A General Theory of Representation: Observable Restriction, Identification, and the Two Sources of Information Loss

Every scientific model, simulation, digital twin, measurement, map, machine learning model, CAD model, and BIM model is fundamentally a representation of reality, and every such representation loses information relative to the system it represents. This paper develops a mathematical framework, grounded in the Formal Theory of Manifestation (FTM), for describing exactly how that loss occurs. We give a formal definition of a deterministic representation and show that every such representation loses information through two distinct mechanisms: Observable Restriction (the choice of which observables to retain) and Identification (the merging of distinguishable states via a quotient), and that these two mechanisms are not interchangeable in order: restriction is necessarily the first stage whenever the identification in question represents observational indistinguishability. This paper does not claim novelty for the underlying mathematical theorems: the minimality of a generated σ-algebra, the categorical first isomorphism theorem in exact categories, and the Shannon entropy chain rule are all classical. The novelty lies in recognizing that, read simultaneously as statements about representation, these results jointly force a structural decomposition that, to our knowledge, has not previously been identified as such: every deterministic representation factors canonically and uniquely as Observable Restriction followed by Identification (Theorems 8.2, 7.2, 9.1), and this decomposition admits a domain-independent quantitative language, the Restriction Index (RI) and Identification Index (II), applicable uniformly across CAD, BIM, GIS, robotics, medical imaging, knowledge graphs, computer vision, and machine learning. We situate the framework against sufficient-statistics theory, rate-distortion theory, and the information bottleneck method, give explicit axioms for the effective-dimension functions underlying RI and II, and illustrate the framework with a single running example (a digital camera's image pipeline) alongside eight domain-specific worked examples in uniform format.