Log-Periodic √2 Signatures in the CMB: Predictions, Search Strategy, and Resolution of Planck Anomalies in the Fractal-Spectral Framework

The fractal-spectral framework makes a specific, parameter-free prediction for the CMB: log-periodic modulations of the temperature power spectrum with period ln√2 ≈ 0. 347 in log-multipole space. No other cosmological model predicts this particular period — it is determined by the √2 scaling ratio of the fractal-temporal Lagrangian, not fitted to data. The modulation arises from the (√2) ⁿ texture tower in the primordial power spectrum, transferred to the angular power spectrum through the standard radiation transfer function. The predicted signal is strongest at low multipoles (ℓ ≲ 100), where Silk damping has not yet washed out the fractal structure, and has an amplitude hierarchy where the m-th harmonic is suppressed by (√2) ^−m. The paper establishes natural connections — presented as suggestive, not claimed as explanations without full statistical analysis — between the predicted modulation and three established Planck anomalies. The low-ℓ power deficit (~5–10% less power than ΛCDM at ℓ ≲ 30) can be partially produced by destructive interference of the log-periodic modulation at appropriate phase. The hemispherical power asymmetry (A ≈ 0. 07 at ~3σ) receives a contribution from the coupling between the modulation and a spatial τ-gradient. The Cold Spot is addressed through local texture field enhancement. A detection methodology is proposed based on MCMC fitting of a modulated ΛCDM spectrum to Planck data, with explicit detection and rejection criteria. The amplitude is constrained to ≲5% by existing Planck data (larger modulations would already have been detected) but could be arbitrarily small — the amplitude depends on the undetermined coefficient β₁, whose first-principles derivation remains an open problem. Forecasts for upcoming missions indicate that the signal should be detectable at >5σ if present at the predicted amplitude: Simons Observatory (2025+) provides ~3× sensitivity improvement, LiteBIRD (2028+) provides full-sky polarization, and CMB-S4 (2030+) provides an order-of-magnitude noise reduction. The CMB provides the cleanest and most decisive test of the entire fractal-spectral framework.