Karlsson's Redshift Periodicity as an Efimov Spectrum: A Zero-Parameter Prediction from Vacuum Mode Structure in Isothermal Halos

For over fifty years, the Karlsson periodicity—a log-periodic spacing of ∆ log10 (1 + z) = 0.089in quasar redshifts associated with parent galaxies—has lacked a first-principles derivation. Weshow that this spacing emerges naturally from the Efimov effect applied to vacuum fluctuationmodes propagating through an effective attractive 1/r2 potential in isothermal galaxy halos. Theisothermal density profile ρ ∝ 1/r2 , observed across galaxy types, creates this potential for vacuummodes. The Efimov theorem [1] guarantees thatp such a potential produces a log-periodic spectrumof bound-state energies with ratio α = exp(π/ g − 1/4), where g is the effective coupling. We arguethat g = 6 × (2π)2 = 24π 2 , counting 6 independent spatial metric components (via a gravitationalAharonov-Bohm argument) each contributing a geometric factor (2π)2 from horizon mode-counting.This gives log10 (α) = 0.0887, matching the observed 0.089±0.005 at 0.06σ with zero free parameters.Monte Carlo testing against 8 consensus peak positions yields p < 0.00002. We present the derivationchain, clearly labeling each step as observed, proven, standard, hypothesis, or motivated, and discussthe open problem of rigorously deriving the (2π)2 factor.