Trace-Class Criteria for Degenerate Quartic Weyl Operators: Zero-Point Stabilization and the Admissible Set of Quantum Quartic Exponential Families

Let h be a real polynomial of degree at most four on the phase space of n bosonic modes and let H = Op^W(h) be its Weyl quantization. We study the admissible set Θ̄ = {θ : Tr e−H_θ < ∞} of the quantum exponential family whose sufficient statistics are the phase-space monomials of degree ≤ 4. The quartic part Q₄ of h must be positive semidefinite, but when Q₄ is degenerate the classical Weyl symbol can be unbounded below while e−H is still trace class (zero-point stabilization), and conversely a position-space Born–Oppenheimer heuristic can predict admissibility where it fails. We prove an operator-valued Berezin–Lieb sandwich in which the corridor variables are treated semiclassically and the transverse variables remain quantum, a gauge-transformed sliced Golden–Thompson bound, a localized Jensen inequality, and necessity bounds without gauge transformations. For fibers of nilpotent type we show that admissibility, polynomial growth of the eigenvalue counting function, and the Gibbs dichotomy are equivalent, and that the criterion is semialgebraic. We give an explicit counterexample to the naive Born–Oppenheimer conjecture whose admissible set is exactly |κ| ≤ 2 (while the classical admissible set is empty). In two-cluster examples where two zero-point energies act simultaneously we determine the sharp threshold √2 and, in an example without U(1) symmetry, prove admissibility below 2 and nonadmissibility from √6 on, the numerics suggesting the threshold √6. The remaining open problems are reduced to the spectral theory of bilinear couplings of two metaplectic representations of sl(2, ℝ). Files: paper2_jst.pdf (manuscript, 20 pages); verify_paper2.py (Python 3 script that reproduces the numerical values reported in the paper; requires numpy, scipy and sympy; about 2 minutes on a standard machine; comments are in Japanese).