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

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 GGUT. Unlike traditional approaches, the group GGUT 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)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 B = (MT,Σ,\,ω,\,δ_ε,\,(2/,\,/2)) contains no free parameters; the counterterms are fixed by the AdS₄ geometry. Color majorant: |ΦG(Γ)| ≤ C₂(G)E/2\|d\|(G)W( G)L/2 — the weight of any Feynman graph is bounded by a polynomial in the group invariants. Non-collapsing radius of convergence: ε₀² < (ccntK₀κ(G)ᵖ)⁻¹ ≡ g₀²(G) > 0 for any finite set of candidate groups GGUT. Pointwise unification: g₁(s₎ = g₂(s₎ = g₃(s₎ = gGUT with the two-loop β-function. Numerical Verification All algebraic statements have been verified symbolically in Google Colab (SymPy): Base of the color majorant (rules F1–F3): residual ≤ 3.3× 10⁻¹⁶ for $SU(3)$ and $SU(4)$. Numerical cross-check against the benchmarks 3.28±0.14, 4.21±0.25 via GEVP mass extraction. Parameters for the supercomputer stage: L≥16, Lₜ≥32, Ncfg≥500, 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≠ 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 GGUT becomes the sole experimental input.