The pole-free Weil quadratic form: a simple even ground state

Fix (λ>1) and let (QWλ) be the Weil quadratic form on (L²([λ⁻¹,λ],d*u)) in the normalization of Connes–Consani–Moscovici. We study the pole-free form (QWλ∘) obtained by deleting the rank-two contribution (W0,2) associated with the poles at (0) and (1). We prove that the lower-bounded self-adjoint operator (Aλ∘) associated with (QWλ∘) has compact resolvent and a simple lowest eigenvalue, whose normalized eigenfunction is real, strictly positive almost everywhere, and invariant under (u↦ u⁻¹). The proof is organized around a Lévy–Khintchine representation of the archimedean term, irreducibility of the corresponding Dirichlet form, preservation of positivity under the bounded prime-power perturbation, and the Perron–Frobenius–Faris theorem. The distinction between the pole-free operator and the full Weil operator is maintained throughout.