F–S–O: Observation Horizons, Task Recoverability, and Realization Cost

Abstract

The Function–Structure–Operator (F–S–O) framing separates a specified task, a channel-to-state structure, and the operations available to a system. This note develops a finite-dimensional, first-order interpretation in which observation is represented by an explicit readout map. Available channel perturbations form a subspace; its quotient by the readout kernel describes distinguishable perturbations. Standard linear algebra then separates unavailable directions, hidden available directions, and redundant inputs. Task recoverability is characterized by a kernel inclusion and an equivalent rank deficiency, while a positive-definite cost metric yields a minimum-cost realization formula. Constructive counterexamples show why an invertible activation derivative need not preserve visible rank, why equal rank does not imply equal realization cost, and why instantaneous invisibility need not persist over time. An exhaustive example with four nodes and five channels evaluates all 32 sensor subsets: measuring one target channel costs 1 unit, whereas complete channel recovery requires a minimum of 3 units. The contribution is a bounded methodological synthesis and an executable collection of examples, rather than a new rank theorem or an empirically established advantage over existing sensor-design methods. Sixteen deterministic checks and three supplementary calculation groups accompany the manuscript. They verify specified constructions, not cross-domain predictive validity. Scope and materials. A methodological technical note with an editable manuscript, executable Python supplement, exact sensor enumeration, constructive counterexamples, and bibliographic records. No new rank theorem, completed C2/C3 benchmark, or empirical advantage over existing methods is claimed. AI assistance. Grok was used in an earlier research handoff. OpenAI ChatGPT/Codex assisted the subsequent mathematical audit, code preparation and execution, literature checking, manuscript drafting, translation, and document preparation. The calculations use explicit synthetic models; they are not measured experimental data. AI tools are not authors or contributors. The package is prepared for final review by the named human author and does not assert that this review has already occurred. Rights. CC BY 4.0 is proposed for the deposited work; the accompanying original code is additionally offered under MIT. The final release is subject to author confirmation.