The Semantic Interpolation Theorem: A Quantitative Conceptual Framework for Ontology Stabilization in Large Semantic Systems

The Semantic Interpolation Theorem formalizes a structural behavior exhibited by large semantic systems when operating under conditions of ontology sparsity. Rather than treating generative instability as stochastic hallucination, the theorem demonstrates that semantic systems engage in a predictable stabilization process: when a concept lacks sufficient indexed grounding, the system imports structure from semantically adjacent high‑density domains to preserve continuity. This behavior—semantic interpolation—is measurable, recurrent, and cross‑model consistent. The paper introduces the Semantic Interpolation Function (SIF) as a quantitative framework for operationalizing interpolation intensity across four variables: ontology sparsity, neighbor convergence, pattern recurrence, and completion coherence. Through a simulated multi‑model experimental design, the paper illustrates how interpolation emerges as an infrastructural semantic normalization mechanism, shaping how emerging concepts become legible prior to institutional ontology formation.