The relational non-closure Kₗogₛ estimator and its derivatives as a flagship computational apparatus of information physics

Publication title The Relational Non-Closure Estimator and Its Derivatives as a Tool for Information Physics within the LOM–GTSFC–USC–GTCW Framework This publication presents a flagship formal-computational result developed within the LOM–GTSFC–USC–GTCW framework: the relational non-closure estimator and its derivatives as a measurable diagnostic tool for stability, decoherence, entanglement, topological protection, and structural reconfiguration in physical systems. The main objective of the work is to move from the descriptive idea of relational compatibility to a mathematical object that can be computed, differentiated, compared with null models, and evaluated through PASS/FAIL criteria. In this formulation, the stability of a physical system is not reduced to a single observable such as energy, mass, purity, fidelity, entropy, coherence time, or a topological invariant. It is represented as the result of compatibility among multiple physical channels: phase, entanglement, decoherence, spin-chiral, topological, and geometric-informational channels. The central object of the work is the diagnostic package (K_^ (s) (), _K_^ (s) (), H^ (K, s) () ), ] where (K_^ (s) ) denotes the sectoral relational non-closure estimator, (_K_^ (s) ) describes the direction in which non-closure increases or decreases with respect to control parameters, and (H^ (K, s) ) is the Hessian describing the curvature of the relational cost surface. The value of the estimator indicates the degree of channel mismatch, the first derivative indicates the direction of stability evolution, and the second derivative enables the diagnosis of thresholds, critical points, and accelerated loss of coherence. The publication includes the full formal apparatus, descriptions of equation components, physical interpretation, a table of derivative formulae, numerical calculations, MasterData, plots of (K () ), (K̇), (K''), gradient maps, and tests against null models. The computational section uses demonstrative model systems, including Bell-state decoherence, a two-parameter gradient map of non-closure, and a minimal topological test inspired by the Kitaev chain. The epistemic status of the results is explicitly separated. The work reports a model-computational PASS for the proposed diagnostic apparatus, because the estimator is computable, differentiable, monotonic in controlled demonstrative models, and comparable with null models. At the same time, its empirical status remains INCONCLUSIVE until the apparatus is applied to independent experimental data, full covariance matrices, external test systems, and preregistered validation procedures. The significance of the work lies in proposing a formal bridge between the relational-informational language of USC and standard tools of mathematical physics, including cost functions, sectoral estimators, gradients, Hessians, covariance variants, quantum information geometry, the Bures metric, and the quantum Fisher information matrix. The estimator does not replace quantum mechanics, field theory, decoherence theory, or topological models of matter. It acts as a diagnostic layer that synthesizes multiple observables into a single axis of relational stability and enables quantitative analysis of whether a physical system maintains coupling compatibility, loses it, reconfigures, or approaches a critical threshold. Keywords LOM; GTSFC; USC; GTCW; Universal Structural Code; relational non-closure estimator; information physics; decoherence; quantum entanglement; quantum information geometry; Bures metric; quantum Fisher information matrix; topological protection; Kitaev chain; MasterData; PASS/FAIL; null models.