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We prove that the IPB98(y,2) empirical scaling law for energy confinement time in tokamak plasmas is determined by the Gauss sum structure of the multiplicative group (Z/30Z)*, where 30 = 2 x 3 x 5 is the primorial of the first three primes. The symmetric distance matrix on the 8 coprime residues of Z/30Z admits an exact palindromic block decomposition into 4x4 even and odd sectors. The Legendre character chi_5 = (./5) is purely even because 5 = 1 mod 4, while chi_3 = (./3) is purely odd because 3 = 3 mod 4. This separation follows from Euler's criterion applied to the primorial factorization. The Gauss sum g(chi_5) satisfies |g(chi_5)| = sqrt(5) by the standard algebraic theorem |g(chi_p)|^2 = p. The IPB98 prefactor is C = phi(30)/|g(chi_5)| = 8/sqrt(5) = 3.57771, matching the experimental fit C_fit = 3.5779 to 0.005% -- a factor of 173x closer than the previously assumed sqrt(4*pi) = 3.5449. All four scaling exponents are determined by the primorial structure: denominators {phi(m), p_max, phi(m)-1, phi(m)/2} = {8, 5, 7, 4} and numerators {-p_max, +1, -p_min, -p_max} = {-5, +1, -2, -5}. We prove via discriminant analysis that sqrt(5) is absent from the splitting field of both characteristic polynomials. The irrationality sqrt(5) enters the scaling law exclusively through the Gauss sum normalization, not through eigenvalue algebra. We compute exact integer determinants for the palindromic blocks at m = 30 (4x4), m = 210 (24x24), and m = 2310 (240x240, 155-digit determinants). At all levels, the p_max-exclusion principle holds: the largest prime is absent from both block determinant factorizations. We discover a polynomial hierarchy: for any prime set S, R(S u {q}) is exactly quadratic in q with universal constant term -1 and leading coefficient 1 - 2R(S). The single-prime formula R(p) = -((p-1)^2 - 2)/2 is verified at 14 primes. The a-recurrence a_{k+1} = a_k * p^2 + b_k * p is verified at all 4 transitions. At the fifth primorial m = 30030 (2880x2880 blocks), the hierarchy breaks: R(30030) = -48317287/3, a non-integer. The numerator 48317287 is prime; the denominator is p_min = 3. The p_max-exclusion principle continues to hold (13 does not divide either block determinant). The 3-adic valuation sequence v_3(R) = {0, 3, 4, 2, -1} goes negative at m = 30030, marking a definitive phase boundary. The exact 3-adic fracture mechanism is identified via p-adic Gaussian elimination (mod 3^k, converged at k=3): both blocks share GF(3) nullity 943, but mod-9 elimination reveals 4 pivots of valuation 2 in the odd block versus 3 in the even block. This single unpaired deep pivot produces v_3(det_odd) = 947 and v_3(det_even) = 946, the first polarity inversion in the hierarchy, yielding v_3(R) = -1. The value 947 is prime; 946 = 2 x 11 x 43. Every prime dividing m = 30030 except p = 3 has valuation zero in R(30030) -- p-adic purity. All results are verified computationally: 88 assertions across 5 primorials, 0 failures, runtime approximately 3 minutes on a consumer PC (hierarchy_termination_proof.py). R(30030) is verified against 13 independent prime moduli. Companion papers: Spectral Isotropy and the Exact Temperature of the Prime Gas, DOI: 10.5281/zenodo.19156532; The Prime Column Transition Matrix Is a Boltzmann Distribution at Temperature ln(N), DOI: 10.5281/zenodo.19076680; Separable Gyro-Bohm Scaling for Fusion Energy Confinement: Resolving the Isotope Sign Conflict, DOI: 10.5281/zenodo.19117880.