Large-Scale Structure and Thermodynamics in Fractal-Spectral Spacetime: Cosmic Web Correlations, Phase Transitions, and Log-Periodic Signatures

What does fractal-spectral spacetime predict for the largest structures in the universe and for the thermodynamics of matter within them? For the cosmic web, the framework predicts a galaxy two-point correlation function slope γ = 3 − √2 ≈ 1. 586, derived from the fractal structure of spacetime — consistent with observed values (γₒbs ≈ 1. 6 ± 0. 05) without parameter fitting. The distinctive prediction beyond ΛCDM is a log-periodic modulation of ξ (r) with the universal period Δln r ≈ 18. 1, detectable with next-generation galaxy surveys. The matter power spectrum acquires √2-periodic corrections localized around fractal characteristic scales, additive to ΛCDM (preserving BAO features and the standard shape). The void size distribution follows a power-law index √2 − 4 ≈ −2. 59, consistent with observed values (−2. 5 to −2. 7). For thermodynamics, phase transitions are reinterpreted as frequency synchronization cascades in the collective vibrational modes of matter. The paper honestly reports a tension: naive fractal critical exponents (ν ≈ 0. 707, γ ≈ 1. 500) disagree with precision 3D Ising measurements (ν = 0. 6300, γ = 1. 2372) by 12–21%, with a sign change in α. The resolution is that fractal corrections must be perturbative — standard universality classes are preserved with small log-periodic modulations around the standard critical behavior. The testable prediction is not modified exponents but log-periodic corrections to scaling near Tc, with the same universal period ln√2. Fast radio burst dispersion measures receive a fractal correction δDM/DM ~ 10⁻⁵ from temporal gradient structure along the sightline — consistent with recent baryon detection results (Connor et al. 2025) but undetectable at current precision. Falsification criteria are explicit: if Euclid/LSST find no log-periodic modulation at amplitude A₁ > 0. 005 in the correlation function, the large-scale structure predictions are ruled out. If precision critical exponent measurements show no log-periodic modulations at the 10⁻⁴ level, fractal corrections to thermodynamics are negligible.