Negative and Fractional Types in the Fidelity Framework

Abstract: Our Native Type Universe (NTU) has been detailed through five previous papers establishing the substrate our framework's compilation pipeline targets across multiple hardware platforms. We have found in the course of that work a deeper reach this foundation makes available: negative and fractional types as native first-class constructs. James and Sabry established these dualities in 2012; Chen and Sabry later developed their categorical interpretation in compact closed categories. These dualities have practical benefit for compute modalities in our Fidelity Framework where extant general purpose compute framings lack the substrate to host them as native constructs. We see practicality with these type forms in preserving decidability and principal types through the abelian-group algebraic pattern Kennedy's dimensional types establish. The resulting isomorphisms would admit new, concise forms of resolution within our novel lowering strategy, and we sketch a notional Clef language syntax that would admit rational dimensional exponents into our algebra. We trace the implications across several problem spaces these type forms would open to our compilation and verification disciplines: Bayesian inference where fractional types would express conditioning obligations, quantum computation (and simulations) where negative types would provide the type-level adjoint, and finally adiabatic computation where the combined discipline would express Hamiltonian deformation as a reversible constraint-propagation process. The inherent structure of our NTU together with the supporting framework appears well-suited to problem spaces that current software ecosystems do not directly address, while keeping approachable development ergonomics and mature tooling aligned with operational guarantees the framework aspires to provide.