Principia Cognitia: Axiomatic foundations

This paper introduces Principia Cognitia, a comprehensive axiomatic framework that formalizes and extends the MLC/ELM (Metalanguage of Cognition/External Language of Meaning) duality currently under review in Cognitive Science. Building on empirical foundations established in companion work, we present a unified mathematical framework for cognitive processes across biological and artificial systems. The theory centers on a cognitive triad ⟨S, 𝒪, Rᵣel⟩ comprising semions (discrete meaning-bearing states), operations (physically realizable transformations), and relations (emergent structural constraints). We provide substrate-invariant formalization of cognitive operations, thermodynamically grounded processing constraints following Landauer's principle, and operationalizable metrics for temporal synchronization and prediction error minimization. Key contributions include: (1) formal axioms grounding cognition in physical reality while maintaining substrate independence; (2) theorems on temporal synchronization, narrative thresholds, and dual language regulation; (3) the "KilburnGPT" gedankenexperiment demonstrating computational invariance across radically different substrates; and (4) three falsifiable experimental protocols (MPE-1, SIT-1, VGE-1) for empirical validation. This framework bridges symbolic and connectionist approaches, offering a mathematical foundation for cross-disciplinary cognitive research with immediate applications to AI alignment, human-machine collaboration, and the formal analysis of transformer architectures. The work establishes cognition as a precise object of formal inquiry, comparable to how mathematics formalized number or physics formalized motion.