywords:** semantic invariance; expression realization; constrained enumeration; formal semantics; paraphrase; equivalence relations; finite automata; information theory; functional preservation; ICERT.