T0-9: Binary Decision Logic Theory - Formal Specification

Formal Framework

Axiom

Axiom T0: ∀S (SelfReferential(S) ∧ Complete(S)) ⟹ EntropyIncrease(S)

Definitions

Definition T9.1 (Encoding State Space): $$\mathcal{S}=\{(v_r,{\mathcal{F}}_k,b_{prev})\mid v_r\in {\mathbb{N}}_0,{\mathcal{F}}_k\subseteq \{F_i{\}}_{i=1}^{\infty },b_{prev}\in \{0,1\}\}$$

Definition T9.2 (Constraint Set): $$\mathcal{C}=\{C_{no11},C_{cover},C_{unique},C_{info}\}$$ where:

• $C_{no11}:b_{k-1}=1\Rightarrow b_k=0$
• $C_{cover}:\forall n\in \mathbb{N},\exists \{b_i\}:n=\sum b_i{F}_i$
• $C_{unique}:\forall n,!\exists \{b_i\}:n=\sum b_i{F}_i$
• $C_{info}:{\min }_{\{b_i\}}I(\{b_i\})$

Definition T9.3 (Decision Function): $$D:\mathcal{S}\times \mathcal{C}\to \{0,1\}$$ $$D((v_r,F_k,b_{prev}),\mathcal{C})=1_{F_k\le v_r}\cdot 1_{b_{prev}=0}$$

Definition T9.4 (Information Content): $$I:\{0,1{\}}^{\ast }\to {\mathbb{R}}^{+}$$ $$I(\{b_i{\}}_{i=1}^k)=|\{i:b_i=1\}|+\sum _{i:b_i=1}{\log }_2(i)$$

Core Theorems

Theorem T9.1 (Greedy Optimality): $$\forall n\in \mathbb{N},\arg \underset{\{b_i\}}{\min }I(\{b_i\})=\text{Greedy}(n)$$ where Greedy(n) applies D iteratively from largest Fibonacci number.

Proof: Let $E_g=\text{Greedy}(n)$ and suppose $\exists E^{\prime }\ne E_g:I(E^{\prime })<I(E_g)$.

1. Let $k^{\ast }=\max \{k:E_k^{\prime }\ne E_{g,k}\}$
2. At position $k^{\ast }$: $F_{k^{\ast }}\le v_r$ but $E_{k^{\ast }}^{\prime }=0$ while $E_{g,k^{\ast }}=1$
3. $E^{\prime }$ must represent $F_{k^{\ast }}$ using $\{F_i{\}}_{i<k^{\ast }}$
4. By Fibonacci recurrence: $F_{k^{\ast }}=F_{k^{\ast }-1}+F_{k^{\ast }-2}$
5. Minimum positions needed: $|E^{\prime }|\ge |E_g|+1$
6. $I(E^{\prime })\ge I(E_g)+1$

Contradiction. Therefore Greedy is optimal. □

Theorem T9.2 (Decision Determinism): $$\forall S_1,S_2\in \mathcal{S},S_1=S_2\Rightarrow D(S_1,\mathcal{C})=D(S_2,\mathcal{C})$$

Proof: $D$ is defined by explicit logical conditions on state components. No randomness in evaluation. Therefore deterministic. □

Theorem T9.3 (Convergence to T0-8 Minimum): $$\underset{k\to \infty }{\lim }{D}^k(S_0,\mathcal{C})={\psi }_{min}$$ where ${\psi }_{min}$ is the T0-8 minimal information state.

Proof:

1. T0-8 identifies Fibonacci-Zeckendorf as unique minimum
2. $D$ implements Zeckendorf algorithm exactly
3. For finite $n$, algorithm terminates in $O(\log n)$ steps
4. Result is unique Zeckendorf representation
5. This matches T0-8 variational minimum

Convergence established. □

Complexity Analysis

Theorem T9.4 (Decision Complexity): $$\text{Time}(D(S,\mathcal{C}))=O(1)$$

Proof: Operations in $D$:

• Comparison $F_k\le v_r$: $O(1)$
• Check $b_{prev}=0$: $O(1)$
• Multiplication of indicators: $O(1)$ Total: $O(1)$. □

Theorem T9.5 (Encoding Complexity): $$\text{Time}(\text{Encode}(n))=O(\log n)$$

Proof:

1. Number of Fibonacci numbers $\le n$: $k=O({\log }_{\varphi }n)=O(\log n)$
2. Each requires one decision: $O(1)$ per decision
3. Total: $O(\log n)\cdot O(1)=O(\log n)$. □

Stability Properties

Theorem T9.6 (Local Stability): $$\forall ϵ>0,\exists \delta >0:|n-m|<\delta \Rightarrow d_H(D^{\ast }(n),D^{\ast }(m))<ϵ$$ where $d_H$ is Hamming distance and $D^{\ast }$ is complete encoding.

Proof:

1. Small value changes affect only local Fibonacci positions
2. Fibonacci gaps grow exponentially
3. Changes confined to $O(1)$ positions
4. Hamming distance bounded by constant Stability proven. □

Theorem T9.7 (Self-Correction): $$\forall S\in \mathcal{S},\text{Invalid}(S)\Rightarrow \text{Valid}(D^k(S,\mathcal{C}))$$ for some finite $k$.

Proof:

1. Invalid states violate $C_{no11}$
2. $D$ enforces $b_k=0$ after $b_{k-1}=1$
3. Finite number of positions to correct
4. Each iteration moves toward validity Self-correction established. □

Optimality Characterization

Theorem T9.8 (Shannon Bound Achievement): $$\underset{n\to \infty }{\lim }\frac{I(D^{\ast }(n))}{{\log }_{2}n}=\frac{1}{{\log }_2\varphi }$$

Proof:

1. Shannon entropy for no-11 sequences: $H={\log }_2\varphi$
2. Fibonacci growth: $F_k\sim {\varphi }^k/\sqrt{5}$
3. Positions needed: $k\sim {\log }_{\varphi }n$
4. Information density: $I/\log n\to 1/\log \varphi$ Shannon bound achieved. □

Theorem T9.9 (Unique Optimal Strategy): $$\forall D^{\prime }:\mathcal{S}\times \mathcal{C}\to \{0,1\},D^{\prime }\ne D\Rightarrow \exists n:I(D^{\prime \ast }(n))>I(D^{\ast }(n))$$

Proof:

1. Zeckendorf representation is unique
2. Any deviation from greedy violates uniqueness or optimality
3. Non-greedy choices require more positions
4. Therefore $D$ is uniquely optimal Uniqueness proven. □

Parallel Architecture

Theorem T9.10 (Parallel Decomposition): $$D^{\ast }(\cdot )=D_{odd}^{\ast }(\cdot )\parallel D_{even}^{\ast }(\cdot )$$ where $\parallel$ denotes parallel composition.

Proof:

1. $C_{no11}$ only couples adjacent positions
2. Odd positions: $\{1,3,5,...\}$ independent
3. Even positions: $\{2,4,6,...\}$ independent
4. Groups can be processed in parallel Decomposition valid. □

Implementation Specification

Algorithm T9.1 (Formal Decision Procedure):

Input: n ∈ ℕ
Output: B = {b_i} ∈ {0,1}*

1. k ← max{i : F_i ≤ n}
2. v_r ← n
3. b_prev ← 0
4. for i from k down to 1:
   5. b_i ← D((v_r, F_i, b_prev), C)
   6. if b_i = 1:
      7. v_r ← v_r - F_i
   8. b_prev ← b_i
9. return {b_i}

Correctness: By construction, implements $D$ exactly.

Variational Formulation

Theorem T9.11 (Variational Equivalence): $$D=\arg \underset{f:\mathcal{S}\times \mathcal{C}\to \{0,1\}}{\min }\mathcal{L}[f]$$ where $\mathcal{L}[f]={\mathbb{E}}_{n\sim p(n)}[I(f^{\ast }(n))]$.

Proof:

1. Minimize expected information over value distribution
2. Constraint: must produce valid encodings
3. Lagrangian: $L=I+\lambda \cdot \text{Violations}$
4. Optimal solution: greedy algorithm
5. This is exactly $D$ Variational equivalence proven. □

Category-Theoretic Structure

Theorem T9.12 (Functor Property): $D$ induces a functor $\mathcal{D}:\mathbf{Val}\to \mathbf{Enc}$ where:

• $\mathbf{Val}$ is category of values with ordering
• $\mathbf{Enc}$ is category of encodings with prefix order

Proof:

1. Object mapping: $n\mapsto D^{\ast }(n)$
2. Morphism mapping: $\le \mapsto ⪯$ (prefix)
3. Identity preservation: $D^{\ast }(n)=D^{\ast }(n)$
4. Composition preservation: order-preserving Functor structure established. □

Formal Synthesis

The decision function $D$ provides the algorithmic bridge between T0-8's variational principle and concrete binary choices. Key formal results:

1. Optimality: $D$ achieves global minimum through local greedy choices
2. Complexity: $O(\log n)$ time with $O(1)$ per decision
3. Stability: Robust under perturbations, self-correcting
4. Uniqueness: Only strategy achieving Shannon bound
5. Parallelizability: Perfect efficiency for odd/even decomposition

Central Identity: $$\boxed{D((v_r,F_k,b_{prev}),\mathcal{C})=1_{F_k\le v_r}\cdot 1_{b_{prev}=0}}$$

This completes the formal specification of binary decision logic under entropy-driven evolution.

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