Definability, Compression, and Truth in a Layered Account of Computation

We provide two separate and important results. First, we formalize a provable separation betweensyntactic and semantic computation for AI learning systems by virtue of the existence of hallucinations.We define run-level “hallucination” under a contract κ= (Lκ,Evalκ,winκ), where truth of an assertedproposition is graded by an external evaluator Evalκ within a fixed decision window. This separationallows us to prove the Transparency Impossibility Theorem: there is no total computable procedurethat, from a single run’s activation log, produces a finite, tape-transparent provenance deciding whetherthe run’s assertion is a hallucination without invoking Evalκ or a Halting/Oracle equivalent. The proofis a halting-encoded diagonal reduction. Rice’s theorem (Rice, 1953) and Tarski’s undefinabilitytheorem (Tarski, 1956) provide independent, complementary impossibility results — Rice rulesout global deciders for nontrivial extensional properties of programs; Tarski rules out an internaltruth predicate for sufficiently expressive Lκ — which we treat as background and supportingmotivation rather than as part of the activation-based reduction itself. Second, in this work we presenta layered account of computation and meaning. The base layer captures effective methods by Turingmachines (Turing, 1937; Church, 1936a). The next layers treat definability, truth, and semantic fixedpoints (Gödel, 1962; Tarski, 1956; Kripke, 1975). Then, we then connect these layers to compression,and description length, which act as practical limits on representation and inference (Li & Vitányi,2008). The aim is clarity about limits. However, we do not enlarge the class of computable sets.Instead, we separate internal effective acceptance from externally grounded acceptance with finitetranscripts. This separation lets us ask when explanation should work, and when it must fail. Rice’stheorem marks undecidable semantic properties that matter for explanation (Rice, 1953). Buildingon this, five corollaries organize the space: completeness, incompleteness, undefinability, groundedacceptance under budgets, and compression ceilings. Each yields a concrete probe or prediction. Theresult is a theoretical framework Church-Turing-Kripke-Meyer (CTKM) to explain how systems canproduce correct but non-derivable behavior without implying hypercomputation. The frameworkalso provides a falsification route insofar as we propose a Diophantine test to refute claims that crossthe classical boundary. Additionally, we offer a conditional description-length lens to mark whenfinite grounding changes acceptance without changing computability. In short, the framework keepscomputability classical while making the role of semantics and resources explicit and testable.