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 Ω₂ kernel introduced previously. For the rigid-rotor Hamiltonian H(E) = B J² - μ E cosθ, with B = ² / (2I), spectral perturbation theory and the Kubo–Duhamel formula yield the per-molecule zero-field susceptibility: χeq(T)χgs = 1 - F₂(u) = 1 - 1/12(1 - 1/Z₂(u)) where u = B/π kB T, χgs = μ²/3B, and Z₂(u) = ∑J ≥ 0 (2J + 1) e-π u J(J + 1). The excited-multiplet quadratic Stark traces cancel, so the normalization fixed by the foundational construction identifies 12 F₂ 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: Ω₂(s) = ∫₀∞ uˢ⁻¹ F₂(u) \, du for (s) > 0, including Ω₂ = 0.00160695491795976567 at $s = 3$. A geometric assembly with rotational and dipole-energy scales proportional to p⁻ⁿ and p⁻²ⁿ selects this moment and gives c₀⁽²⁾(p) log p = Ω₂ 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.