FOX Theory (Branch B): The Nucleon as Torus Knot T(2,3)

We study the nucleon as a topological soliton of the FOX scalar field, a pre-geometric framework in which a dimensionless scalar field φ = Φ/Φ_MAX ∈ [0,1] couples to gravity through the Palatini action (Branch A, Zenodo 2026). The central result is a dimensionless identity mφ · r_n = (1−φ_m)/φ_m = 4, exact to 10⁻⁸ and independent of the field amplitude Φ_MAX (Theorem B2). This identity simultaneously encodes Newton's constant (G_eff = G), the stasis value (φ_m = 1/5), and the residual boson mass mφ = 877 MeV, which agrees with the vector meson mass m_ρ = 770 MeV to 14% without parameter adjustment. Under the working hypothesis H4a — that the nucleon is a torus knot T(p,q), motivated by π₃(SO(3)) = ℤ but not yet derived from the equation of motion — Theorem T10 establishes that T(2,3) is the unique admissible torus knot satisfying proper closure (gcd = 1), three-colorability (Z₃), and minimal writhe. Five exact invariants are derived under H4a: writhe = 2, φ_in = exp(−4π) ≈ 3.49 × 10⁻⁶, φ_m·(p+q) = 1, V_trefoil(exp(2iπ/3)) = 1, and the quarter-wave identity λ_res/r_n = π/2. By analogy with the instanton tunneling amplitude in Yang-Mills theory, φ_in = exp(−S_inst) where S_inst = π(p+q−1) is the Chern-Weil instanton action of the T(p,q) bundle (Proposition CW, conditional on H4a, g²=4π, BPS saturation, and WKB identification). The BPS parameter β = (mφ/m_W)² = 200/π ≈ 63.7, formerly a free numerical input, is now exactly constrained by three independent results. The resonance conjecture T9 (φ_m = 1/(p+q)) is supported by a BPS scaling argument and a numerical computation giving ω₀ = 954 MeV at 9% above target mφ = 877 MeV; a complete derivation requires the coupled scalar-gauge system with explicit f(φ), which defines Program FOX V4. A hierarchy of exotic solitons T(p,q) is predicted: T(3,4) at 1315 MeV, T(2,9) and T(5,6) at 2192 MeV. All results in Sections IV–VI are conditional on H4a. Explicit limits and open questions are listed in Table V. This paper constitutes Branch B of the FOX program, a structured framework under construction.