A theory T* is a model companion of a theory T if T* is model‑complete and every model of T can be embedded into a model of T* (equivalently, T and T* have the same universal consequences), so T* serves as a canonical companion capturing existentially closed behavior relative to T.

Principle

A model companion consolidates the existential consequences of T into a model‑complete theory: it completes T from the model‑theoretic perspective by ensuring that embeddings into models of the companion reflect existential properties and that the companion eliminates existential indeterminacy.

Demonstration

The theory of algebraically closed fields of a fixed characteristic is the model companion of the theory of fields with that characteristic: every field embeds into an algebraically closed field and ACF is model‑complete, so ACF is a canonical companion of the theory of fields.

Misapplication

Assuming every consistent theory has a model companion; in fact existence can fail — taking T and treating any model‑complete extension as its companion without checking the preservation of universal consequences can produce an incorrect claim of companionship.

Consequence

When a model companion exists, it provides a robust framework for analyzing T's models via model‑complete methods: existentially closed models become prime objects, quantifier reduction for certain formulas becomes available, and transfer of existential properties from T to T* is systematic.

Reversal

A theory that extends T but is not model‑complete (or that does not preserve T's universal theory) is not a model companion; such an extension may add structure but fails to provide the canonical model‑theoretic closure embodied by a companion.

Boundary

A companion need not exist and is not unique unless specified up to logical equivalence; model companion relates only theories with the same universal part and does not automatically guarantee quantifier elimination or that every embedding is an elementary embedding.

Semantic Tension

Model companion is often conflated with model completion; the tension lies in the degree of 'completeness' required: companions must be model‑complete and share universal consequences with T, whereas completions add the stronger requirement that every model of T embeds as a substructure into a model of the completion.

Synthesis

A model companion of T is a model‑complete theory T* that captures the existentially closed behavior compatible with T's universal axioms, serving as the model‑theoretic closure that makes existential properties decidable inside a canonical companion theory.