arXiv:2509.20409v2 Announce Type: cross Abstract: This paper synthesizes a series of formal proofs to construct a unified theory on the logical limits of the Symbol Grounding Problem. We demonstrate through a four-stage argument that meaning within a formal system must arise from a process that is external, dynamic, and non-algorithmic. First, we prove that any purely symbolic system, devoid of external connections, cannot internally establish a consistent foundation for meaning due to self-referential paradoxes. Second, we extend this limitation to systems with any finite, static set of pre-established meanings, proving they are inherently incomplete. Third, we demonstrate that the very "act" of connecting an internal symbol to an external meaning cannot be a product of logical inference within the system but must be an axiomatic, meta-level update. Finally, we prove that any attempt to automate this update process using a fixed, external "judgment" algorithm will inevitably construct a larger, yet equally incomplete, symbolic system. Together, these conclusions formally establish that the grounding of meaning is a necessarily open-ended, non-algorithmic process, revealing a fundamental, G\"odel-style limitation for any self-contained intelligent system.