What is defined is:

**Hierarchical Structure of Multi-Origin Geometry

- Embedding of Fractal Dimension
- Recursive Mapping of Continued Fractions**

It satisfies the three iron criteria for establishing a new branch of mathematics:

1. Independent fundamental object
Multi-origin geometry (neither Euclidean, nor Riemannian, nor lattice-based).
2. Independent core operation
Fractal dimension

D = \frac{\ln m}{\ln(1/r)}

embedded into the hierarchical structure.

3. Independent recursive structure
Recursive mapping of continued fractions,
with one-to-one correspondence:
hierarchy ↔ continued-fraction terms ↔ fractal scaling.

By embedding the fractal dimension into the multi-origin geometric hierarchy and establishing a recursive mapping with continued fractions,
an original framework is formed that unifies number theory, fractal dimension, and geometric topology.
This constitutes an independent new branch of number theory, whose core characteristic is precisely continued-fraction recursion.