The Semantic Closure Impossibility Boundary: A Non-Closure Theorem for Reference in Irreversible Symbolic Systems (SC-Ω / SNC-Ω)

This paper proves a structural impossibility result: no irreversible, semantically closed symbolic system can preserve stable reference indefinitely. Reference is defined as diachronic identifiability — the capacity of present symbols to be compared to the conditions under which their referents were originally fixed. The result assumes only (1) irreversible evolution, (2) symbolic generation from prior symbols, (3) semantic closure (no external grounding injection), and (4) persistence of symbolic activity. No assumptions are made about agents, cognition, belief, truth, interpretation, linguistics, information theory, or empirical noise. The core theorem (Semantic Non-Closure Impossibility, SNC-Ω) establishes that reference preservation requires weak reversibility, irreversibility destroys the structural conditions required for comparison, and symbolic derivation alone cannot restore non-circular reference. Therefore semantic self-containment is impossible. The result is representation-independent and invariant under reparameterization of symbols or meta-languages. The boundary concerns predicate exhaustion, not error. Under closure and irreversibility, the predicate “this symbol refers to the same thing as before” becomes undefined rather than false. Systems may remain syntactically coherent and operationally active after reference is exhausted. SC-Ω is formally distinct from Admissibility Boundary Science (ABS). ABS governs admissible continuation of systems; SC-Ω governs admissible reference. Together they delimit independent structural limits on execution and meaning. The paper classifies SC-Ω as a non-closure boundary law, analogous in structure (though not in mechanism) to Gödel’s incompleteness theorem, Turing’s halting theorem, and Shannon’s capacity theorem. It introduces no empirical claims and proposes no applications. It establishes a boundary condition on the internal preservability of reference in symbolic systems.