8.1 Core Variables and Stability Model

Core Variable Definition

Liquidity Variables

P(t) = Total funds in the liquidity pool at time t I(t) = Fund inflow rate at time t O(t) = Fund outflow rate at time t N(t) = Number of active participants at time t

Resonance Dimension Parameters

Dᵢ = Investment amount of the i-th dimension (i=1,2,3,4) Tᵢ = Resonance cycle of the i-th dimension (1,7,15,30 days) Rᵢ = Resonance amplification rate of the i-th dimension (0.5%,5%,13%,30%)

Network Structure Parameters

Cᵢⱼ = Participant i's contribution to regional consensus for participant j Wₖ = Value weight of the k-th prosperity node α = Prosperity pool allocation ratio (20%)

Basic System Stability Model

Liquidity Balance Equation

The basic stability of the system can be described by the liquidity balance equation:

dP(t)/dt = I(t) - O(t)

Where:

I(t) = Σ[i=1 to 4] λᵢ(t) · Dᵢ · Nᵢ(t) (inflow rate) O(t) = Σ[i=1 to 4] μᵢ(t) · Dᵢ · (1 + Rᵢ) · Nᵢ(t - Tᵢ) (outflow rate)

Stability Conditions

Critical Stability Condition

The necessary condition for maintaining system stability is:

P(t) >= Σ[i=1 to 4] Σ[s=t to t+24h] Oᵢ(s)

That is, the liquidity pool funds must be able to cover all due payments in the next 24 hours.

Long-term Stability Condition

lim[T->∞] (1/T) · ∫[0 to T] [I(t) - O(t)]dt >= 0

Stability Analysis Under Participation Scale

In small-scale situations, the system exhibits exponential growth characteristics:

N(t) = N₀ · e^(r·t)

Where the growth rate is mainly driven by the regional consensus mechanism:

r = Σ[i=1 to 5] βᵢ · R_zone,i - δ

βᵢ are the expansion coefficients for each region, δ is the natural attrition rate.

Stability Analysis: At small scale, the system is highly dependent on new user growth, with high volatility.

Medium-scale System

The system enters the S-shaped growth phase, following the Logistic model:

dN/dt = rN(1 - N/K)

Where K is the system capacity upper limit, related to BSC network processing capacity.

Stability Characteristics:

• Growth rate gradually slows down but becomes more stable
• Phoenix restart mechanism begins to play a regulatory role
• Prosperity node mechanism provides additional stability

Large-scale System

The system enters a dynamic equilibrium state, where the number of participants oscillates around the equilibrium point:

N(t) = Neq + A · sin(ωt + φ) · e^(-γt)

Where Neq is the balanced number of participants, γ is the damping coefficient.

Stability Guarantee: The Phoenix restart mechanism ensures long-term stability of the system at large scale.