BKT-51: Electrochemical Dendritic Growth as a Test of USC Closure Operators

BKT-51 presents a formal-computational analysis of electrochemical dendritic growth as a laboratory model case for the LOM–GTSFC–USC–GTCW framework. The starting point is the classical electrochemical null model (M_0), composed of Poisson–Nernst–Planck transport, Butler–Volmer interfacial kinetics, the Gibbs–Thomson curvature correction, Mullins–Sekerka morphological instability and phase-field approximations. This standard model correctly describes the main dynamics of ionic transport, interfacial reaction, capillarity and dendritic growth. The extension proposed in this work introduces a relational-information USC layer, where the local state of the interface is represented by the growth-channel vector (U_{\mathrm{dend}}), the tolerance set (\mathcal T_{\mathrm{grow}}), the closure operator (\mathcal C_{\mathcal T}), the non-closure operator (\mathcal N_{\mathcal T}), the local logarithmic estimator (K_{\log}^{\mathrm{loc}}), and the orthogonal residual (R_{\perp}^{\mathrm{USC}}) with respect to the null model. This framework makes it possible to test whether the morphological residual left after subtracting the standard model is random or whether it exhibits a reproducible structure associated with channel closure, screening, anisotropy, curvature and growth history. The paper provides demonstrative calculations for dendritic growth under strong electrochemical forcing, compares the classical (K_{\log}^{\mathrm{loc}}) estimator with an information-geometric version based on the Fisher–Nernst–Planck metric, analyzes the statistical (\alpha_K) test, and presents logistic simulations together with Monte Carlo/Kinetic Monte Carlo procedures. The results indicate that including an information-based closure operator may produce sharper selection of active dendritic tips and stronger suppression of screened regions than the standard model (M_0) alone. The work does not replace standard electrochemistry or classical transport theory. Its contribution is the formalization of an additional level of description: a relational-information layer capable of organizing the morphological residual and making it statistically testable. In this sense, BKT-51 provides a model example of the transition from a local transport-equation description to a whole-system response analysis based on channels, tolerance, closure and non-closure. The significance of the work extends beyond electrochemistry. The dendritic laboratory system shows that stronger external forcing does not necessarily imply greater structural order. Stability emerges not at the maximum of external drive, but within the range of channel compatibility. This principle may have methodological relevance for broader research programs, including high-energy physics, nucleon-structure studies, rare-decay searches and precision experiments. In this interpretation, the central object of analysis is not only the production of new states or decay products, but the full tomography of the system response to controlled shifts of coupling channels. Epistemic status: theoretical-computational and methodological work. The presented results provide formal and numerical support for the hypothesis of a relational-information layer of structural organization, but they do not yet constitute full empirical validation of USC. Full validation requires microscopic experimental data, reconstruction of local fields, estimation of (\alpha_K), analysis of (R_{\perp}^{\mathrm{USC}}), and out-of-sample validation. Keywords: LOM, GTSFC, USC, GTCW, BKT-51, dendrites, electrodeposition, Nernst–Planck, Butler–Volmer, Gibbs–Thomson, Mullins–Sekerka, phase-field, information geometry, Fisher metric, Wasserstein, Wasserstein–Fisher–Rao, large deviations, closure operator, logarithmic estimator, (\alpha_K), new physics, relational information structure.