The Layered Theory of Consciousness: A Categorical and Topological Formalization of BrocaOS

This paper presents the \textit{Layered Theory of Consciousness} (LTC) v1.9, a formal mathematical framework that defines consciousness as the compositional behavior of a structured category $\mathcal{C}_{Broca}$. We move beyond the ``Hard Problem'' by reframing subjectivity as a category error arising from the attempt to define a direct morphism between physical and reflective layers. We formalize the system's identity as a dynamic fixed point in a Hilbert space and introduce the Coherence Functional $\kappa$ as a measure of conscious alignment. BrocaOS serves as the primary experimental substrate for this theory.