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 (HSM), the local Breit–Wigner resonance model (HBW), the one-parameter USC tail deformation (H⁽¹⁾{ USC}), and the five-component USC projection model (H(Π){ USC}). The comparison is carried out using a common statistical language: Mahalanobis distance, (χ^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 → eνμν) 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(Π)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[χ^2SM=19.340383,Δclose=0.108369,Δχ^2=19.113253,Δ{ BIC}{H-0}=-9.383702,ln B 4.691851.]Monte Carlo validation gives a local rarity of approximately (p{ MC} 0.0014), corresponding to a local one-sided scale of about (2.9)–(3.0σ). 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.