Quartic Gauss sums over primes with a fixed numerator

David, Dunn, Hamieh and Lin proved that the normalized quartic Gauss sums g̃₄(c) over primary primes c ≡ β (mod 4) of Z[i], twisted by (c/|c|)^ℓ and weighted by the von Mangoldt function, have smoothed sums O(X5/6+ε) (arXiv:2306.11875, Theorem 1.1). We show that their argument gives the bound X5/6+ε(1+|ℓ|)3/2+ε for g̃₄(ν, c) with any fixed nonzero numerator ν ∈ λ⁻²Z[i]. The only new ingredient is a local identity at the primes π dividing ν. It splits off powers of π with no condition linking ν and π, and so removes these primes from the coprimality conditions in the Type I Dirichlet series: When ord_π ν ≡ 1 or 3 (mod 4), the removal is a single equation whose determinant has no zeros for Re s > 3/4. When ord_π ν ≡ 0 or 2, it reduces to DDHL’s identities (8.6) and (8.8). After that, DDHL’s Proposition 4.3 and Suzuki’s relation (4.39) apply as stated, Type II goes through verbatim, and DDHL §9 is not needed. As a corollary the sharp sums are o(X) for every fixed numerator. This proves the hypothesis on fixed numerators used in Paper VIII of the author’s series, the input for equidistribution modulo 2π of the argument of the quartic Bernoulli number. The local identity is checked numerically from the definition in 927 cases, plus 1352 cases where the modulus shares primes with the numerator. The local identity, its four cases and the removal of the smoothing are formalized in Lean 4 (doi:10.5281/zenodo.23882652).