Nonperturbative Glueball Spectrum in History-Dependent Gravity: A No-Go Theorem for Scalar Ladder Approximations (Paper XXXIV)

This is the 34th paper in the History-Dependent Gravity (HDG) series, part of the "Temporal Nonlocality and Fundamental Physics" community. We present a nonperturbative calculation of the Yang–Mills glueball spectrum within the History-Dependent Gravity (HDG) framework using a scalar ladder Bethe–Salpeter kernel. By projecting the universal HDG memory kernel onto specific JPCJPC channels via gauge-invariant projectors, we treat all channels on equal footing without phenomenological bias (CJ=1CJ=1). We demonstrate that this scalar approximation fails to reproduce the Yang–Mills glueball spectrum even at the qualitative level. While bound states appear in all channels, the resulting mass hierarchy is severely distorted: tensor states are underbound, pseudoscalar states are systematically mis-positioned, and a clear spectral inversion occurs (e.g., M2−+<M2++M2−+<M2++), in direct contradiction with lattice QCD. We prove that this failure is structural and cannot be cured by modifying the radial dependence of the kernel. A scalar interaction kernel enforces a universal, incorrect binding pattern due to the loss of angular-momentum-dependent interference effects. Our results establish a generalized no-go theorem for scalar interaction kernels in confining gauge theories, demonstrating that the glueball spectrum is a direct probe of the spin–spin and spin–orbit components of the non-Abelian interaction. The restoration of the correct spectrum requires the explicit inclusion of the full tensorial and topological (ϵμνρσϵμνρσ) vertex structures, which will be constructed in the companion paper (Paper XXXV).