The Polynomial n(k) as a Dimension of an Irreducible Representation of the Exceptional Group F4: n(k) = dim V_F4(k·ω₄)

We provide the definitive answer to the origin of the exact polynomial n(k) = (103k⁴ − 370k³ + 101k² + 478k)/12. The polynomial is the dimension of an irreducible representation of the exceptional Lie group F₄ with highest weight proportional to k: n(k) = dim V_F₄(k·ω₄), where ω₄ is the minimal fundamental representation of dimension 26. The formula is verified analytically (Weyl formula for 24 positive roots) and numerically (LiE, GAP) for k = 1,…,10. The result explains all previously established connections: the Deligne series, the 691 congruence, linear representations via modular forms, and the SU(26) hierarchy. This is the final paper in a series of six works on the polynomial n(k). Languages: Russian, English. License: UAL v1.0.