Renormalization-Group Fixed Point and Spectral Compression in History-Dependent Glueball Dynamics

We study the renormalization-group origin of spectral compression in the History-Dependent Gravity (HDG) framework for non-Abelian bound-state dynamics. A scale-dependent Bethe–Salpeter kernel is parametrized as Kk=K0+cmix(k)KHDGKk=K0+cmix(k)KHDG, where KHDGKHDG represents a memory-induced momentum-gradient operator. Using a minimal functional renormalization-group (FRG) projection of the Wetterich equation, we derive an effective flow equation ∂tcmix=θcmix−Acmix2∂tcmix=θcmix−Acmix2. The coefficients are evaluated using the nonperturbative infrared propagator scaling Dk(p)∼p−4Dk(p)∼p−4, previously obtained from the infrared fixed-point analysis of the HDG gluon sector. The projected FRG flow exhibits a non-trivial infrared attractive fixed point at cmix∗=θ/A≃3.01cmix∗=θ/A≃3.01 within the two-operator truncation. This value coincides with the saturation region of the spectral compression observed in the numerical Bethe–Salpeter analysis of our companion work (Paper XXXVI). Furthermore, we show that the fixed-point HDG deformation suppresses the infrared singular structure of the kernel and drives it toward a compact operator class along the regulated trajectory, producing a discrete bound-state spectrum. The remaining deviation of scalar glueball ratios from lattice values is interpreted as a limitation of the single-channel truncation and motivates the inclusion of coupled tensor RG flows.