FCFG-A: An Arithmetic Extension of Fractal-Continued-Fraction Geometry

Author: Zhang Suhang (Luoyang, Henan)

Abstract

Within the framework of Fractal-Continued-Fraction Geometry (FCFG), this paper elevates finite continued fractions from "numerical objects" to "arithmetic recursive systems," and establishes a strict isomorphism between them and geometric recursive systems. We define arithmetic recursive systems, recursive operator chains, and structural isomorphism, propose the FCFG-A isomorphism axiom, and prove that arithmetic recursion and geometric recursion are structurally isomorphic in the finite-order case. We further discuss the positions of the Euclidean algorithm, Diophantine approximation, quadratic irrationals, Ramanujan continued fractions, and the continued-fraction expansions of the constants e and \pi within this framework.

Keywords: Fractal-Continued-Fraction Geometry; arithmetic recursion; continued fractions; recursive operators; structural isomorphism

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1 Introduction

Fractal-Continued-Fraction Geometry (FCFG) establishes an isomorphism between finite continued fractions and finite self-similar fractals:

r_n = S_n

where r_n is the n-th order finite continued fraction, and S_n is the global similarity ratio of the n-th order finite self-similar fractal. This isomorphism shows that arithmetic recursion and geometric recursion correspond to the same structure in the finite-order case.

However, the existing FCFG mainly treats r_n as a numerical value, without explicitly unfolding its arithmetic recursive structure. The goal of this paper is: to elevate continued fractions from "numerical values" to "arithmetic recursive systems," and to establish a strict isomorphism between them and geometric recursive systems. This work is called FCFG-A, i.e., the arithmetic extension of FCFG.

This paper does not claim to establish a new discipline independent of FCFG, but rather to make the arithmetic side explicit, so that the isomorphism axiom of FCFG obtains an arithmetic foundation.

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2 Preliminaries: Core Review of FCFG

2.1 Finite Continued Fractions

Let a_1,a_2,\dots,a_n be a sequence of positive integers. Define the n-th order finite continued fraction:

r_n = \cfrac{1}{a_1+\cfrac{1}{a_2+\cdots+\cfrac{1}{a_n}}}

Denote r_n\in\mathbb{Q}.

2.2 Finite Self-Similar Fractals

Let F_0 be the initial figure. Recursively scale by the ratio 1/a_k and replicate a_k copies, obtaining F_k. Define the global similarity ratio of F_n as S_n.

2.3 FCFG Isomorphism Axiom

r_n = S_n

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3 Arithmetic Recursive Systems

Definition 3.1 (Arithmetic recursive system)

Let a_1,\dots,a_n be a sequence of positive integers. Define the arithmetic recursive system:

\mathcal{A}_n = (a_1,\dots,a_n)

corresponding to the continued fraction r_n.

Definition 3.2 (Recursive operator)

For a positive integer a, define the recursive operator:

T_a(x) = \frac{1}{a+x}

Then the arithmetic recursive system can be expressed as an operator chain:

r_n = T_{a_1}\circ T_{a_2}\circ\cdots\circ T_{a_n}(0)

Definition 3.3 (Arithmetic recursive structure)

Call the operator chain

\mathcal{T}_n = (T_{a_1},T_{a_2},\dots,T_{a_n})

the n-th order arithmetic recursive structure.

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4 FCFG-A Isomorphism Axiom

Axiom 4.1 (FCFG-A isomorphism axiom)

For any n-th order arithmetic recursive system \mathcal{A}_n, there exists an n-th order geometric recursive system \mathcal{G}_n such that:

\mathcal{A}_n \cong \mathcal{G}_n

and

r_n = S_n

where \cong denotes recursive structural isomorphism, i.e., the operator chain and the scaling chain correspond one-to-one at the recursive levels.

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5 Structural Isomorphism Theorem

Theorem 5.1 (Arithmetic–geometric recursive isomorphism)

The arithmetic recursive structure \mathcal{T}_n is isomorphic to the geometric recursive structure \mathcal{R}_n.

Proof:

The geometric recursive structure is defined as the scaling chain:

\mathcal{R}_n = (R_{a_1},R_{a_2},\dots,R_{a_n})

where R_{a_k} denotes scaling by the ratio 1/a_k and replicating a_k copies.

Define the mapping:

\phi: T_{a_k} \mapsto R_{a_k}

Then \phi preserves recursive levels and composition order:

\phi(T_{a_1}\circ\cdots\circ T_{a_n})

= R_{a_1}\circ\cdots\circ R_{a_n}

By the FCFG isomorphism axiom, r_n=S_n, so \phi is a structural isomorphism. \square

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6 Basic Property Theorems

Theorem 6.1 (Convergence correspondence)

r_n converges if and only if S_n converges.

Proof: By Axiom 4.1, r_n=S_n, so convergence is equivalent. \square

Theorem 6.2 (Periodicity correspondence)

If a_k is a periodic sequence, then r_n is a quadratic irrational, and S_n has self-similar periodicity.

Proof: A periodic continued fraction converges to a quadratic irrational. By Axiom 4.1, S_n is also a quadratic irrational. The periodicity of the geometric recursion is directly determined by the periodicity of a_k. \square

Theorem 6.3 (Boundedness correspondence)

If a_k is bounded, then the arithmetic recursive depth is controllable, and the geometric recursive dimension is finite.

Proof: Boundedness of a_k means that the scaling ratio at each level is bounded, so the recursive depth is controllable. The geometric dimension is given by \log N/\log a, which is finite under boundedness. \square

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7 FCFG-A Interpretation of Classical Arithmetic Structures

7.1 Euclidean Algorithm

The Euclidean algorithm:

a = bq + r

Recursive remainder-taking corresponds to continued-fraction expansion. FCFG-A interprets this as the geometrization of arithmetic recursion.

7.2 Diophantine Approximation

Continued-fraction convergents:

\frac{p_k}{q_k}

Recursively approximate real numbers, corresponding to layer-by-layer scaling approximation in geometry.

7.3 Quadratic Irrationals

Periodic continued fractions correspond to quadratic irrationals. FCFG-A interprets this as:

Periodic arithmetic recursion ↔ Self-similar geometric recursion

7.4 Ramanujan Continued Fractions

The Ramanujan continued fraction:

\frac{4}{\pi}=1+\cfrac{1^2}{2+\cfrac{3^2}{2+\cfrac{5^2}{2+\cdots}}}

Its coefficient sequence corresponds in FCFG-A to a special class of geometric recursive systems, and the convergence rate is determined by the growth of the coefficients.

7.5 The Constants e and \pi

e=[2;1,2,1,1,4,1,1,6,\dots]

\pi=[3;7,15,1,292,1,1,1,2,\dots]

Their continued-fraction coefficient sequences correspond in FCFG-A to specific recursive geometric structures.

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8 Discussion

8.1 Relationship with FCFG

FCFG-A is the explicit arithmetic side of FCFG. FCFG handles the isomorphism relation; FCFG-A unfolds the arithmetic recursive structure.

8.3 Limitations

· Only applicable to positive coefficients, finite order, and convergent cases;

· Not all arithmetic recursions can be geometrized;

· Currently a framework-level work, requiring further rigorous theorem support.

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9 Conclusion

Within the FCFG framework, this paper establishes a strict isomorphism between arithmetic recursion and geometric recursion. The main results include:

1. The operator-chain definition of arithmetic recursive systems;

2. The FCFG-A isomorphism axiom;

3. The arithmetic–geometric recursive isomorphism theorem;

4. The convergence, periodicity, and boundedness correspondence theorems;

5. The FCFG-A interpretation of classical arithmetic structures.

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References

[1] Original framework materials of FCFG.

[2] Classical theory of continued fractions and Diophantine approximation.

[3] Research related to Ramanujan continued fractions.

[4] Fractal geometry and self-similar structures.