Three Irreducible Modes of a Temporal Substrate: Conceptual and an Information-Geometric Prototype

This paper proposes a unified exploratory framework in which time is the fundamental substrate of physical reality, organized into three irreducible organizational modes — latent (unstructured potential), free (directed propagating relation), and bound (stable self-sustaining configuration). These modes emerge from a minimal generative rule consisting of expansion, contraction, and relation. Representing the rule as probability weights on a statistical manifold, Amari's uniqueness theorem for monotone Riemannian metrics under Markov morphisms derives the Fisher information metric. Under the square-root embedding, the resulting 2-simplex becomes the round 2-sphere (Fisher sphere), providing a derived — rather than postulated — configuration space for a modal field η. Coupling this field to spacetime geometry via a nonlinear sigma-model action with a carefully constructed modal potential yields canonical equations of the form G_μν = 8πκ T_μνη, where the stress-energy tensor incorporates harmonic-map gradients and potential energy on the Fisher sphere. In three distinct regimes, the system recovers limiting behaviors consistent with general relativity (bound mode), dynamical dark energy with evolving equation of state (latent mode), and massless propagation (free mode). An information-geometric curvature evolution law on the sphere further links modal flow to latent-mode dynamics, while a metastable trough in the potential naturally accommodates “incomplete binding” as a candidate for dark matter. The framework originated as an attempt to give coherent conceptual structure to intuitive metaphors about cycling, seeds, frequency as a master dial, stratified forces, and the boundaries where established physics feels incomplete. The mathematics was developed as an AI-assisted consistency test (the AI formalised the statistical-manifold construction, derived the Fisher geometry via Amari's theorem, and verified the canonical equations and limits). An ignition-probability estimate matches the observed matter density to within 3%. This work is presented as a completed exploratory prototype rather than a finished theory. Open problems include deriving the curvature evolution law directly from the action, full quantization and mapping to Standard Model particle content, and robustness studies of the modal potential. It is offered as a minimal, bottom-up scaffold for further discussion, simulation, and testing against ongoing and future data from DESI, JWST, LISA, LSST, and pulsar timing arrays. Notes on Scope: This framework began as conceptual exploration of time as fundamental substrate and three irreducible modes (latent, free, bound). The mathematics was developed as an AI-assisted consistency test. It is presented as a completed exploratory prototype — internally coherent and anomaly-motivated — rather than a finished theory. Open derivations and robustness questions are stated explicitly. It does not replace general relativity, quantum field theory, or the Standard Model in their established domains but seeks a deeper interpretive layer at their boundaries.