BKT–37B Annex B to BKT–37 Extension of the Mathematical-Physical-Logical Apparatus and Flagship Numerical Calculations of USC

BKT–37B: Annex B to BKT–37 — Extension of the Mathematical, Physical, and Logical Apparatus and Leading Numerical Calculations of USC BKT–37B v4.3 is a formal mathematical, physical, and logical extension of the main BKT–37 article. Its purpose is to organize the USC apparatus so that every new theoretical object has a defined mathematical type, physical interpretation, relation to the null model, epistemic status, and potential falsification path. The document translates the intuitive notions of compatibility, closure, and non-closure into the language of a typed state space, tolerance metric, defect vector, null-model projector, orthogonal residual, Fisher information, and PASS/FAIL protocol. The central contribution of the annex is the correction and strengthening of the tensor apparatus. The non-closure tensor is formulated as the proper mixed operator (D^{a}{\ b}=\Delta^{a}G{bc}\Delta^{c}), whose trace satisfies ({\rm Tr}D=\delta^2). The closure tensor is extended from a local rank-one diagnostic estimator to the multichannel Gram operator (C_{\rm multi}=e^{-\delta^2/2}SWS^\dagger G). As a result, the USC apparatus is not reduced to a single scalar compatibility indicator, but can describe a multichannel structure involving the metric, correlations, defect directions, stabilizing channels, and effective test modes. The second part of the annex organizes the leading numerical calculations. The hydrogen–deuteron metrological block uses the ionization energy of hydrogen and the binding energy of the deuteron to demonstrate the separation between the surface scale and the deep nuclear scale. In particular, the deuteron binding energy (B_d=2.22456637\ {\rm MeV}) gives the contrast (B_d/E_{\rm ion}(H)=1.63589886298\times 10^5). The annex also presents the logarithmic indices (K^{\log}{H}) and (K^{\log}{d}), while explicitly stating that they are ordering estimators, not independent physical proofs. A separate nuclear block includes the chain (^{108}{\rm Xe}\to{}^{104}{\rm Te}\to{}^{100}{\rm Sn}), the alpha-decay width (\Gamma_\alpha), and the barrier index (\Theta_\alpha), treated as controlled structural anchors of the apparatus rather than as direct validation of USC. The main conclusion of the annex is deliberately controlled: BKT–37B v4.3 strengthens USC as a formal, testable, and reproducible framework for analyzing residuals with respect to a null model, but it does not present USC as an empirically confirmed fundamental theory. Version v4.3 defines the conditions under which a new formal language may become testable physical content: an explicit null model, covariance, orthogonal residual, Fisher identifiability, information penalty, out-of-sample testing, and a reproducible MasterData record are required. In this sense, Annex B is the tensor-metrological core of further BKT–37 tests and a technical foundation for future USC analyses in hadronic physics, nuclear physics, and high-energy data. Keywords: BKT–37B; USC; PJM–GTWSSF–USC–GTCW; Universal Structural Code; non-closure tensor; closure tensor; Mahalanobis metric; tolerance metric; orthogonal residual; Fisher information; Kullback–Leibler; BIC; EFT; Wilson loops; null model; hydrogen; deuteron; deuteron binding energy; alpha decay; (^{104}{\rm Te}); (^{100}{\rm Sn}); PASS/FAIL; MasterData; nuclear physics; hadronic physics; residual analysis.