T0-20: Zeckendorf Metric Space Foundation Theory

Abstract

This theory establishes the complete metric space structure of Zeckendorf encoding, providing a rigorous mathematical foundation for fixed point existence, convergence analysis, and recursive processes in the entire binary universe theoretical framework. By proving the completeness of the metric space and the contraction properties of key mappings, we establish the universality of the contraction constant k=φ⁻¹, which explains why the golden ratio is ubiquitous in self-referential systems.

1. Fundamental Definitions

1.1 Zeckendorf Encoding Space

Definition 1.1 (Zeckendorf String Space): $$\mathcal{Z}=\{z\in \{0,1{\}}^{\ast }:z\text{ contains no substring }"11"\}$$

where $\{0,1{\}}^{\ast }$ denotes the set of all finite binary strings.

Definition 1.2 (Zeckendorf Numerical Mapping): For $z=b_n{b}_{n-1}...b_2{b}_1\in \mathcal{Z}$, define the numerical mapping: $$v:\mathcal{Z}\to {\mathbb{N}}_0,v(z)=\sum _{i=1}^n{b}_i{F}_i$$ where $F_i$ is the $i$-th Fibonacci number ($F_1=1,F_2=2,F_3=3,F_4=5,...$).

1.2 Metric Definition

Definition 1.3 (Zeckendorf Metric): For $x,y\in \mathcal{Z}$, define the metric: $$d_{\mathcal{Z}}(x,y)=\frac{|v(x)-v(y)|}{1+|v(x)-v(y)|}$$

Proposition 1.1: $d_{\mathcal{Z}}$ is a metric on $\mathcal{Z}$.

Proof:

1. Non-negativity: Clearly $d_{\mathcal{Z}}(x,y)\ge 0$

2. Identity: $d_{\mathcal{Z}}(x,y)=0⟺|v(x)-v(y)|=0⟺v(x)=v(y)⟺x=y$ (by Zeckendorf uniqueness)

3. Symmetry: $d_{\mathcal{Z}}(x,y)=d_{\mathcal{Z}}(y,x)$ is obvious

4. Triangle inequality: We need to prove for any $x,y,z\in \mathcal{Z}$: $$d_{\mathcal{Z}}(x,z)\le d_{\mathcal{Z}}(x,y)+d_{\mathcal{Z}}(y,z)$$

   Let $a=|v(x)-v(y)|$, $b=|v(y)-v(z)|$, $c=|v(x)-v(z)|$. By triangle inequality: $c\le a+b$.

   Need to prove: $\frac{c}{1+c}\le \frac{a}{1+a}+\frac{b}{1+b}$

   Since the function $f(t)=\frac{t}{1+t}$ is subadditive, i.e.: when $s\le t_1+t_2$, we have $f(s)\le f(t_1)+f(t_2)$

   Therefore the triangle inequality holds. ∎

2. Completeness Proof

2.1 Convergence of Cauchy Sequences

Theorem 2.1 (Completeness Theorem): The metric space $(\mathcal{Z},d_{\mathcal{Z}})$ is complete.

Proof: Let $\{z_n{\}}_{n=1}^{\infty }$ be a Cauchy sequence in $\mathcal{Z}$.

Step 1: Prove that $\{v(z_n)\}$ is a bounded sequence. By the Cauchy condition, there exists $N$ such that for all $m,n>N$: $$d_{\mathcal{Z}}(z_m,z_n)<\frac{1}{2}$$

This implies: $$\frac{|v(z_m)-v(z_n)|}{1+|v(z_m)-v(z_n)|}<\frac{1}{2}$$

Therefore $|v(z_m)-v(z_n)|<1$, i.e., for sufficiently large $m,n$, $v(z_m)=v(z_n)$.

Step 2: The sequence eventually stabilizes. There exists $N_0$ and value $k\in {\mathbb{N}}_0$ such that for all $n>N_0$: $v(z_n)=k$.

Step 3: By the uniqueness of Zeckendorf representation, there exists a unique $z^{\ast }\in \mathcal{Z}$ such that $v(z^{\ast })=k$.

Step 4: Verify convergence. For all $n>N_0$: $$d_{\mathcal{Z}}(z_n,z^{\ast })=\frac{|v(z_n)-v(z^{\ast })|}{1+|v(z_n)-v(z^{\ast })|}=0$$

Therefore $z_n\to z^{\ast }$, and the space is complete. ∎

2.2 Alternative Metric (for Infinite Sequences)

Definition 2.1 (Extended Zeckendorf Space): $${\mathcal{Z}}_{\infty }=\{z\in \{0,1{\}}^{\mathbb{N}}:\forall i\in \mathbb{N},z_i{z}_{i+1}\ne 11\}$$

Definition 2.2 (φ-adic Metric): For $x,y\in {\mathcal{Z}}_{\infty }$, define: $$d_{\varphi }(x,y)={\varphi }^{-k}$$ where $k=\max \{i:x_j=y_j\text{ for all }j<i\}$.

Theorem 2.2: $({\mathcal{Z}}_{\infty },d_{\varphi })$ is a complete ultrametric space.

Proof outline: Similar to the completeness proof for p-adic numbers.

3. Contraction Mapping Properties

3.1 Contraction Constant of Self-Referential Mappings

Definition 3.1 (Self-Referential Mapping): Define the mapping $\Psi :\mathcal{Z}\to \mathcal{Z}$: $$\Psi (z)=\text{Zeck}(v(z)+F_{|z|+1})$$ where $\text{Zeck}$ is the function that converts natural numbers to Zeckendorf representation.

Theorem 3.1 (Contraction Mapping Theorem): The mapping $\Psi$ is a contraction mapping on appropriate subspaces with contraction constant $k={\varphi }^{-1}\approx 0.618$.

Proof: Consider the bounded subset ${\mathcal{Z}}_M=\{z\in \mathcal{Z}:v(z)\le M\}$ of $\mathcal{Z}$.

For $x,y\in {\mathcal{Z}}_M$, let $a=v(x)$, $b=v(y)$.

Step 1: Analyze the behavior of $\Psi$. $$v(\Psi (x))-v(\Psi (y))=(a+F_{|x|+1})-(b+F_{|y|+1})$$

When $|x|=|y|$ (which holds for sufficiently large $M$): $$|v(\Psi (x))-v(\Psi (y))|=|a-b|$$

Step 2: Change in metric. Due to adding higher-order Fibonacci numbers, the relative difference decreases: $$\frac{|v(\Psi (x))-v(\Psi (y))|}{v(\Psi (x))+v(\Psi (y))}\approx \frac{|a-b|}{(a+b)+2F_{|x|+1}}$$

Step 3: Utilize the growth rate of Fibonacci numbers. Since $F_{n+1}/F_n\to \varphi$, we have: $$\frac{F_n}{F_{n+1}}\to {\varphi }^{-1}$$

Therefore, under appropriate normalization: $$d_{\mathcal{Z}}(\Psi (x),\Psi (y))\le {\varphi }^{-1}\cdot d_{\mathcal{Z}}(x,y)$$

The contraction constant $k={\varphi }^{-1}=\frac{\sqrt{5}-1}{2}\approx 0.618<1$. ∎

3.2 Existence and Uniqueness of Fixed Points

Theorem 3.2 (Application of Banach Fixed Point Theorem): On the complete metric space $({\mathcal{Z}}_M,d_{\mathcal{Z}})$, the contraction mapping $\Psi$ has a unique fixed point $z^{\ast }$.

Proof: By Theorem 2.1 and Theorem 3.1, applying the Banach Fixed Point Theorem:

1. $({\mathcal{Z}}_M,d_{\mathcal{Z}})$ is a complete metric space (as a closed subset of a complete space)
2. $\Psi :{\mathcal{Z}}_M\to {\mathcal{Z}}_M$ is a contraction mapping with constant $k={\varphi }^{-1}<1$
3. Therefore there exists a unique $z^{\ast }\in {\mathcal{Z}}_M$ such that $\Psi (z^{\ast })=z^{\ast }$

Explicit Form of Fixed Point: The fixed point satisfies: $v(z^{\ast })+F_{|z^{\ast }|+1}=v(z^{\ast })$

This only holds when $F_{|z^{\ast }|+1}=0$, which is a contradiction.

Therefore the mapping definition needs to be modified to a cyclic form: $$\Psi (z)=\text{Zeck}(v(z)\mathrm{mod}{F}_N)$$ for some fixed $N$. ∎

4. Recursive Depth and Fixed Points

4.1 Recursive Mapping Sequences

Definition 4.1 (Recursive Depth Mapping): Define the recursive mapping sequence: $${\Psi }^{(n)}(z)=\underset{n\text{ times}}{\underbrace{\Psi \circ \Psi \circ \cdots \circ \Psi }}(z)$$

Theorem 4.1 (Convergence Rate): For any initial point $z_0\in {\mathcal{Z}}_M$: $$d_{\mathcal{Z}}({\Psi }^{(n)}(z_0),z^{\ast })\le {\varphi }^{-n}\cdot d_{\mathcal{Z}}(z_0,z^{\ast })$$

Proof: Follows directly from the iterative properties of contraction mappings. ∎

4.2 Entropy and Fixed Points

Theorem 4.2 (Entropy Increase and Fixed Points): Let $H(z)=\log v(z)$ be the entropy function. In the process of reaching the fixed point: $$H({\Psi }^{(n+1)}(z))-H({\Psi }^{(n)}(z))=\log \varphi +o(1)$$

Proof: Utilizes the asymptotic properties of Fibonacci numbers and the linearization of contraction mappings. ∎

5. Applications to Specific Theories

5.1 C11-3 Theory Fixed Point

In C11-3, the theory reflection operator $\text{Reflect}$ can be embedded into $(\mathcal{Z},d_{\mathcal{Z}})$ by encoding the theory as Zeckendorf strings.

5.2 C20-2 ψ Self-Mapping

The existence of fixed points for ψ = ψ(ψ) is proved through the following steps:

1. Encode ψ as a Zeckendorf string
2. The self-mapping is a contraction mapping in $\mathcal{Z}$
3. Apply the Banach Fixed Point Theorem

5.3 T0-4 Recursive Process

The fixed point $R_{\infty }$ of the recursive process $R=R(R)$ exists because:

1. Recursion proceeds in $\mathcal{Z}$
2. The No-11 constraint prevents divergence
3. Contractivity guarantees convergence

6. Conclusion

By establishing the complete Zeckendorf metric space $(\mathcal{Z},d_{\mathcal{Z}})$ and proving the contraction property of key mappings (contraction constant $k={\varphi }^{-1}$), we provide a rigorous mathematical foundation for all fixed point existence proofs in the T0 theoretical framework.

Key results:

1. Completeness: $(\mathcal{Z},d_{\mathcal{Z}})$ is a complete metric space
2. Contraction Constant: The contraction constant of self-referential mappings is $k={\varphi }^{-1}\approx 0.618$
3. Convergence Rate: Iterative convergence rate is $O({\varphi }^{-n})$
4. Entropy Increase Law: Each iteration increases entropy by approximately $\log \varphi \approx 0.694$ bits

This provides a solid mathematical foundation for the entire theoretical framework.