T0-6: System Component Interaction Theory - Formal Specification
Formal System Definition
Language L₆
- Constants: 0, 1, F₁, F₂, F₃, ...
- Variables: C₁, C₂, ..., Cₙ (components)
- Functions: κ: C × C → [0,1], τ: C × C → ℕ, Φ: C × C × T → Z
- Relations: ↔ (interaction), → (information flow), ≤ (capacity ordering)
- Operators: ⊕ (composition), ⊗ (coupling), ∇ (gradient)
Axioms
A6.1 (Inherited from T0-5):
∀S closed: ∑ᵢ Eᵢ(t) + ∑_{flows} = ∑ᵢ Eᵢ(0) + ∫₀ᵗ Γ(τ)dτ
A6.2 (Coupling Symmetry):
∀i,j: κᵢⱼ = κⱼᵢ
A6.3 (Coupling Bounds):
∀i,j: 0 ≤ κᵢⱼ ≤ 1
A6.4 (Information Quantization):
∀ information z: z ∈ {0, F₁, F₂, F₃, ...}
Core Definitions
D6.1: Interaction Channel
𝒾ᵢⱼ ≡ (Cᵢ ↔ Cⱼ, κᵢⱼ, τᵢⱼ)
where:
- Cᵢ, Cⱼ ∈ Components
- κᵢⱼ ∈ [0,1]
- τᵢⱼ ∈ ℕ
D6.2: Information Packet
Pᵢⱼ(t) ≡ (z, ε, φ)
where:
- z ∈ Zeckendorf
- ε ≥ 0
- φ ∈ {i→j, j→i}
D6.3: Coupling Strength
κᵢⱼ ≡ min(Fₖᵢ, Fₖⱼ) / max(Fₖᵢ, Fₖⱼ)
D6.4: System State Vector
σ(t) ≡ (E₁(t), E₂(t), ..., Eₙ(t))
where ∀i: 0 ≤ Eᵢ(t) ≤ Fₖᵢ - 1
Fundamental Theorems
T6.1: Safe Exchange Condition
Safe(Cᵢ → Cⱼ, z) ⟺
(NoConsecutiveOnes(sᵢ - z) ∧
NoConsecutiveOnes(sⱼ + z) ∧
z ≤ κᵢⱼ × min(Fₖᵢ, Fₖⱼ))
Proof Structure:
- Define pre-states: sᵢ, sⱼ in Zeckendorf form
- Apply exchange operation
- Verify no-11 preservation
- Check capacity constraints
- Confirm entropy balance
T6.2: Bandwidth Theorem
Bᵢⱼ = (κᵢⱼ × min(Fₖᵢ, Fₖⱼ)) / τᵢⱼ
Proof Structure:
- Maximum transfer per unit time = min(Fₖᵢ, Fₖⱼ)
- Effective transfer = κᵢⱼ × maximum
- Rate = effective / delay
T6.3: Transmission Loss
ΔS_loss = z × (1 - κᵢⱼ) × log₂(τᵢⱼ + 1)
Proof Structure:
- Coupling imperfection: (1 - κᵢⱼ)
- Temporal degradation: log₂(τᵢⱼ + 1)
- Combined loss proportional to z
T6.4: Error Propagation Bound
ε_max = F_⌊log_φ(z)⌋ × (1 - κᵢⱼ)
where φ = (1+√5)/2
Proof Structure:
- Decompose z in Zeckendorf
- Identify largest Fibonacci term
- Apply coupling error factor
Coupling Dynamics
T6.5: Dynamic Coupling Evolution
κᵢⱼ(t+1) = κᵢⱼ(t) + α × (Φᵢⱼ(t)/F_min) × (1 - κᵢⱼ(t))
where α ∈ (0,1)
Proof Structure:
- Reinforcement from successful flows
- Saturation at κᵢⱼ = 1
- Fibonacci normalization
T6.6: Optimal Coupling
κᵢⱼ* = √((Fₖᵢ × Fₖⱼ)/(Fₖᵢ + Fₖⱼ)²)
Proof Structure:
- Define loss function L = ∑ᵢⱼ ΔS_loss(κᵢⱼ)
- Compute ∂L/∂κᵢⱼ = 0
- Verify minimum condition
Network Properties
T6.7: Network Stability
Stable(G) ⟺ λ_max(𝐊) < 1
where 𝐊 = [κᵢⱼ]ₙₓₙ
Proof Structure:
- System dynamics: s(t+1) = 𝐊·s(t) + Γ(t)
- Stability requires bounded growth
- Spectral radius condition
T6.8: Broadcast Conservation
Broadcast(Cᵢ → {Cⱼ}, z):
Eᵢ^after = Eᵢ^before - z
∑ⱼ≠ᵢ Eⱼ^after = ∑ⱼ≠ᵢ Eⱼ^before + z - ε_broadcast
Proof Structure:
- Sender depletion
- Receiver accumulation
- Overhead accounting
Synchronization
T6.9: Critical Coupling for Synchronization
Sync(Cᵢ, Cⱼ) ⟺ κᵢⱼ > |Fₖᵢ - Fₖⱼ|/(Fₖᵢ + Fₖⱼ)
Proof Structure:
- Phase dynamics in Zeckendorf space
- Kuramoto-like equations
- Critical threshold derivation
T6.10: Collective Mode Frequencies
f_m = F_m / ∑ᵢ Fₖᵢ for m ∈ ℕ
Proof Structure:
- Define collective entropy
- Fourier decomposition
- Fibonacci frequency spacing
Information Integrity
T6.11: Consistency Condition
Consistent({Cᵢ}) ⟺ ∀i,j: H(zᵢ | zⱼ) = 0
Proof Structure:
- Define conditional entropy
- Zero condition implies determinism
- Protocol requirements
T6.12: Recovery Redundancy
Recoverable(z, n) ⟺ redundancy ≥ ⌈log_φ(n)⌉
Proof Structure:
- Information distribution strategy
- Recovery threshold analysis
- Fibonacci optimal encoding
Optimization
T6.13: Optimal Load Distribution
Eᵢ* = Fₖᵢ × (E_total / ∑ⱼ Fₖⱼ)
Proof Structure:
- Proportional to capacity
- Entropy minimization
- Natural equilibrium
T6.14: Minimum Loss Routing
Path* = argmin_p ∑_{(i,j)∈p} (1/κᵢⱼ) × τᵢⱼ
Proof Structure:
- Path loss function
- Graph optimization
- Dijkstra variant
Safety Properties
T6.15: Deadlock Freedom
DeadlockFree(S) ⟺ ∃i: Eᵢ < Fₖᵢ/2
Proof Structure:
- Deadlock characterization
- Half-capacity invariant
- Progress guarantee
T6.16: Livelock Prevention
LivelockFree(S) ⟺ priority_i = Fₖᵢ × (1 - Eᵢ/Fₖᵢ) forms strict order
Proof Structure:
- Priority function definition
- Ordering prevents cycles
- Progress guarantee
Security
T6.17: Isolation Guarantee
κᵢⱼ = 0 ⟹ I(Cᵢ; Cⱼ) = 0
Proof Structure:
- Zero coupling blocks flow
- Mutual information vanishes
- Complete isolation
T6.18: Leakage Bound
I_leak ≤ ∑ⱼ κᵢⱼ × log₂(Fₖⱼ)
Proof Structure:
- Per-channel leakage
- Sum over channels
- Controllable through coupling
Composition
T6.19: Compositional Safety
S = S₁ ⊕ S₂ ⟹ λ_max(𝐊_S) ≤ max(λ_max(𝐊_{S₁}), λ_max(𝐊_{S₂}))
Proof Structure:
- Block matrix structure
- Spectral bound preservation
- Safety inheritance
T6.20: Scalability
Overhead(n) = O(log_φ(n))
Proof Structure:
- Fibonacci addressing
- Routing path length
- Logarithmic growth
Completeness
Metatheorem: Theory Completeness
T0-6 is complete for component interaction:
1. All safe exchanges characterized (T6.1)
2. All transmission properties bounded (T6.2-T6.4)
3. All coupling dynamics specified (T6.5-T6.6)
4. All network behaviors determined (T6.7-T6.10)
5. All safety properties guaranteed (T6.15-T6.18)
Consistency
Consistency with Prior Theory
T0-6 ∧ T0-5 ∧ T0-4 ∧ T0-3 ∧ T0-2 ∧ T0-1 is consistent:
- Preserves entropy conservation (T0-5)
- Maintains Zeckendorf encoding (T0-4)
- Respects no-11 constraint (T0-3)
- Uses Fibonacci capacities (T0-2)
- Operates in binary universe (T0-1)
Central Result
Master Equation for Component Interaction
ΔIᵢⱼ = κᵢⱼ × min(Fₖᵢ, Fₖⱼ) × (1 - ΔS_loss/z)
This equation fully characterizes information transfer between components, incorporating:
- Coupling strength (κᵢⱼ)
- Capacity constraints (min function)
- Entropy loss factor (1 - ΔS_loss/z)
The theory provides a complete, minimal framework for safe component interaction in self-referential systems with Fibonacci-quantized capacities and Zeckendorf encoding.
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