Ground, Inheritance, and Indiscernibility in Monadic Systems

We study first-order structures with a countable family of unary predicates and an element satisfying a ground condition: every predicate instantiated elsewhere is also instantiated at the ground. We establish an Inheritance-Bound Principle: the ground’s predicate profile is an upper bound on every element’s predicate profile. This ceiling is canonical—any two inheritance-maximal elements share the same profile. The Inheritance Bound is the governing conceptual insight of the paper; its complete formal articulation is a Representation Theorem giving a full classification of monadic inheritance-bounded structures. When the ground satisfies a set Γ of predicates, models are classified up to isomorphism by the fiber cardinalities of the profile map, and every admissible profile distribution is realized by some model. The automorphism group decomposes as a product of symmetric groups on profile classes. The zero-information case (Γ = ∅) is the extremal endpoint, yielding a Property Collapse Theorem: all predicate extensions are empty, the full symmetric group acts as the automorphism group, no element is definable, and models are classified by cardinality alone. This collapse produces a language-relative failure of monadic discernibility, generating entities that are absolutely indiscernible relative to the language’s monadic resources. A robustness analysis establishes that collapse is stable under three variations (predicate arity, formula enrichment, language cardinality) and fails under two specific weakenings (minimality-based grounding, per-coordinate subsumption). A Bifurcation Theorem shows that any system with a zero-information ground must be predicate-empty under subsumption, descriptively unsubsumed, or a singleton—these cases exhaust the possibilities under the stated conditions.