Class-Field Generators from Quadratic-Phase Lemniscatic Sums

We study a normalized series with coefficients chi_4(m) m sech(pi m/2) and quadratic phases exp(-2 pi i x m^2). It converges exponentially. For every odd conductor q, its complete numerator family generates the ring class field H_q of the order Z + q Z[i]. For every odd prime p, each value h(a/p) with p not dividing a individually generates H_p. The family theorem follows from finite Fourier analysis and elliptic zero-pole counts; the inert-prime theorem uses a uniformizer and a congruence at two. For split primes at least thirteen, an integral theta translation on a degree-two quotient determines three torsion moments from one nonzero invariant. The resulting local quadratic polynomials at the conjugate primes have resultant 32, so at least one yields a uniformizer. The prime five is treated separately. We also obtain denominator and norm laws, real Eisenstein generators at inert primes, explicit polynomial families, and finite composite-conductor examples. Individual generation for arbitrary composite conductors is not asserted.