We derive all seven nuclear magic numbers {2, 8, 20, 28, 50, 82, 126} from closed-form expressions for the two shell-model parameters conventionally fit to data, with zero adjusted parameters and zero false positives. The central potential and spin-orbit coefficients share a common architecture D = numerator/(2π)^d, where d counts the independent angular dimensions of the operator (d=2 for L², d=1 for L·S). The derived values are D_ll = cos(π/5)/(2π)² = φ/(8π²) = 0.02049 and D_ls = −ln(2)/(2π) = −0.1103, where φ = (1+√5)/2 is the golden ratio. Both numerators are derived within Infoton Theory: cos(π/5) = φ/2 arises from pentagonal screening geometry (Theorem 113), and ln(2) from the phase frequency of the first prime p₁ = 2 (Axiom 5). The denominators (2π)^d are the Fourier normalization on d angular dimensions of the phase torus. Uniqueness is proven: the pair (d=2, n=5) is the unique choice yielding all seven magic numbers. The uniqueness of n=5 follows from a cyclotomic constraint: for no other integer n≥3 does cos(π/n) belong to ℚ(φ), as the minimal polynomial degree φ(2n)/2 must divide [ℚ(φ):ℚ] = 2, which only n=5 satisfies (where φ(10)/2 = 2). The uniqueness of d=2 is verified by exhaustive scan. The result is empirically discriminating: all published empirical parametrizations (Nilsson 1955 light and heavy, Ring-Schuck 2004, Bengtsson-Ragnarsson) fall outside the 7/7 acceptance window, achieving only 5/7 or 6/7. Monte Carlo analysis drawing 10⁵ pairs uniformly from the minimal bounding box containing all published values yields 7/7 in only 0.07% of cases (approximately 1 in 1,400). Sign inversion of D_ls yields only 3/7 magic numbers. The number K=7 of distinct gap levels is not an external input: it is determined by the first algebraic degeneracy in the spectrum, where the transitions 2p₁/₂ → 1g₉/₂ and 3p₁/₂ → 2g₉/₂ share the symbolic expression 1 − 18D_ll + 3D_ls. The seven highest distinct energy gap levels match the seven experimental magic numbers exactly. Every element is traced to axioms or proven theorems: the factor 2 (Pauli exclusion) from phase exclusion on p₁=2, the degeneracies 2j+1 from 3D projection, cos(π/5)=φ/2 from Axiom 3, the pentagon (n=5) from the five-level cascade (Theorem 112), the (2π)^d normalization from the phase torus, the degree bound d≤2 from Theorem 64, ln(2) from p₁ phase, and the negative sign of D_ls from confinement-to-attraction (Theorem 136). Verification script included.