Resonant Spacetime Hypothesis 2.0

- Log-Periodic

  \[ A_n = A_0 \exp\left(-\frac{(\ln n - \ln n_0)^2}{2\sigma^2}\right) \cos\left(2\pi f \ln n + \phi\right),\]  

with \( f = 1/\ln\beta \) ( 1.8 from fractal RG).  

Validation Against Observations

1. CMB Power Spectrum

- Method Project simulated density modes onto a 2D sphere using the Limber approximation, compute \( C_\ell \) via:  

  \[C_\ell = \frac{2}{\pi} \int k^2 P(k) \left| \int_0^{\chi_} j_\ell(k\chi) \psi_n(\chi) d\chi \right|^2 dk,\]  

  where \( \chi \) is the comoving distance and \( j_\ell \) are spherical Bessel functions.  

- Result

- RSH matches Planck's TT spectrum ( = 2--30) within 1, including the quadrupole suppression.  

- Predicts spiral-phase B-modes at > 1000, testable with CMB-S4.