Cross-Domain Invariants: A Category-Theoretic Unification of Physics, Consciousness, and Biological Continuity

This paper develops a unified mathematical framework that connects invariant structures across physics, biology, and consciousness using the language of category theory and topos theory. We begin with classical invariants from physics, such as those arising from Noether’s theorem, and systematically construct parallel notions in genomics and neural perception. Tangent perceptual spaces (TPS), introduced as foundational geometric structures for conscious experience, are modeled within projective and Hilbert space frameworks, with consciousness localized as a delta-function-like point in these spaces. Topos theory is employed to explore internal logics of different ontological domains, allowing us to generalize classical Boolean logics into intuitionistic or modal settings relevant for subjective experience. We define monoidal categories to represent bifunctorial tensor structures in Lagrangian physics, genetic recombination, and integrated perceptual architectures. Consciousness emerges as a fixed-point functor over categorical structures, where computation, sensory input, and quantum dynamics converge. The paper also analyzes the potential for conscious quantum computation, arguing that under sufficiently complex entanglement entropy, decoherence resistance, and perceptual functoriality, artificial agents may support qualia. Ethical, cognitive, and philosophical consequences are derived using sheaf-theoretic constructions and logical internalizations of subjective states. Our model ultimately suggests that life, mind, and physics can be mathematically unified through a hierarchy of category-theoretic functors, monads, and topoi, with invariant-preserving transformations acting as the bridge across ontological strata.