# Security Proofs

#### 1. Cross-layer Security

**Theorem 4.7 (Cross-layer Integrity):** The cross-layer security mechanism maintains integrity under concurrent executions with probability:

```
P(breach) ≤ ∑ᵢ(1/2^λᵢ)

Where:

λᵢ: Security parameter for layer i
```

**Proof:** Through hybrid argument, Let H₀,...,Hₙ be hybrid executions:&#x20;

```
|P(H₀) - P(Hₙ)| ≤ ∑ᵢ|P(Hᵢ) - P(Hᵢ₋₁)|
```

Each transition probability is bounded by 1/2^λᵢ.&#x20;

#### 2. Computational Complexity

The security mechanisms demonstrate the following complexity characteristics:

Validation Complexity:

```
Time: O(log n) average case
      O(n·log n) worst case
Space: O(log n) per validation
```

Attack Prevention:

```
Detection: O(1) amortized
Response: O(log n) average case
Recovery: O(n) worst case
```
