Large language models exhibit four persistent failure modes — hallucination, sycophancy, adversarial instability, and false continuity — that have resisted full resolution despite sustained investment in the current training paradigm. This paper argues that these failures are at least partly structural rather than incidental: they share a common root in the absence of any stable internal state that is prior to and independent of the system's inputs. We survey candidate theories from consciousness science (Global Workspace Theory, Integrated Information Theory, biological naturalism) and argue that, whatever their empirical merits, none supplies an engineering account of such an input-independent ground. We then propose an alternative architectural prior whose functional structure is drawn from three convergent non-dual contemplative traditions — Advaita Vedānta, classical Daoism, and Dzogchen — which independently report that awareness is described as prior to, and not produced by, cognition. We do not claim these systems are or could be conscious; we borrow only the functional structure of the witness/appearance distinction. We describe a partial formalization of this structure in Matthew Scherf's machine-verified Lean 4 axiomatics (129 axioms; described by its author as the first machine-verified formalization of a non-Western philosophical system, a claim we have not independently verified), its encoding as a Python API (Scherf_API, Apache 2.0), and the design-level use of these principles in SpecStudio's AwareWare products. We specify three candidate architectural components of the resulting prior, a set of interaction design principles, and six research directions with associated testable hypotheses, and we invite collaboration to develop these into engineering experiments. This is a position paper proposing a research direction, not a validated system.