The Grounding Barrier: Internal Closure, Design Space, and the Fragility of Expressiveness

A representational system with a distinguished base faces a three-way tension: the base can be blank, internally closed, or the system can be nontrivial—but not all three. If the base is required to determine every representational distinction from within the system’s own resources (internal closure), it must bound the system’s full expressive ceiling. If it is also contentless, that ceiling is empty. The system is expressively vacuous. This paper proves this principle exactly in one regime—monadic first-order representational systems with a distinguished ground element—and establishes that neither arity extension nor multi-system composition circumvents it. We first prove that internal closure is equivalent to inheritance-maximality (the GroundVisibility Completeness Theorem): the ground’s profile determines the model up to isomorphism if and only if every instantiated predicate is instantiated at the ground. Without this condition, representational leakage occurs—structurally distinct models become indistinguishable from the ground’s perspective, and no invariant computable from ground-visible data can recover the difference. We then decompose the conditions generating collapse into a minimal unsatisfiable set (the Grounding Trilemma), extend the result to relational languages via diagonal reduction, and prove a Constraint Inheritance Lemma showing that under injective ground-preserving bridges, the closure and nullness constraints propagate freely while expressiveness is fragile at every interface. The results bear on the Symbol Grounding Problem insofar as Harnad’s notion of semantic grounding is read as requiring internal closure—that the foundation accounts for all representational content from within. Under that reading, the trilemma applies. A corollary for AI system design: within the internal-closure framework, a nontrivially expressive grounded system necessarily has a contentful foundation—structural bias is not a contingent defect but the cost of closure, and expressiveness scales with foundational content. Accounts of grounding that rely on external determination (supervenience, causal laws, interpretive functions) avoid the trilemma but relocate grounding facts outside the object language; whether this yields a general opacity constraint is open.