Given Distinction: Why the Space of Structural Reasoning Failure Is Finite and Classifiable

Papers I and II of this series supply the empirical content of the Convergent Semantic Architecture: CLCA recovers the constraint geometry's axes and operators from sixteen languages of thirteen independent lineages; UFA classifies every entry of a 101-fallacy inventory under five structural violation types. This paper argues why the violation space has this five-part form, and states the claim that the five are complete as a conjecture defended against the strongest candidates for a further condition. The argument does not depend on the specific inventory of AOML (which axes, which operators, which methods, which levels) but on the structural conditions that any distinction-making system must maintain to remain coherent. Any system capable of distinguishing one thing from another must maintain four conditions: what kind of distinction is being made (category), how distinctions compose (licensing), what validates a distinction (method), and at what level of abstraction the distinction operates (stratification). These four conditions yield four primitive violation types. The opposition topology that makes all four possible is a precondition whose misrepresentation yields the fifth. On the reading given in §3, philosophical traditions developed over 2,400 years without coordination have each identified one or more of these conditions; that none identified a condition outside them is the hypothesis a coded survey would test. The convergence is consistent with the structural argument. The argument establishes the form of the constraint geometry; CLCA and oAOML (operational AOML, engineering failure analysis) supply its content from independent substrates. Together, the four papers provide one connected account: what the constraints are (Paper I), what happens when they are violated (Paper II), why the violation space takes the form it does (this paper), and what truth consists of on each face of the geometry (Paper IV).