Sectorized Observable Framework

This paper defines the static object language of the Sectorized Observable Framework (SOF). A finite SOF realization consists of a finite-dimensional complex Hilbert space, a complete marked orthogonal sectorization, and a declared labelled observable alphabet. The construction is formal relative to these choices; it is not a classification theorem and does not assert that every source has a unique or canonical sectorization. The framework keeps its linear and multiplicative layers distinct. The marked sector algebra, observable operator system, and sector-enriched operator system are recorded separately from the positive-word algebra, observable star-closure, and sector-enriched star-closure. These closures answer different questions and do not recover generator labels, route data, word length, or first-hit depth. Routed products and full words form one finite-filtration branch, while any Lie/Hall carrier and filtration must be registered independently. The theorem layer establishes strict operator and independently enriched Lie/Hall categories. Under strict morphisms, conjugation identifies matched multiplication closures by exact algebra or star-isomorphisms, preserves matched operator, route, word, and Lie/Hall witnesses, and makes target first-hit depths non-increasing. Equality of support and depth is reserved for strict equivalence rather than arbitrary strict embedding. The results concern static objects and carrier-qualified functoriality. They do not provide a weak deformation category, moving observable fields, wall classification, compiler contract, application census, universal completion theorem, or an unconditional bridge between operator/word and Lie/Hall depth. Version 2.1 adds an exact marked finite-realization family, a finite represented positive-closure certificate, and a strict Boolean-path control. The paper-owned conformance record covers six modular actions, seventeen bounded triangle-group actions, fifty-seven marked realization and saturation receipts, three equal-operator/distinct-label witnesses, and one exact four-state Path/Route/W witness. These computational artifacts are source-addressed Computational Certificates and are not premises of the abstract theorems. The results remain carrier-qualified. They do not establish canonical sectorization, a general finite-representation classification, generic positive/star-closure separation, Lie/Hall accessibility or Lie depth, deformation or wall semantics, graph-Laplacian, surface, Hecke, moduli, Selberg, or automorphic claims. Source repository: https://github.com/dooven-prime/rime-lite Paper VIII of the Representation-Induced Mechanics and Evolution (RIME) program. Independent researcher; no institutional affiliation.