Sectorized Observable Framework

Part VIII of the RIME program. Papers IV--VII repeatedly introduce structures of the same form: sectors, observable families, projected blocks, support shadows, commutator shadows, accessibility depth, jets, walls, and completion principles. Paper VIII identifies the mathematical object to which these constructions are attached. We define the Sectorized Observable Framework (SOF) as an analysis paradigm for sectorized observables built from a chosen space, a compatible sectorization, and an observable family. In finite dimension, this means a Hilbert space decomposed into orthogonal sectors together with observables acting on that space. We then define strict SOF morphisms, the strict SOF category, and the natural accessibility constructions: direct support, commutator survival, first-depth accessibility, and the accessibility jet. The main result is a naturality theorem: the RIME accessibility observables are canonical constructions on SOF data. Under strict SOF equivalences they are preserved exactly; under strict embeddings they are preserved on the embedded image, with first-depth accessibility satisfying the expected monotonicity. Spectral charts and accessibility walls are likewise natural once the corresponding smooth spectral SOF family has been specified. Paper VIII marks the transition from result-oriented papers to object-oriented theory. Rubik, quantum gates, Markov systems, graph systems, and Yang-like filtration systems are treated as different instances of the same sectorized observable architecture rather than separate theories. Source repository: https://github.com/dooven-prime/rime-lite This work is part of the Representation-Induced Mechanics and Evolution (RIME) research program. Foundational archive DOI: 10.5281/zenodo.21108197