The Planck Core: A Log-Periodic Resolution of Gravitational Singularities

This work presents a complete analytical and numerical resolution of gravitational singularities within the Log–Sinus–Polylog (LSP) conformal framework.
It demonstrates that the Ricci curvature in General Relativity remains finite once a small log-periodic modulation is introduced into the metric’s conformal factor.
The resulting structure — the Planck Core — acts as a self-regulated attractor, replacing classical singularities with bounded oscillatory equilibria.

The article provides:

• A full derivation of the Boundedness Theorem, showing that
  ∣RE(a)∣≤R0exp⁡[6cω/(m+2q)]|R_E(a)| \le R_0 \exp[6c\omega/(m+2q)]∣RE(a)∣≤R0exp[6cω/(m+2q)].

• A physical interpretation of the destructive interference condition cω≈1/ ⁣6c\omega \approx 1/\!\sqrt{6}cω≈1/6.

• Numerical simulations confirming transition from divergence to a stable log-periodic oscillation RE(t)R_E(t)RE(t).

• A concise discussion on the conformal and quantum feedback mechanisms leading to curvature stabilization.

Figures 1–2 visualize the dynamical evolution of curvature and the phase-space structure of the Planck Core.
All results are derived within classical GR plus conformal modulation — no external quantum gravity assumptions are required.

Keywords: singularity resolution, log-periodic cosmology, bounded curvature, Planck core, emergent gravity