Evidence for Log-Periodic Modulation in High-Redshift Compact Object Abundance Consistent with Cyclic Condensate Collapse

We analyze the redshift distribution of high-$z$ galaxies and active galactic nuclei identified in early JWST data, and investigate the presence of periodic structure in the variable $x=\ln(1+z)$. A baseline-corrected unbinned frequency analysis reveals a statistically significant peak corresponding to a spacing $\Delta x \simeq 0.34$, suggesting an approximately log-periodic pattern in the redshift distribution. A periodic structure in $x$ implies a preferred scaling ratio in $(1+z)$, which may be interpreted as a realization of discrete scale invariance. We discuss the possibility that such behavior arises from cyclic condensate dynamics in Bose--Einstein condensate (BEC) cosmology. In the Fukuyama--Morikawa--Tatekawa framework, repeated collapse and re-formation episodes of a self-interacting condensate occur over characteristic timescales of order several $10^8$ years. When mapped into redshift space, this temporal periodicity naturally translates into an approximately constant spacing in $\ln(1+z)$. While the observed frequency is not interpreted as a sharp theoretical prediction, its magnitude is quantitatively consistent with the intrinsic cycle timescale of QCD-axion motivated condensate dynamics. The present analysis therefore provides observational support for cyclic BEC cosmology as a viable dynamical origin of log-periodic structure in the high-redshift universe.