Comparative Analysis

1. Performance Comparison Framework

The comparative analysis employs a formal evaluation framework:

C(s₁,s₂) = ∑ωᵢ·M(s₁,s₂,mᵢ)

Where:

s₁,s₂: Systems under comparison
mᵢ: Individual metrics
ωᵢ: Metric weights

2. Benchmark Results

Throughput Comparison:

Metric	HASC	Polkadot	Cosmos	Solana
Core TPS	2,000	1,000	1,500	1,800
Peak TPS	100,000	50,000	40,000	65,000
Latency (s)	2-8	6-12	5-10	3-8
Energy Eff.	95%	75%	80%	70%

Security Metrics:

Metric	HASC	Traditional PoS	Hybrid
Byzantine Tol.	33%	33%	33%
Slash Eff.	99.9%	95%	97%
Recovery Time	2.5s	10s	7s

3. Resource Efficiency Analysis

The system demonstrates significant improvements in resource utilization:

Theorem 5.4 (Resource Optimization): The resource utilization function R(u) achieves optimal efficiency when:

∇R(u) = 0 subject to:
∑uᵢ ≤ U_max
uᵢ ≥ 0 ∀i

Where:

u: Resource allocation vector
U_max: Maximum available resources

Proof: Through Karush-Kuhn-Tucker conditions:

∂L/∂uᵢ = 0
μᵢ·uᵢ = 0
μᵢ ≥ 0

Where L is the Lagrangian function