This paper introduces a novel framework replacing traditional string theory with Möbius strip wavelets. This approach resolves the dimension explosion problem while naturally encoding spinors.
The Möbius stripwavelet is a fractal, self-similar structure that encodes the fundamental geometry of the universe:
- It forms an infinity-like structure, creating black holes and white holes depending on position and direction within the loop.
- The wave propagating along the strip follows a conformal cyclic cosmology process.
Each complete loop of the strip represents an aeon, and each side switch acts as a conformal transformation:
\[ \Phi (t) = \lim_{n \to \infty} \Psi (t + nT) \]
where \( \Phi (t) \) represents the evolution of the wave along the stripwavelet, conformally mapping cycles.
Geodesics in this framework are described as orthogonal spiraling fibrations on a hypersphere, mapped to two dimensions.
These geodesics converge at strip inversions, leading to big bang-like events in subsequent cycles.
\[ G_n = \sum_{i=1}^{n} \Gamma (P_i) \times \Omega (t) \]
To derive the wave function along the Möbius strip, we define the fundamental mode of the wavelet as:
\[ \psi (x,t) = A e^{i(kx - \omega t)} \]
where \( A \) is the amplitude, \( k \) is the wave number, and \( \omega \) is the angular frequency.
Given the Möbius topology, we enforce the periodic boundary conditions:
\[ \psi (x + L, t) = \psi (-x, t) \]
which modifies the eigenstates accordingly.
Each cycle through the Möbius strip enacts a conformal transformation on the wave function. This can be modeled as:
\[ \Psi_{n+1} = T \Psi_n \]
where \( T \) is a transformation matrix satisfying conformal symmetry conditions.
The geodesics follow orthogonal fibrations described in higher-dimensional space:
\[ x^2 + y^2 + z^2 + w^2 = R^2 \]
where the collapse to a singularity occurs at the inversion points:
\[ \lim_{t \to 0} \nabla \cdot G = \delta (t) \]
This theory provides an elegant, self-consistent framework that unifies fundamental physics under a cyclic, fractal Möbius strip-based geometry.