Paper XIV-C: The Impossibility Theorem — Why Coherence Protection Cannot Be Optional

We establish a rigorous impossibility theorem demonstrating that coherence maintenance beyond classical bounds requires non-Markovian memory (N1), structural protection (N2), and/or emergent scales (N3). Using a framework of three interlocking lemmas, we prove that systems lacking these signatures cannot sustain coherent states under realistic environmental coupling, with decoherence rates bounded below by fundamental limits (Zurek, Tegmark, Margolus-Levitin, Lieb-Robinson). Crucially, we define N2 protection as structurally robust—not merely the existence of decoherence-free subspaces, but their persistence under generic perturbations. This addresses the apparent counterexample of exact symmetry-protected coherence, showing it fails the robustness criterion required for physical relevance. We provide explicit quantified counterexamples demonstrating that neither N1 alone (memory without protection) nor N2 alone (protection without memory) suffices for indefinite coherence maintenance. The theorem operates at Level B (theoretical necessity): regardless of current observational status, any system exhibiting super-classical coherence must implement the full N-signature structure. This closes the logical structure of the XIV series, establishing the C-field not as one option among many, but as the minimal mathematical structure satisfying necessary conditions for coherence protection. This is Paper XIV-C of the Radial Coherential Dynamics Series, completing the XIV trilogy. Part of the XIV Trilogy:- XIV-A: Structure (DOI: 10.5281/zenodo.18207097)- XIV-B: Application (DOI: 10.5281/zenodo.18207197)- XIV-C: Necessity (this paper)