Curvature-Level Crossings of Riemann Zeta Argand Curves: Smooth Random Jets, Joint Center–Level Averaging, and a Jensen Squeeze

This paper studies the asymptotic frequency with which the signed curvature of the Riemann zeta Argand curve gamma_sigma(t) = zeta(sigma + i t), with 1/2 < sigma < 1, crosses a prescribed level k. The apparent singularity caused by inverse powers of the absolute value of zeta prime is removed by introducing a globally twice continuously differentiable, denominator-free crossing function. Its simple zeros correspond exactly to transverse curvature-level crossings, while points at which zeta prime vanishes are automatically excluded. The paper proves that every prescribed curvature level is crossed with positive linear lower density. For the random Euler-product model, a prime-block Fourier argument establishes a smooth Schwartz density for the full logarithmic jet. This validates a finite, strictly positive Kac-Rice intensity at every level and reveals an exact cancellation near small values of the absolute value of zeta prime. On the deterministic side, a symmetric holomorphic complexification couples the levels k and -k and reduces the fixed-center problem to a logarithmic no-escape condition. A far-right Cartan anchor then gives stretched-exponential small-value bounds after averaging over the center sigma. These estimates yield unconditional raw, unregularized convergence of N_(sigma,k)(T) / T to the corresponding random Euler-product intensity, locally in L1 with respect to sigma and jointly locally in L1 with respect to sigma and k. Further results include maximal convergence over multiplicative height windows, an exact almost-everywhere lower envelope, recovery at every prescribed center through shrinking center averages, and sub-Weibull logarithmic Orlicz bounds. Two countermodels identify the remaining obstruction. Ambient two-variable Ronkin control does not control a resonant symmetry trace, and even a uniform far-right anchor with exponential center-averaged tails does not imply logarithmic control on a named center fiber. Accordingly, the raw limit at an individually prescribed value of sigma remains open. The paper isolates a moving-normal-divisor logarithmic-moment conjecture whose proof would close this fixed-center problem. For nonzero k, these crossing statistics also describe the generic cusp intensity of the normal offsets of the zeta Argand curve having radius 1 / |k|.