Mathematical universes developed as formalizations of relationality. These are organized by a 2×2 grid of internal character (process versus structure) and validation target (mathematical correspondence versus empirical correspondence). See The Five Mathematical Systems for the full architecture.
Process × mathematical correspondence
The semioverse hierarchy formalizes processual dynamics — signs, interaction, agency — and validates against established mathematical theorem.
- Semiotic Universe — the mathematical bedrock: a complete Heyting algebra with modal closure and typed lambda calculus
- Interactive Semioverse — extends the Semiotic Universe with things, interaction terms, and footprints
- Agential Semioverse — extends the Interactive Semioverse with agent profiles, tool signatures, and skill calculus
Structure × mathematical correspondence
- Quasicrystalline Hypertensor Topos — structural mathematical universe defined by trace categories, fiberwise Heyting-modal algebras, and generative closure
Process × empirical correspondence
- Dynamical Universe — derived from empirical processual phenomena; formalizes evolution, directed transformation, and irreversibility
Structure × empirical correspondence
- Spectral Universe — derived from empirical structural phenomena; formalizes measurement, spectral decomposition, and discrete outcomes
Historical
- Generative Fibered Recognition Trace Universe — deprecated; superseded by QCHTTopos