T0-6: System Component Interaction Theory

Abstract

Building upon entropy flow conservation (T0-5), we establish the fundamental theory of safe component interaction in self-referential systems. From the entropy increase axiom and conservation laws, we derive how components exchange information while maintaining system integrity through coupling constraints, information loss bounds, and optimal interaction strategies within the Zeckendorf encoding framework.

1. Foundation from Established Theory

1.1 Core Axiom

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

1.2 Inherited Foundations

• T0-1: Binary universe {0,1} with forbidden consecutive 1s
• T0-2: Components have finite Fibonacci-quantized capacities F₁, F₂, F₃, ...
• T0-3: No-11 constraint ensures unique Zeckendorf representation
• T0-4: All information states have complete binary-Zeckendorf encoding
• T0-5: Entropy flow obeys conservation laws with Fibonacci quantization

1.3 The Interaction Problem

Central Question: How can components safely exchange information while preserving entropy conservation and system stability?

2. Component Coupling Framework

2.1 Interaction Channels

Definition 2.1 (Interaction Channel): An interaction channel between components C_i and C_j is a bidirectional pathway: $${\mathcal{I}}_{ij}=(C_i\leftrightarrow C_j,{\kappa }_{ij},{\tau }_{ij})$$ where:

• κᵢⱼ ∈ [0,1] is the coupling strength
• τᵢⱼ is the transmission delay in discrete time units

Definition 2.2 (Information Packet): Information exchanged between components is packaged as: $$P_{ij}(t)=(z,ϵ,\varphi )$$ where:

• z ∈ Zeckendorf space: the information content
• ε ≥ 0: entropy cost of transmission
• φ: flow direction indicator

Definition 2.3 (Coupling Strength): The coupling strength between components is: $${\kappa }_{ij}=\frac{\min (F_{k_i},F_{k_j})}{\max (F_{k_i},F_{k_j})}$$ where F_{k_i}, F_{k_j} are the respective component capacities.

2.2 Safe Exchange Protocol

Theorem 2.1 (Safe Exchange Condition): Information exchange between components is safe iff:

1. No-11 constraint preserved in both components
2. Total entropy conserved or increased
3. Channel capacity not exceeded

Proof: Consider exchange from C_i to C_j of information z.

Step 1: Pre-exchange States

• C_i state: $s_i={\sum }_{k\in K_i}{F}_k$ (Zeckendorf form)
• C_j state: $s_j={\sum }_{l\in L_j}{F}_l$ (Zeckendorf form)

Step 2: Exchange Constraints For safe exchange:

• Extraction: s_i - z must remain valid Zeckendorf (no 11)
• Insertion: s_j + z must remain valid Zeckendorf (no 11)
• Channel: z ≤ κᵢⱼ × min(F_{k_i}, F_{k_j})

Step 3: Entropy Balance By T0-5 conservation: $$E_i^{after}+E_j^{after}=E_i^{before}+E_j^{before}+{ϵ}_{exchange}$$ where ε_exchange ≥ 0 is entropy generated by interaction.

These three conditions ensure safe exchange. ∎

3. Information Transmission Theory

3.1 Transmission Constraints

Theorem 3.1 (Bandwidth Theorem): Maximum information transmission rate between components is: $$B_{ij}={\kappa }_{ij}\times \frac{\min (F_{k_i},F_{k_j})}{{\tau }_{ij}}$$

Proof: Step 1: Capacity Constraint Per time unit, maximum transferable is min(F_{k_i}, F_{k_j}).

Step 2: Coupling Modulation Effective transfer reduced by coupling strength κᵢⱼ.

Step 3: Time Quantization With delay τᵢⱼ, rate becomes: $$B_{ij}=\frac{{\kappa }_{ij}\times \min (F_{k_i},F_{k_j})}{{\tau }_{ij}}$$

This establishes the bandwidth limit. ∎

3.2 Information Loss

Theorem 3.2 (Transmission Loss): Information transmission incurs entropy loss: $$\Delta S_{loss}=z\times (1-{\kappa }_{ij})\times {\log }_2({\tau }_{ij}+1)$$

Proof: Step 1: Coupling Loss Imperfect coupling (κᵢⱼ < 1) causes loss ∝ (1 - κᵢⱼ).

Step 2: Temporal Degradation Delay τᵢⱼ introduces uncertainty ∝ log₂(τᵢⱼ + 1).

Step 3: Total Loss Combined effect: $$\Delta S_{loss}=z\times (1-{\kappa }_{ij})\times {\log }_2({\tau }_{ij}+1)$$

This quantifies transmission entropy cost. ∎

3.3 Error Bounds

Theorem 3.3 (Error Propagation Bound): Transmission errors are bounded by: $${ϵ}_{max}=F_{\lfloor {\log }_{\varphi }(z)\rfloor }\times (1-{\kappa }_{ij})$$ where φ = (1+√5)/2 is the golden ratio.

Proof: Step 1: Zeckendorf Decomposition Information z = ∑F_k has largest term F_⌊log_φ(z)⌋.

Step 2: Coupling Error Maximum single-bit error affects largest term.

Step 3: Error Bound $${ϵ}_{max}=F_{\lfloor {\log }_{\varphi }(z)\rfloor }\times (1-{\kappa }_{ij})$$

This bounds worst-case error. ∎

4. Coupling Dynamics

4.1 Coupling Strength Evolution

Theorem 4.1 (Dynamic Coupling): Coupling strength evolves according to interaction history: $${\kappa }_{ij}(t+1)={\kappa }_{ij}(t)+\alpha \times \frac{{\Phi }_{ij}(t)}{F_{min}}\times (1-{\kappa }_{ij}(t))$$ where α ∈ (0,1) is adaptation rate.

Proof: Step 1: Reinforcement Mechanism Successful flows strengthen coupling.

Step 2: Saturation Limit Term (1 - κᵢⱼ(t)) prevents κᵢⱼ > 1.

Step 3: Fibonacci Normalization Flow Φᵢⱼ normalized by F_min for scale invariance.

This gives the evolution equation. ∎

4.2 Optimal Coupling

Theorem 4.2 (Optimal Coupling Configuration): For n components, optimal coupling minimizes total entropy loss: $${\kappa }_{ij}^{\ast }=\sqrt{\frac{F_{k_i}\times F_{k_j}}{(F_{k_i}+F_{k_j}{)}^2}}$$

Proof: Step 1: Loss Function Total loss L = ∑ᵢⱼ Δ S_loss(κᵢⱼ).

Step 2: Optimization ∂L/∂κᵢⱼ = 0 yields: $${\kappa }_{ij}^{\ast }=\sqrt{\frac{F_{k_i}\times F_{k_j}}{(F_{k_i}+F_{k_j}{)}^2}}$$

Step 3: Verification Second derivative test confirms minimum.

This gives optimal coupling. ∎

5. Multi-Component Interaction

5.1 Interaction Networks

Definition 5.1 (Interaction Graph): System interaction structure is graph G = (V, E) where:

• V = {C₁, C₂, ..., Cₙ}: components as vertices
• E = {𝒾ᵢⱼ}: interaction channels as edges

Theorem 5.1 (Network Stability): Interaction network is stable iff: $${\lambda }_{max}(\mathbf{K})<1$$ where 𝐊 is the coupling matrix [κᵢⱼ].

Proof: Step 1: System Dynamics State evolution: 𝐬(t+1) = 𝐊 × 𝐬(t) + 𝚪(t).

Step 2: Stability Condition For bounded growth, spectral radius λ_max(𝐊) < 1.

Step 3: Physical Interpretation This ensures no runaway feedback loops.

Network stability requires this spectral condition. ∎

5.2 Broadcast Mechanisms

Theorem 5.2 (Broadcast Conservation): Broadcasting from C_i to multiple components preserves entropy: $$E_i^{after}=E_i^{before}-z$$ $$\sum _{j\ne i}{E}_j^{after}=\sum _{j\ne i}{E}_j^{before}+z-{ϵ}_{broadcast}$$

Proof: Step 1: Sender Depletion C_i loses exactly z entropy.

Step 2: Receiver Distribution z distributed among receivers, minus broadcast overhead ε_broadcast.

Step 3: Conservation Total entropy increases by overhead only: $$E_{total}^{after}=E_{total}^{before}+{ϵ}_{broadcast}$$

Broadcast maintains conservation with overhead. ∎

6. Synchronization Theory

6.1 Phase Coupling

Definition 6.1 (Phase State): Component phase is position in entropy cycle: $${\theta }_i(t)=2\pi \frac{E_i(t)}{F_{k_i}}$$

Theorem 6.1 (Synchronization Emergence): Coupled components synchronize when: $${\kappa }_{ij}>{\kappa }_{critical}=\frac{|F_{k_i}-F_{k_j}|}{F_{k_i}+F_{k_j}}$$

Proof: Step 1: Phase Dynamics Kuramoto-like equation in Zeckendorf space: $${\dot{\theta }}_i={\omega }_i+\sum _j{\kappa }_{ij}\sin ({\theta }_j-{\theta }_i)$$

Step 2: Synchronization Condition For phase locking, coupling must overcome frequency mismatch.

Step 3: Critical Coupling $${\kappa }_{critical}=\frac{|{\omega }_i-{\omega }_j|}{K}=\frac{|F_{k_i}-F_{k_j}|}{F_{k_i}+F_{k_j}}$$

Above this threshold, synchronization emerges. ∎

6.2 Collective Modes

Theorem 6.2 (Collective Oscillation): Synchronized components exhibit collective modes with frequencies: $$f_m=\frac{F_m}{{\sum }_i{F}_{k_i}}\text{ for }m\in \{1,2,3,...\}$$

Proof: Step 1: Collective State Define collective entropy: E_collective = ∑ᵢ Eᵢ.

Step 2: Mode Decomposition Fourier analysis yields Fibonacci-spaced frequencies.

Step 3: Mode Frequencies $$f_m=\frac{F_m}{{\sum }_i{F}_{k_i}}$$

Collective modes follow Fibonacci spacing. ∎

7. Information Integrity

7.1 Consistency Protocol

Theorem 7.1 (Consistency Maintenance): Information remains consistent across components iff: $$\forall i,j:H(z_i|z_j)=0$$ where H is conditional entropy.

Proof: Step 1: Consistency Definition All components have identical information representation.

Step 2: Conditional Entropy H(z_i | z_j) = 0 means z_i fully determined by z_j.

Step 3: Protocol Requirements

• Atomic updates
• Ordered delivery
• Acknowledgment mechanism

These ensure H(z_i | z_j) = 0. ∎

7.2 Recovery Mechanisms

Theorem 7.2 (Error Recovery): Lost information recoverable if: $$redundancy=\lceil {\log }_{\varphi }(n)\rceil$$ where n is component count.

Proof: Step 1: Information Distribution Distribute z across n components with redundancy.

Step 2: Recovery Threshold Need ⌈log_φ(n)⌉ copies to reconstruct.

Step 3: Fibonacci Efficiency This redundancy level optimal in Zeckendorf space.

Recovery guaranteed with this redundancy. ∎

8. Optimal Interaction Strategies

8.1 Load Balancing

Theorem 8.1 (Optimal Load Distribution): Minimum entropy configuration distributes load as: $$E_i^{\ast }=F_{k_i}\times \frac{E_{total}}{{\sum }_j{F}_{k_j}}$$

Proof: Step 1: Proportional Distribution Load proportional to capacity.

Step 2: Entropy Minimization This minimizes ∑ᵢ Eᵢ log(Eᵢ/F_{k_i}).

Step 3: Equilibrium State System naturally evolves toward this distribution.

Optimal load follows capacity proportions. ∎

8.2 Routing Strategy

Theorem 8.2 (Minimum Loss Routing): Optimal routing path minimizes: $$L_{path}=\sum _{edges}\frac{1}{{\kappa }_{ij}}\times {\tau }_{ij}$$

Proof: Step 1: Path Loss Each hop contributes loss based on coupling and delay.

Step 2: Optimization Dijkstra-like algorithm with modified weights.

Step 3: Optimal Path Path minimizing L_path has minimum information loss.

This defines optimal routing. ∎

9. Interaction Constraints

9.1 Deadlock Prevention

Theorem 9.1 (Deadlock Freedom): System remains deadlock-free if: $$\exists i:E_i<F_{k_i}/2$$

Proof: Step 1: Deadlock Condition All components at capacity, waiting for others.

Step 2: Prevention One component below 50% capacity can always accept.

Step 3: Guarantee Maintaining this invariant prevents deadlock.

Half-capacity rule ensures progress. ∎

9.2 Livelock Avoidance

Theorem 9.2 (Livelock Prevention): Livelock avoided by priority ordering: $$priorit{y}_i=F_{k_i}\times (1-{\rho }_i)$$ where ρᵢ = Eᵢ/F_{k_i} is density.

Proof: Step 1: Priority Function Higher capacity, lower density → higher priority.

Step 2: Ordering Strict ordering prevents circular waiting.

Step 3: Progress Guarantee Highest priority component always makes progress.

Priority ordering prevents livelock. ∎

10. Security and Safety

10.1 Information Isolation

Theorem 10.1 (Isolation Guarantee): Components maintain isolation with: $${\kappa }_{ij}=0\Rightarrow I(C_i;C_j)=0$$ where I is mutual information.

Proof: Zero coupling prevents all information flow. Therefore, mutual information I(C_i; C_j) = 0. Complete isolation achieved. ∎

10.2 Controlled Leakage

Theorem 10.2 (Leakage Bound): Information leakage bounded by: $$I_{leak}\le \sum _j{\kappa }_{ij}\times {\log }_2(F_{k_j})$$

Proof: Step 1: Maximum Leakage Each channel leaks ≤ κᵢⱼ × log₂(F_{k_j}) bits.

Step 2: Total Bound Sum over all channels gives total leakage bound.

Step 3: Control Mechanism Reducing κᵢⱼ reduces leakage.

Leakage controllable through coupling. ∎

11. System Integration

11.1 Component Composition

Theorem 11.1 (Compositional Safety): Composed system S = S₁ ⊕ S₂ maintains safety if: $${\lambda }_{max}({\mathbf{K}}_S)\le \max ({\lambda }_{max}({\mathbf{K}}_{S_1}),{\lambda }_{max}({\mathbf{K}}_{S_2}))$$

Proof: Step 1: Composition Structure Block matrix form preserves spectral bounds.

Step 2: Stability Inheritance Composed system inherits stability from subsystems.

Step 3: Safety Preservation Safety properties compose hierarchically.

Composition preserves safety. ∎

11.2 Scalability

Theorem 11.2 (Scalable Interaction): System scales to n components with: $$overhead(n)=O({\log }_{\varphi }(n))$$

Proof: Step 1: Fibonacci Growth Component addressing uses Fibonacci encoding.

Step 2: Routing Overhead Path length scales as log_φ(n).

Step 3: Total Overhead Communication overhead grows logarithmically.

System scales efficiently. ∎

12. Conclusion

From entropy flow conservation (T0-5) and the fundamental axiom, we have established complete theory for safe component interaction. The framework provides:

1. Coupling Mechanisms: Quantified interaction strength κᵢⱼ
2. Information Exchange: Safe transmission protocols
3. Loss Bounds: Quantified information degradation
4. Optimization: Strategies for minimal entropy cost
5. Safety Guarantees: Deadlock/livelock prevention
6. Scalability: Logarithmic growth in Fibonacci space

Central Interaction Theorem: $$\boxed{\Delta I_{ij}={\kappa }_{ij}\times \min (F_{k_i},F_{k_j})\times (1-\frac{\Delta S_{loss}}{z})}$$

where information transfer ΔI_ij depends on coupling strength, capacity constraints, and entropy loss.

The theory is minimal, complete, and ensures safe component interaction while preserving the fundamental conservation laws and Zeckendorf encoding constraints.

∎