Z_c by number of ramified primes and phase class
all-even row parity → Z_c = 2^{r−1}
all-odd → 0
mixed → 0
Stage 4 — mod-2 Dijkgraaf–Witten Z_c(X̄) for K=ℚ(√(p₁···p_r)), pᵢ≡1 mod 4 (Hirano route). LHS (brute character sum) vs RHS (closed form). Renders emitted JSON only.