Language, Probability, and Geometry as Equivalent Projections of Stabilized Inference

Translator This paper develops a regime-relative account of inference in which language, probability, and geometry arise as distinct representational projections of a common invariant-stable semantic structure. Rather than treating any of these domains as foundational, the framework takes invariant stabilization under reflexive decoding as the sole semantic primitive and analyzes representational form as the result of projection that forgets stabilization history and transient refinement. Within this perspective, linguistic reasoning, statistical inference, and geometric representation are shown to be equivalent at the level of stabilized inferential behavior, while remaining inequivalent as representational systems. The equivalence is structural rather than constructive and holds only relative to decoding regimes and projection operators. Differences between domains, including asymmetries and partial correspondences, are explained as consequences of projection loss rather than as failures of inference. The framework clarifies why probabilistic models can successfully emulate language-like reasoning without possessing semantic understanding, why geometric structure frequently emerges from statistical constraints, and why no single formalism can subsume the others without loss. It also provides a principled account of the limits of representation-based reasoning and of semantic failure modes arising from non-invariant constructions. The contribution is explanatory rather than reductive. No domain is reduced to another, and no representational level is privileged. Instead, the paper situates language, probability, and geometry within a unified taxonomy of stabilized inference, offering a structural basis for understanding their persistent correspondences and intrinsic asymmetries.