Equal-Ratio Moran Complex Dimensions

Abstract

The equal-ratio Moran dimension extends to a log-periodic ladder of complex solutions.

Theorem 1.1 (The Moran equation holds on the complex dimension ladder).

$$D=\frac{\log M}{k\log \phi },s_n=D+\frac{2\pi in}{k\log \phi },n\in \mathbb{Z},M\cdot {\phi }^{-k{s}_n}=1.$$

Proof. Machine-checked in Lean as D5/S3/Constants/MoranComplexDimensions.moran_complex_dimension (✓ std3). ∎

Source. Repository-derived.

Commentary.

For positive natural M and k and real phi greater than one, the real Moran dimension is log M divided by k log phi. Adding each integer multiple of 2 pi i divided by k log phi leaves the complexified equal-ratio Moran equation unchanged, producing its log-periodic tower of solutions.

References

• Truth anchor: D5/S3/Constants/MoranComplexDimensions.moran_complex_dimension