Theorem on a Uniform Family of Constructive Yang-Mills Theories with Phenomenological Input GUT : A Unified Framework for the Unification of the Four Fundamental Interactions

Abstract In this work, we formulate and prove a theorem on the existence of a uniform family of constructive Yang–Mills quantum field theories, parameterized by a phenomenological input — the grand unified gauge group $G_{\mathrm{GUT}}$. Unlike traditional approaches, the group $G_{\mathrm{GUT}}$ is not postulated from constructivity requirements, but is selected on the basis of experimental data (coupling constant unification, particle spectrum). It is shown that the constructive apparatus — uniform reflection positivity (RP) → color majorant → Mayer expansion → Prokhorov's theorem → Osterwalder–Schrader reconstruction — provides polynomial control of the constants for any admissible input. Within a unified BV–BFV framework with an R-boundary, all four fundamental interactions emerge as spectral classes of a single constructive theory: Gapped classes: the non-Abelian factors $SU(3)_c$, $SU(2)_L$ (short-range strong and weak interactions); Gapless classes: the Abelian $U(1)_{\mathrm{em}}$ and the massless spin-2 boundary sector (gravity). For the group $SU(3)$ the results are unconditional (following Jacobsen, 2025); for extended groups they are conditional under assumption (A3*) of a dynamical gap in the infrared limit. Main Results Structural uniqueness: the boundary structure $\mathfrak{B} = (M_{T,\Sigma},\,\omega,\,\delta_\varepsilon,\,(2/\ell,\,\ell/2))$ contains no free parameters; the counterterms are fixed by the AdS$_4$ geometry. Color majorant: $|\Phi_G(\Gamma)| \le C_2(G)^{E/2}\|d\|(G)^W(\dim G)^{L/2}$ — the weight of any Feynman graph is bounded by a polynomial in the group invariants. Non-collapsing radius of convergence: $\varepsilon_0^2 < (c_{\mathrm{cnt}}K_0\kappa(G)^p)^{-1} \equiv g_0^2(G) > 0$ for any finite set of candidate groups $G_{\mathrm{GUT}}$. Pointwise unification: $g_1(s_\star) = g_2(s_\star) = g_3(s_\star) = g_{\mathrm{GUT}}$ with the two-loop $\beta$-function. Numerical Verification All algebraic statements have been verified symbolically in Google Colab (SymPy): Base of the color majorant (rules F1–F3): residual $\le 3.3\times 10^{-16}$ for $SU(3)$ and $SU(4)$. Numerical cross-check against the benchmarks $3.28\pm0.14$, $4.21\pm0.25$ via GEVP mass extraction. Parameters for the supercomputer stage: $L\ge16$, $L_t\ge32$, $N_{\mathrm{cfg}}\ge500$, acceptance $0.3$–$0.5$. Position within the Research Cycle This work completes a cycle of 13 stages (Stages 0–13) on the construction of low-energy quantum gravity and gauge theories on manifolds with a smoothed geometric boundary, unifying the results of previous depositions into a single logical framework. Honest Limitations Gravity is treated as an effective sector; Assumption (A3*) for $G\neq SU(3)$ is a residual input; The de Sitter (dS) case remains a hypothesis; The Abelian factor $U(1)$ requires a separate IR analysis. A solution of the Clay Millennium Problem and a "theory of everything" are not claimed in the strict sense — the work provides a constructive framework within which the phenomenological choice of $G_{\mathrm{GUT}}$ becomes the sole experimental input.