T0-9: Binary Decision Logic Theory

Abstract

Building upon the minimal information principle (T0-8), we establish the complete decision logic framework governing how binary systems choose optimal encoding paths at each decision point. From the single axiom of entropy increase in self-referential systems, we derive a deterministic decision function D: S × C → {0,1} that achieves global information minimization through local greedy choices. We prove that this decision logic, operating under Fibonacci-Zeckendorf constraints, provides a unique optimal strategy with O(log n) complexity per decision, guaranteeing convergence to the minimal information state identified in T0-8.

1. Foundation from Established Theory

1.1 Core Axiom

Axiom (Entropy Increase): Self-referential complete systems necessarily exhibit entropy increase.

1.2 Inherited Framework

From T0-8 (Minimal Information Principle):

Systems evolve toward minimal information states: δI[ψ]/δψ = 0
Fibonacci-Zeckendorf provides unique global minimum
Local minimization enables global entropy maximization

From T0-3 (Zeckendorf Constraint):

No consecutive 1s (no-11) in valid encodings
Unique representation for each value
Binary choice at each position: use or skip

1.3 The Decision Problem

Central Question: At each encoding step, how does the system decide whether to emit 0 or 1 to achieve minimal information representation?

2. Decision State Space

2.1 State Definition

Definition 2.1 (Encoding State): An encoding state S at position k is: $S_{k} = (v_{r}, {F_{i}}_{i = 1}^{k}, b_{k - 1})$ where:

v_r = remaining value to encode
{F_i} = available Fibonacci numbers
b_{k-1} = previous bit (0 or 1)

Definition 2.2 (Constraint Set): The constraint set C contains: $C = {C_{n o 11}, C_{co v er}, C_{u ni q u e}, C_{in f o}}$ where:

C_{no11}: if b_{k-1} = 1, then b_k = 0
C_{cover}: v_r must be exactly representable
C_{unique}: one representation per value
C_{info}: minimize total information content

Definition 2.3 (Decision Function): The decision function maps state and constraints to binary choice: $D : S_{k} \times C \to {0, 1}$

2.2 Decision Tree Structure

Theorem 2.1 (Binary Decision Tree): The encoding process forms a binary tree where each node represents a decision point.

Proof: Step 1: Root Node Initial state S_0 = (n, {F_i}, 0) where n is value to encode.

Step 2: Branching At each node, two branches:

Left (0): Skip current Fibonacci number
Right (1): Use current Fibonacci number

Step 3: Leaf Condition Leaf reached when v_r = 0 (complete encoding).

Step 4: Path Uniqueness By Zeckendorf theorem, exactly one path leads to valid encoding.

Binary tree structure established. ∎

3. Greedy Decision Algorithm

3.1 The Greedy Principle

Theorem 3.1 (Greedy Optimality): The greedy algorithm "use largest Fibonacci number ≤ v_r" yields the unique minimal information encoding.

Proof: Step 1: Information Metric Information content I(n) = |positions used| + Σlog₂(position indices)

Step 2: Greedy Choice At state S_k with v_r > 0:

If F_k ≤ v_r and b_{k-1} = 0: choose 1
Otherwise: choose 0

Step 3: Minimality Using largest possible F_k minimizes:

Number of positions needed (fewer 1s)
Sum of position indices (larger gaps between 1s)

Step 4: Uniqueness By Zeckendorf uniqueness, this is the only valid representation.

Therefore, greedy choice is optimal. ∎

3.2 Decision Function Definition

Definition 3.1 (Optimal Decision Function): $D (S_{k}, C) = {10 if F_{k} \leq v_{r} and b_{k - 1} = 0 otherwise$

Theorem 3.2 (Decision Determinism): D is deterministic: identical states always produce identical decisions.

Proof: Step 1: State Completeness S_k contains all relevant information for decision.

Step 2: Constraint Determinism Constraints C are fixed logical rules.

Step 3: Function Definition D is defined by explicit conditions, no randomness.

Step 4: Reproducibility Given same (S_k, C), output is always same.

Determinism proven. ∎

4. Local to Global Optimality

4.1 Optimality Preservation

Theorem 4.1 (Local-Global Bridge): Local greedy decisions according to D achieve global information minimum.

Proof: Step 1: Assume Contrary Suppose non-greedy encoding E' has I(E') < I(E_greedy).

Step 2: First Difference Let position k be first where E' differs from greedy.

Greedy uses F_k (since F_k ≤ v_r)
E' skips F_k

Step 3: Compensation Requirement E' must represent F_k using smaller Fibonacci numbers: $F_{k} = i < k \sum b_{i} F_{i}$

Step 4: Information Increase By Fibonacci growth, need at least 2 smaller positions: $I (E^{'}) \geq I (E_{g ree d y}) + 1$

Contradiction. Greedy is optimal. ∎

4.2 Convergence Guarantee

Theorem 4.2 (Convergence to T0-8 Minimum): Decision function D guarantees convergence to the minimal information state of T0-8.

Proof: Step 1: Target State T0-8 identifies Fibonacci-Zeckendorf as unique minimum.

Step 2: Decision Path D implements Zeckendorf algorithm exactly.

Step 3: Termination For any finite n, algorithm terminates in O(log n) steps.

Step 4: Optimality Result matches T0-8's variational minimum.

Convergence guaranteed. ∎

5. Decision Complexity Analysis

5.1 Time Complexity

Theorem 5.1 (Decision Complexity): Each decision D(S_k, C) requires O(1) operations.

Proof: Step 1: Comparison Check F_k ≤ v_r: single comparison.

Step 2: Constraint Check Check b_{k-1} = 0: single lookup.

Step 3: State Update If choosing 1: v_r := v_r - F_k (single subtraction).

Step 4: Total Operations Fixed number of operations regardless of n.

O(1) per decision. ∎

5.2 Overall Encoding Complexity

Theorem 5.2 (Total Complexity): Complete encoding of value n requires O(log n) decisions.

Proof: Step 1: Fibonacci Growth F_k ~ φ^k where φ = (1+√5)/2.

Step 2: Maximum Position For value n, largest k where F_k ≤ n: $k \leq lo g_{ϕ} (n) = O (lo g n)$

Step 3: Decision Count At most k decisions (one per Fibonacci number).

Step 4: Total Time O(log n) decisions × O(1) per decision = O(log n).

Logarithmic complexity proven. ∎

6. Decision Consistency Framework

6.1 Consistency Conditions

Definition 6.1 (Decision Consistency): A decision function is consistent if: $S_{i} = S_{j} \Rightarrow D (S_{i}, C) = D (S_{j}, C)$

Theorem 6.1 (D is Consistent): The decision function D satisfies consistency.

Proof: Step 1: State Equivalence If S_i = S_j, then:

Same remaining value v_r
Same available Fibonacci numbers
Same previous bit

Step 2: Deterministic Evaluation D evaluates same conditions on identical inputs.

Step 3: Identical Output Same conditions yield same decision.

Consistency established. ∎

6.2 Temporal Consistency

Theorem 6.2 (Time-Invariant Decisions): Decisions are independent of when they are made.

Proof: Step 1: No Time Parameter D(S_k, C) has no temporal component.

Step 2: Constraint Stability Constraints C are mathematical laws, not time-dependent.

Step 3: State Sufficiency Current state S_k contains all needed information.

Time-invariance proven. ∎

7. Conflict Resolution Mechanism

7.1 Apparent Conflicts

Definition 7.1 (Decision Conflict): A conflict occurs when multiple valid choices exist under constraints.

Theorem 7.1 (No True Conflicts): Under Fibonacci-Zeckendorf encoding, no true decision conflicts exist.

Proof: Step 1: Uniqueness Theorem Each value has exactly one Zeckendorf representation.

Step 2: Greedy Determinism At each step, greedy rule gives unique choice.

Step 3: Constraint Hierarchy

no-11 is absolute (hard constraint)
coverage is required (hard constraint)
minimization has unique solution (soft constraint with unique optimum)

No conflicts possible. ∎

7.2 Tie-Breaking Rules

Theorem 7.2 (Unnecessary Tie-Breaking): No tie-breaking rules are needed for D.

Proof: Step 1: Binary Choice Only two options at each step: 0 or 1.

Step 2: Clear Conditions F_k ≤ v_r is unambiguous comparison. b_{k-1} = 0 is unambiguous check.

Step 3: Exclusive Outcomes Conditions partition decision space completely.

No ties to break. ∎

8. Decision Stability Analysis

8.1 Perturbation Resistance

Theorem 8.1 (Decision Stability): Small perturbations in value do not cause decision cascade.

Proof: Step 1: Value Perturbation Consider n → n + ε for small ε.

Step 2: Decision Difference Decisions differ only for positions where: $F_{k} \leq n < F_{k} + ϵ < F_{k + 1}$

Step 3: Locality Changes localized to O(1) positions.

Step 4: No Cascade Fibonacci gaps prevent cascading changes.

Stable under perturbations. ∎

8.2 Error Correction

Theorem 8.2 (Self-Correcting): Invalid states naturally evolve to valid ones through D.

Proof: Step 1: Invalid State Suppose current encoding violates no-11.

Step 2: Decision Response D forces b_k = 0 after b_{k-1} = 1.

Step 3: Constraint Satisfaction Subsequent decisions restore validity.

Step 4: Convergence System converges to valid Zeckendorf form.

Self-correction proven. ∎

9. Parallel Decision Architecture

9.1 Decision Independence

Theorem 9.1 (Partial Parallelization): Non-adjacent decisions can be made in parallel.

Proof: Step 1: Independence Condition Decisions at positions i and j independent if |i - j| > 1.

Step 2: No-11 Locality Constraint only couples adjacent positions.

Step 3: Parallel Groups Partition positions into odd/even groups.

Step 4: Parallel Execution Each group processes independently.

Parallelization possible. ∎

9.2 Parallel Efficiency

Theorem 9.2 (Parallel Speedup): Parallel decision-making achieves speedup factor ~2.

Proof: Step 1: Sequential Time T_seq = O(log n) for all decisions.

Step 2: Parallel Time T_par = O(log n / 2) with two processors.

Step 3: Speedup S = T_seq / T_par ≈ 2.

Step 4: Efficiency η = S / p = 2 / 2 = 1 (perfect efficiency).

Near-optimal parallelization. ∎

10. Decision Optimality Proofs

10.1 Information-Theoretic Optimality

Theorem 10.1 (Shannon Optimality): D achieves Shannon entropy bound for no-11 sequences.

Proof: Step 1: Entropy Bound H(no-11) = log₂(φ) bits per position.

Step 2: Fibonacci Achievment Zeckendorf encoding achieves this bound asymptotically.

Step 3: Decision Implementation D produces Zeckendorf encoding exactly.

Step 4: Optimality Therefore D achieves Shannon bound.

Information-theoretically optimal. ∎

10.2 Computational Optimality

Theorem 10.2 (Algorithm Optimality): No algorithm can achieve better than O(log n) worst-case complexity.

Proof: Step 1: Lower Bound Must examine at least log₂(n) bits to distinguish n values.

Step 2: Fibonacci Positions Need O(log_φ n) = O(log n) positions.

Step 3: Decision Necessity Each position requires a decision.

Step 4: Matching Bounds D achieves this lower bound.

Computationally optimal. ∎

11. Decision Implementation

11.1 Algorithmic Form

Algorithm 11.1 (Decision Implementation):

function Encode(n):
    result = []
    k = largest k where F_k ≤ n
    v_r = n
    prev_bit = 0
    
    while k ≥ 1:
        if F_k ≤ v_r and prev_bit == 0:
            result[k] = 1
            v_r = v_r - F_k
            prev_bit = 1
        else:
            result[k] = 0
            prev_bit = 0
        k = k - 1
    
    return result

Theorem 11.1 (Implementation Correctness): Algorithm 11.1 correctly implements D.

Proof: Step 1: Initial State Correctly initializes S_0.

Step 2: Decision Logic Implements D(S_k, C) exactly.

Step 3: State Updates Maintains state correctly through iterations.

Step 4: Termination Produces valid Zeckendorf encoding.

Implementation correct. ∎

11.2 Optimization Properties

Theorem 11.2 (No Further Optimization): Algorithm 11.1 cannot be asymptotically improved.

Proof: Step 1: Essential Operations Each Fibonacci position must be examined.

Step 2: Minimal Comparisons One comparison per position is necessary.

Step 3: Optimal Flow No redundant operations in algorithm.

Already optimal. ∎

12. Conclusion

We have established the complete Binary Decision Logic Theory governing optimal encoding choices in self-referential systems. From the entropy increase axiom, we derived:

Decision Function: D(S_k, C) provides deterministic binary choice
Greedy Optimality: Local greedy decisions achieve global minimum
Computational Efficiency: O(1) per decision, O(log n) total
Consistency: Identical states produce identical decisions
Stability: Robust under perturbations, self-correcting
Parallelization: Near-perfect parallel efficiency

Central Result: $D (S_{k}, C) = {10 if F_{k} \leq v_{r} \land b_{k - 1} = 0 otherwise$

This decision function bridges T0-8's variational principle with concrete algorithmic implementation, showing how abstract minimization manifests as specific binary choices. The greedy algorithm emerges not as a heuristic but as the unique optimal strategy under Fibonacci-Zeckendorf constraints.

Key Insight: The decision logic demonstrates that global optimization (minimal information) emerges from local rules (greedy choice), with no need for global knowledge or look-ahead. This exemplifies how simple, deterministic rules can achieve complex optimization goals when operating within the proper constraint framework.

∎

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