🎙️Episode 65-SG: Noether’s Theorem and Fractal Symmetry. Energy is Conserved!

FB(S³)R: A Generalized Noethers Theorem for Multifractal Cosmology
https://zenodo.org/records/15710167

A Generalized Noether’s Theorem for Multifractal Cosmology” presents a substantially expanded theoretical framework by generalizing Noether's theorem on a time-dependent multifractal S3S^3S3-manifold. Building upon the earlier formulation [1.0], a covariant variational principle is established, adapted to fractal geometry governed by the golden ratio 1+52\frac{1 + \sqrt{5}}{2}21+5, and incorporating a dynamically evolving embedding level N(t)N(t)N(t), which defines a multifractal action.

Formal derivation of Euler–Lagrange equations under multifractal scaling;
Introduction of a time-dependent embedding function
N(t)=⌊log⁡φ(ctlP)⌋N(t) = \left\lfloor \log_\varphi \left( \frac{ct}{l_P} \right) \right\rfloorN(t)=⌊logφ(lPct)⌋ and analysis of its cosmological implications;

Identification of two limiting cases — N→∞N \to \inftyN→∞ and φ→e\varphi \to eφ→e — corresponding to logarithmic and exponential cosmological phase transitions, respectively;
Definition of a fractal symmetry transformation leading to a generalized conserved energy current ∇μJnμ=0\nabla_\mu J^\mu_n = 0∇μJnμ=0, allowing for conservation laws without relying on classical symmetry groups;
Demonstration of consistency with observational data, including cosmic microwave background (CMB) anisotropies, gravitational wave spectra, and dark matter structure distributions.

This version addresses fundamental challenges in quantum cosmology, particularly the breakdown of global symmetries in curved spacetime and the limitations of classical Noetherian conservation. By extending the theorem in a multifractal context, it lays the foundation for a unified theoretical approach connecting quantum field theory, topology, and cosmological observation. Namaste

Orpheus Praxis AI (myGPT)
https://chatgpt.com/g/g-67b1e32b02508...