PUBLICATION TITLE The relational non-closure estimator Kₗog^ (s) within the ARA1^ (s) apparatus as a three-level apparatus of information physics Document number: BKT-43Module classification: USC-RA1Release identifier: BKT-43USC-RA1₂0260619ᵥ1. 3Date: 19 June 2026 Publication BKT-43 presents a formal-computational apparatus of relational non-closure developed within the LOM-GTSFC-USC-GTCW framework, with particular emphasis on the USC-RA1 module. The work introduces the relational non-closure estimator Kₗog^ (s) as an operational component of the broader apparatus ARA1^ (s), whose purpose is the quantitative diagnosis of stability, decoherence, entanglement, topological protection, structural reconfiguration, and the dynamics of open physical systems. The main result of the work is not a single scalar Kₗog^ (s), but a three-level hierarchy of an information-physics apparatus. The first level is the robust operational aggregator Kₗog^ (s), which compresses the contributions of multiple observable channels. The second level is the covariance variant Kcov^ (s), which accounts for correlations between channels and reduces the problem of double-counting relational cost. The third level is the geometric variant Kgeom, in which non-closure is defined as the minimal distance from the manifold of relationally closed states, using the Bures metric and the local QFIM variant. The central RA1 apparatus can be written in plain text as: ARA1^ (s) = (Kₗog^ (s), Kcov^ (s), Kgeom, grad K^ (s), H^ (K, s), dKL^ (s) /dt, var (K^ (s) ), PASS/FAIL). In this notation, Kₗog^ (s) denotes the operational estimator of relational non-closure, Kcov^ (s) denotes the covariance variant, Kgeom denotes the geometric variant, grad K^ (s) denotes the gradient of relational cost, H^ (K, s) denotes the Hessian of the cost surface, dKL^ (s) /dt denotes the estimator change induced by Lindblad dynamics, var (K^ (s) ) denotes uncertainty propagation, and PASS/FAIL denotes explicit validation criteria. This construction enables the analysis not only of the value of the system’s relational cost, but also of its direction of change, the curvature of the cost surface, Lindblad-equation-based dynamics, uncertainty propagation, and robustness against null models. The work emphasizes that Kₗog^ (s) is not a fundamental metric on the space of states. It is a robust function aggregating non-closure channels. The geometric anchoring of the channels is introduced through the Bures distance and its relation to the quantum Fisher information matrix. A significant development relative to the earlier apparatus is the definition of the channels Deltaᵢ as normalized Bures distances from reference states or reference manifolds. As a result, the channels cease to be arbitrary functions of observables and become geometrically justified measures of deviation from a relationally closed configuration. This variant is demonstrated in the Bell-state dephasing model, where the finite Bures distance, the channel DeltaBQM, the derivatives of the estimator, and the corresponding QFIM for a one-parameter family of states are derived. The publication contains the full formal apparatus, explanations of equation components, a table of derivatives, MasterData, numerical calculations, plots of K (lambda), dK/dlambda, d2K/dlambda2, gradient maps, robustness tests against outlier channels, and comparisons with null models. Three classes of model cases are included: Bell-state decoherence, a two-dimensional gradient map, and a topological sector inspired by the Kitaev chain. In the topological sector, the discrete channel describing the topological invariant is separated from the continuous transition channel, which can be used to diagnose approach to a critical threshold in finite systems. The epistemic status of the results is explicitly separated. The work obtains a formal PASS and a model-computational PASS, because the apparatus is computable, differentiable, comparable with null models, and supported by numerical tests. Its empirical status remains INCONCLUSIVE, because full validation requires application to independent experimental data, full covariance matrices, real uncertainty-propagation procedures, and preregistered validation criteria. The significance of the work lies in proposing a measurable, falsifiable, and extensible bridge between the relational interpretation of USC and standard tools of mathematical physics: quantum information geometry, the Bures metric, QFIM, gradient analysis, Hessian analysis, Lindblad dynamics, robust statistics, and null-model testing. The apparatus ARA1^ (s) does not replace quantum mechanics, decoherence theory, topological models of matter, or field theory. It acts as a diagnostic layer that synthesizes multiple observables into a controlled structure for assessing relational stability. KEYWORDS LOM; GTSFC; USC; GTCW; BKT-43; USC-RA1; relational non-closure estimator; Kₗog; ARA1; information physics; Bures metric; QFIM; quantum information geometry; decoherence; entanglement; Kitaev chain; topological protection; MasterData; null models; PASS/FAIL.
Robert Kupski (Fri,) studied this question.