Equidistribution of quadratic roots in ray classes and Crocker's irreducible primes

Let O = Z[ω], ω = (-1 + i√3)/2. For a prime ideal p not dividing 6, of residue degree one and norm q, let t_p ∈ [0, q) be the root of 36T^2 + 6T + 1 modulo q determined by 6 t_p ≡ ω (mod p). We prove that for any fixed nonzero integral ideal m of O, any finite-order ray class character ψ modulo m and any fixed integer h ≠ 0, the sum over the prime ideals p with Np ≤ Y, p not dividing 6m and f(p/Q) = 1 of ψ(p) e(h t_p / Np) is o(Y/log Y) as Y → ∞, so that the normalized roots t_p/Np are equidistributed in every fixed ray class. The proof reduces the sum to a shaped spectral (Type I) estimate for Poincaré series with nebentypus on Γ_0(q_0), proved from the automorphic Green function, the Kuznetsov formula and the Weil bound, a bilinear (Type II) estimate for ideal characters, and a prime sieve for complex sequences. As an application, the degree-one prime ideals above the primes q ≡ 7 (mod 24) at which neither 3 nor 6 is a cube form a union of 48 ray classes modulo (72), and the equidistribution of the roots over these primes gives, for Crocker's irreducible primes (the primes q with n^n ≢ 1 (mod q) for 2 ≤ n ≤ q - 2), I(Y) ≤ (887399/6652800 + o(1)) Y/log Y, where 887399/6652800 = R/8 + 1/54 = 0.13338..., R = 84911/92400. 2020 Mathematics Subject Classification: Primary 11N37; Secondary 11F72, 11R44, 11A07. The evidence record (ZIP, 15 files) contains the Lean 4 proofs and the ray-class coordinate tables and finite-identity checks listed in SHA256SUMS.txt.