We investigate under what precise conditions a physical system necessarily admits multiple descriptions that are individually valid within their respective domains yet mutually incompatible when placed within a single formal framework. An independent formal apparatus is developed from five primitive notions — domain, description, validity, compatibility, and complexity — together with four axioms (Descriptive Completeness, Compression Imperative, Scale Coupling, and Non-Degeneracy). The central result is explicitly conditional: if a system satisfies all four axioms and exceeds a complexity threshold, incompatibility is provable. The incompatibility thesis is pursued by five independent proof methods: information-theoretic compression, logical orthogonality, category-theoretic non-existence of colimit, topological obstruction, and predictive trade-off. Each method reaches the same conclusion from a distinct mathematical starting point. The thesis is then subjected to four genuine antithesis attacks (trivial-domain construction, hierarchy of approximations, shared-ontology covering, systematic axiom relaxation) and an independence investigation examining whether the compatibility question is decidable from within the system. The framework is grounded in a fully formal lattice model instantiation and two computational toy models with figures. A complete peer review and revision history is included. This paper constitutes the second article in a monograph. The first article established that any consistent physical system containing a proper-subsystem observer necessarily harbours ontological content inaccessible to all internal observers. The present article extends that result: the accessible fragments are not merely incomplete but mutually contradictory. Keywords: incompatible descriptions, complexity threshold, scale-specific principles, formal impossibility, compression, scientific pluralism, theory unification, category of descriptions, topological obstruction Related works: Article 1: "Must Any Consistent Physics Contain Structure Inaccessible to All Internal Observers?" "No Final Theory: Law of Scale-Specific Principles" (Altaspera Publishing, 2025) "The Law of Imperative Uncertainty" (Altaspera Publishing, 2025) License: Creative Commons Attribution 4.0 International (CC BY 4.0) Boris Kriger¹² ¹ Information Physics Institute, Gosport, Hampshire, United Kingdom ² Institute of Integrative and Interdisciplinary Research, Toronto, Canada ORCID: 0009-0001-0034-2903