Abstract

The cognitive sciences are amidst a great convergence on the principles of substrate-neutrality, predictive processing, and information compression, yet this emerging paradigm lacks a unifying mathematical foundation, resulting in interdisciplinary paralysis. This paper introduces *Principia Cognitia* (PC) as the first comprehensive, axiomatic system for this new science of mind. We formalize cognition through the triad of Semions ($S$), Operations ($O$), and Relations ($R$), grounding these primitives in the physical imperative of cognitive compression. The framework provides a constructive, non-magical model for the genesis of its own components, explaining the emergence of operators via a diffusion-based process and the dynamics of the relational architecture via a topological model. We demonstrate that the PC framework is uniquely capable of synthesizing empirical findings from disparate fields, including the geometric structures in transformer residual streams, neural dynamics of event segmentation, and the architectural vulnerabilities of AI systems. The theory provides a mechanistic explanation for complex phenomena such as subliminal learning in AI and mental compartmentalization in humans. We conclude by presenting a program of three falsifiable experiments (the MLC Primacy Experiment, the Substrate Invariance Test, and the Vectorial Genesis Experiment) that test the core axioms. *Principia Cognitia* is thus offered as a rigorous, predictive, and unifying lingua franca required to move the science of mind from a collection of narratives to a cumulative, mathematical theory.