The Stark-Effect Realization of the Ω2 Kernel: Rotational Susceptibility, Multiplet Cancellation, and Discrete Scale Assembly

The rotational Stark effect of a linear polar molecule gives an exact physical realization of the $\Omega_2$ kernel introduced previously. For the rigid-rotor Hamiltonian $H(E) = B J^2 - \mu E \cos\theta$, with $B = \hbar^2 / (2I)$, spectral perturbation theory and the Kubo–Duhamel formula yield the per-molecule zero-field susceptibility: $$\frac{\chi_{\text{eq}}(T)}{\chi_{\text{gs}}} = 1 - F_2(u) = 1 - \frac{1}{12}\left(1 - \frac{1}{Z_2(u)}\right)$$ where $u = \frac{B}{\pi k_B T}$, $\chi_{\text{gs}} = \frac{\mu^2}{3B}$, and $Z_2(u) = \sum_{J \ge 0} (2J + 1) e^{-\pi u J(J + 1)}$. The excited-multiplet quadratic Stark traces cancel, so the normalization fixed by the foundational construction identifies $12 F_2$ both with the fractional thermal loss of ground-state susceptibility and with the total excited-state probability. Two preparations yield this response: thermal equilibrium in the applied field, and adiabatic entry that preserves initially thermal rotational populations. We prove their zero-field coincidence and express the leading cubic separation of their polarizations as a positive variance of quadratic Stark coefficients. We also show that a fixed induced-polarizability background cancels from the calibrated susceptibility difference. The single-rotor response determines the holomorphic Mellin family: $$\Omega_2(s) = \int_{0}^{\infty} u^{s-1} F_2(u) \, du$$ for $\Re(s) > 0$, including $\Omega_2 = 0.00160695491795976567\dots$ at $s = 3$. A geometric assembly with rotational and dipole-energy scales proportional to $p^{-n}$ and $p^{-2n}$ selects this moment and gives $c_{0}^{(2)}(p) \log p = \Omega_2$ for every $p > 1$. Detailed proofs, spectral and assembly truncation bounds, and a finite-temperature-window reconstruction estimate complete the molecular response dictionary. The result extends the physical realization of the Omega construction from circle-based systems to the orientational geometry of a spatial rotor.