BKT–37A Annex A to BKT–37 Key Numerical Calculations of the USC Model, Algebraic Verification, Monte Carlo Tests, and Comparison with ATLAS/CMS Channels

BKT–37A: Annex A to BKT–37 — Key Numerical Calculations of the USC Model, Algebraic Verification, Monte Carlo Tests, and Comparison with ATLAS/CMS Channels This technical annex extends the main BKT–37 article by transforming the discussion of a candidate USC residual from a qualitative interpretive layer into an explicit, reproducible numerical and statistical procedure. The document introduces and compares four model families: the Standard Model null hypothesis (H_{\rm SM}), the local Breit–Wigner resonance model (H_{\rm BW}), the one-parameter USC tail deformation (H^{(1)}{\rm USC}), and the five-component USC projection model (H^{(\Pi)}{\rm USC}). The comparison is carried out using a common statistical language: Mahalanobis distance, (\chi^2) reduction, BIC penalty, approximate Bayes factor, Monte Carlo validation, Fisher-matrix diagnostics, and PASS/FAIL decision criteria. The main empirical context is provided by aggregated ATLAS (WW \to e\nu\mu\nu) observables, with CMS EXO–25–021 used as a control context for high-mass visible dilepton resonance searches. The annex identifies M10 and M14 as the strongest sectoral cases supporting the five-component projection model (H^{(\Pi)}_{\rm USC}), while M18 and M22 provide weaker but structurally consistent support. M16 and M2 favour the simpler one-parameter USC deformation, M13 is better described by a local resonance-type model, and M4 is treated conservatively as marginal after the BIC-sign audit. This classification is essential: the USC projection model is not presented as a global replacement for the Standard Model, but as a candidate diagnostic model for selected multibin residual structures. The leading numerical case is M10, for which the annex reports[\chi^2_{\rm SM}=19.340383,\quad\Delta_{\rm close}=0.108369,\quad\Delta\chi^2=19.113253,\quad\Delta{\rm BIC}{H-0}=-9.383702,\quad\ln B\simeq 4.691851.]Monte Carlo validation gives a local rarity of approximately (p{\rm MC}\simeq 0.0014), corresponding to a local one-sided scale of about (2.9)–(3.0\sigma). The document emphasizes, however, that this is a sectoral computational result, not a global discovery claim. A full global PASS would require a block likelihood with cross-covariances between observables, out-of-sample validation, and independent reproduction in public likelihood workspaces. The scientific role of Annex A is therefore methodological and falsifiable. It provides a controlled numerical framework for testing whether selected residuals, after subtraction of the null model, can be described more efficiently by a structured multicomponent projection than by a pure Standard Model baseline or a local resonance peak. In the broader BKT–37 framework, this annex supplies the computational bridge between the concept of a stable informational proton node and measurable residual diagnostics in high-energy collider data. It does not claim empirical confirmation of USC as a fundamental theory; it defines the statistical and algebraic conditions under which such a claim could eventually become testable. Keywords: BKT–37A; USC; PJM–GTWSSF–USC–GTCW; Standard Model; ATLAS WW; CMS EXO–25–021; orthogonal residual; BIC; Bayes factor; Monte Carlo; Fisher matrix; missing transverse energy; multibin residuals; PASS/FAIL; collider phenomenology.