What makes information physically possible? This research framework marks a significant shift in the evolution of the HCC-S³ programme. Rather than asking how much information a physical system can store, it asks a more fundamental question: under what conditions can a difference between two quantum states become physically real? The framework proposes that information is not a primitive entity, but a derived quantity arising from realised physical distinguishability. In this view, geometry does not merely provide a background for physics—it constrains which distinctions can exist as physically meaningful states. The central conceptual chain is intentionally reversed with respect to classical information theory: Physical Carrier (M, g) → Geometric Constraints → Physical Distinguishability → RPD (M) → Ndist = |RPD (M) | → I = log₂ Ndist. Here, RPD (M) denotes the set of realised physically distinguishable equivalence classes on a compact carrier, while I = log₂ Ndist becomes a consequence rather than an axiom. This approach shifts the focus from counting bits to identifying the geometric and operational conditions that make bits possible in the first place. A key feature of the framework is the introduction of the general geometric functional 𝒢 (M), combining candidate structures such as holonomy, Berry phase, spin structure, curvature invariants, Dirac and Laplace spectra, and spectral gaps. The framework does not claim that any single element determines distinguishability. Instead, it establishes a disciplined research programme in which these geometric structures are investigated as potential constraints on physically admissible differences between quantum states. The framework also introduces an operational definition of distinguishability: two states are physically distinguishable only if there exists a finite-energy, finite-time local operation capable of discriminating them above a prescribed threshold. This deliberately separates physical distinguishability from purely mathematical orthogonality and places experimental accessibility at the centre of the formalism. Importantly, HCC-S³ is presented as the primary compact test carrier, not as an exclusive description of physical reality. The formalism is intentionally written for a general compact manifold M, allowing future comparison with alternative compact geometries. This makes the programme methodologically open while preserving a concrete reference model for mathematical development. Rather than proposing a completed theory, this work establishes a research framework. It introduces precise definitions, axioms, candidate criteria, research questions, and open mathematical problems, providing a structured path toward a future ontology of physical information. The long-term objective is not merely to measure information, but to understand how geometry, quantum structure, and operational constraints give rise to physically realised distinguishability, from which information naturally emerges as a derived concept.
Preece et al. (Tue,) studied this question.