Extension of the Differential Algebraic Framework to the Baryon (Three-Quark) System: A Constructive Approach to Unified Analytic Solutions in Nonrelativistic QCD

Thenonrelativistic three-quark system (baryon) with the Cornell potential V (r) =−αs/r+σr is a fundamental model of low-energy QCD. No general closed-form solution is known. In this work we extend the constructive differential algebraic framework recently developed for the gravitational three-body problem to the baryon system. We define the nonlinear ordinary differential algebraic closure K(baryon)NLODE as a differentially closed field extension constructed by recursively adjoining: (i) fundamental solutions of linearized variational equations around reference two-body Coulomb orbits; (ii) multi-index radical extensions Φ1/p; (iii) roots of unity ωp; (iv) Airy functions, generalized Laplace coefficients B(j)m (e;γ) that incorporate confinement corrections, and elliptic integrals. Weprovide a complete derivation of Γ(baryon)m,k using the multivariate Faà di Bruno formula and projection onto a Fourier–Airy–Laplace basis. Explicit formulas involve Beta functions, generalized color Laplace coefficients B(2)m (e;γ) (which satisfy exponential decay, Proposition B.1), and symmetry factors. Using these coefficients we: • Recover the Y-shaped string (equilateral triangle) and collinear string configurations as special cases where Φm ≡ 0 or Φm reduces to an algebraic number of degree at most five; • Derive the color stability criterion — a QCD analogue of the Gascheau–Routh criterion — showing that the linear confining term enlarges the stability region (Theorem 6.2); • Prove that the equal-mass baryon is not Liouville integrable in general, but exhibits algebraic Arnold diffusion (Theorem 9.4); • Compute the leading 1/αs (post Coulomb) corrections from spin–spin and spin–orbit interactions (Theorems 8.2 and 8.4) and obtain the existence condition for collinear string solutions with spin effects; • Establish an algebraic integrability criterion based on polynomial relations among the Φm polynomials (Theorem 9.3) and prove that for unequal masses the system is generically non integrable (Theorem 9.4). All constructions are algorithmic: we present pseudocode for precomputing Γ(baryon)m,k (Algorithm ??) involving Airy-integral quadratures, and for adaptive homotopy continuation with certified precision (Algorithm 7.3). Complexity analysis yields O(∥M∥·∥Bdmax ∥+SM 3) operations. The framework naturally extends to multiquark systems (N ≥ 4), where the combinatorial coefficients become sums over unordered pairs and are sparse (Theorem 10.1). Quantum speedup via amplitude estimation and HHL is discussed (Theorems 9.1 and 9.2). All previously open problems (convergence radius vs Lyapunov exponent, algebraic Arnold diffusion, existence of a resonant closure, sparsity, HHL feasibility) are settled with rigorous proofs in the main text and appendices.