Geometric Confinement and QCD String Tension from Internal Topological Flux Tubes in the Zhang Effective Metric

Quark confinement—the phenomenon where isolated quarks are never observed and the strong force potential grows linearly with distance—remains one of the most profound unsolved problems in the Standard Model. While lattice QCD confirms the linear potential V(r) ~ σr, an analytical derivation of the string tension σ from first principles is still lacking. In this paper, we demonstrate that confinement naturally emerges from the 15-dimensional geometry within the rotational-order geometric unification framework based on the Zhang Effective Metric. Gluon flux tubes between color charges are identified as topological strings embedded in the internal S³ fibre. We analytically derive the QCD string tension as σ ~ g₄²/r₁². Using the strong infrared coupling g₄ ≈ 2.1 and matching the observed σ ≈ 0.18 GeV² yields an internal radius r₁ ≈ 0.4 fm. The finite mass gap (Paper 66) and Abelian projection ensure the flux tube maintains a finite transverse width. Furthermore, integrating out the transverse quantum fluctuations of the internal S³ string naturally yields the correct Lüscher term (-π/12r), matching lattice QCD. The string action directly yields the Wilson loop area law ⟨W(C)⟩ ~ exp(-σA), and string breaking via dynamical quark pair creation naturally limits the linear potential at r_break ~ 1.2 fm.