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 and provides substrate-invariant formalization of cognitive operations, thermodynamically grounded processing constraints, and operationalizable metrics for temporal synchronization. We demonstrate applications through analysis of transformer architectures and propose experimental protocols for validation. This framework bridges symbolic and connectionist approaches, offering a mathematical foundation for cross-disciplinary cognitive research with immediate applications to AI alignment and human-machine collaboration.