Observation–Feedback Closure I (v1.5.0): A Self-Consistent Relational Reconstruction from Directed Processes and Internal Occurrence Arithmetic to Functional Dimensionality and Conditional Geometric, Temporal, and Topological Extensions

Abstract

This preprint is version v1.5.0 of Observation–Feedback Closure I. It presents a foundational reconstruction and logical restructuring of the OF Closure framework rather than a merely additive extension of v1.4.0. The principal purpose of this version is to repair three foundational dependency problems in the earlier architecture: linear independence and dimensionality were used before a scalar field and vector space had been constructed; real-valued effective counts were introduced without an internal arithmetic route from finite occurrences; and countable density was not adequately distinguished from continuum completeness. The reconstructed dependency chain begins with non-numerical directed elementary observation–feedback processes carrying primitive observation and feedback ports. Finite process complexes and compatible closure witnesses are used to construct complete physical individuals. Generation, termination, and closure-stabilization are subsequently defined as derived functional roles rather than primitive process ports. Equivalence classes of finite occurrences provide an internal realization of the non-negative integers, followed by group completion and localization: N₀ᴼᶠ → ℤᴼᶠ → ℚᴼᶠ. Quantitative refinement stability is introduced separately through the RCS-4 postulate. The internal rational field is then completed mathematically to Q̂ᴼᶠ ≅ ℝ, while the physically realizable spectrum is retained as a potentially proper subset of this mathematical completion. The framework therefore distinguishes finite occurrence structure, rational calibration, countable density, mathematical continuum completion, and physical realizability. Only after these constructions is a controlled linear realization introduced. Under the Three-Channel Linear Nondegeneracy condition, the representations of the three derived functions generate a functional response space of dimension exactly three. This is a theorem about functional response dimensionality; it does not establish that the underlying state space or physical space is three-dimensional. Under a selected positive-definite evaluation form, the functional response space carries an internal O(3) frame-isometry structure. Its externalization as spatial O(3) remains conditional. The Lorentz-type reconstruction is likewise divided into distinct logical levels. The proper-orthochronous isometry group of a selected Lorentz-type form is identified with SO⁺(3,1) as a standard mathematical result. Closure of the OF-derived generators remains a Target, and faithful global identification of an OF-derived representation with SO⁺(3,1) remains a Conditional Target. Version v1.5.0 also records quotient and inverse-limit reconstruction spaces, a conditional pure-topological non-Hausdorff implication, and a conditional minimal two-class residual classification. The bridge from general OF reconstruction data to the non-Hausdorff hypotheses remains unresolved. A structural time-arrow mechanism is retained as a Conditional Target, while gravitational-type and electromagnetic-type constructions remain conditional structural correspondences rather than derived interaction theories. The version includes an explicit logical-role taxonomy, a unique result-status registry, and a complete v1.4.0–v1.5.0 migration record. Earlier archival versions remain part of the historical development of the OF Closure program, but v1.5.0 governs the current definitions, dependencies, notation, and logical status of inherited claims. Results from earlier versions are retained, reformulated, weakened, deferred, or withdrawn according to their present level of justification; unrestricted equivalence or backward compatibility among all versions is not claimed. This release is a working-record version deposited primarily to establish priority and document the systematic development of the OF Closure research program. It provides a reconstructed foundation for subsequent work, not a completed physical theory. No field equations, quantum dynamics, gauge dynamics, gravitational dynamics, interaction laws, quantitative cosmology, or new empirical predictions are derived.