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# Dynamic Supply Adjustment with Single Token (DSAST) Workflow

## Token Details
- **Name**: Elastic Indemnity Token (EIT)
- **Initial Supply**: S₀ = 1,000,000,000 EIT

## 1. Performance Score Calculation

Let P_i be the performance score of validator i, where 0 ≤ P_i ≤ 100.
Let n be the total number of validators.

The average performance score P̄ on day t is calculated as:

P̄_t = (∑_{i=1}^n P_i) / n

graph LR
    A[Validator 1 Score] --> E[Calculate Average]
    B[Validator 2 Score] --> E
    C[Validator 3 Score] --> E
    D[Validator n Score] --> E
    E --> F[Average Performance Score]

## 2. Dynamic Supply Adjustment

Let S_t be the token supply on day t.
The daily supply adjustment factor α_t is defined as:

α_t = {
1.001 if P̄_t > 80
0.999 if P̄_t < 60
1 otherwise
}

The token supply for day t+1 is calculated as:

S_{t+1} = S_t * α_t

Induction proof for supply bounds:
- Base case: S₀ = 1,000,000,000
- Inductive step: Assume S_k is bounded. Then:
0.999 * S_k ≤ S_{k+1} ≤ 1.001 * S_k

Therefore, the supply remains bounded for all t.

graph TD
    A[Average Performance Score] --> B{Score > 80?}
    B -->|Yes| C[Increase Supply by 0.1%]
    B -->|No| D{Score < 60?}
    D -->|Yes| E[Decrease Supply by 0.1%]
    D -->|No| F[No Change in Supply]
    C --> G[New Supply]
    E --> G
    F --> G

## 3. Staking Reward Calculation

Let R be the base annual reward rate (5% or 0.05).
Let A_i be the amount staked by validator i.
Let M_i be the performance multiplier for validator i, defined as:

M_i = P_i / 50

The daily reward for validator i on day t is calculated as:

R_i_t = (A_i * R / 365) * M_i * α_t

graph TD
    A[Staked Amount] --> E[Calculate Base Reward]
    B[Base Rate] --> E
    E --> F[Apply Performance Multiplier]
    C[Performance Score] --> F
    F --> G[Apply Supply Change Factor]
    D[Supply Change Factor] --> G
    G --> H[Daily Reward]

## 4. Reward Distribution

Let Δ_t be the total rewards distributed on day t:

Δ_t = ∑_{i=1}^n R_i_t

If α_t > 1 (supply increasing):
Mint Δ_t new tokens for distribution

If α_t < 1 (supply decreasing):
Distribute Δ_t tokens from the reserve pool

Let RP_t be the reserve pool balance on day t.
RP_{t+1} = RP_t - Δ_t (when α_t < 1)

Assumption check: Ensure RP_t ≥ Δ_t at all times to prevent insufficient funds for reward distribution.

graph TD
    A[Calculate Total Rewards] --> B{Supply Increasing?}
    B -->|Yes| C[Mint New Tokens]
    B -->|No| D[Distribute from Reserve Pool]
    C --> E[Distribute Rewards]
    D --> E
    E --> F[Update Validator Balances]

## 5. Reserve Pool Management

Initial Reserve Pool: RP₀ = 0.2 * S₀ = 200,000,000 EIT

Daily Reserve Pool update:
RP_{t+1} = RP_t + (S_{t+1} - S_t) - Δ_t

This ensures that supply decreases add to the reserve pool, while rewards are deducted.

graph TD
    A[Current Reserve Pool] --> E[Update Reserve Pool]
    B[Supply Change] --> E
    C[Distributed Rewards] --> E
    E --> F[New Reserve Pool Balance]

## 6. Validator Performance Update

Each validator's performance score is updated daily based on their activities and adherence to protocol rules. The exact scoring mechanism would depend on the specific requirements of the indemnification system.

graph TD
    A[Validator Activities] --> B[Update Performance Score]
    B --> C[New Performance Score]
    C --> D[Affects Next Day's Rewards]

## 7. Governance and Parameter Adjustments

The system should include a governance mechanism to adjust parameters such as the base reward rate R, performance score thresholds, and supply adjustment factors. This ensures the system can adapt to changing conditions.

graph TD
    A[Governance Proposal] --> B[Voting Period]
    B --> C{Proposal Passed?}
    C -->|Yes| D[Implement Parameter Changes]
    C -->|No| E[Maintain Current Parameters]

## Formal System Definition

Let t ∈ ℕ represent days since system inception.
Let V = {1, ..., n} be the set of validators.

### State Variables:
- S_t: Token supply on day t
- RP_t: Reserve pool balance on day t
- P_i_t: Performance score of validator i on day t
- A_i_t: Amount staked by validator i on day t

### System Parameters:
- R: Base annual reward rate
- θ_high: Upper performance threshold (80)
- θ_low: Lower performance threshold (60)

### System Equations:
1. P̄_t = (∑_{i∈V} P_i_t) / |V|
2. α_t = 1.001^(P̄_t > θ_high) * 0.999^(P̄_t < θ_low)
3. S_{t+1} = S_t * α_t
4. M_i_t = P_i_t / 50
5. R_i_t = (A_i_t * R / 365) * M_i_t * α_t
6. Δ_t = ∑_{i∈V} R_i_t
7. RP_{t+1} = RP_t + (S_{t+1} - S_t) - Δ_t

### Invariants to maintain:
1. ∀t: RP_t ≥ 0
2. ∀t, ∀i ∈ V: 0 ≤ P_i_t ≤ 100
3. ∀t: S_t > 0

## Potential faulty assumptions and mitigations:

1. **Assumption**: The reserve pool will always have sufficient funds for reward distribution when the supply is decreasing.
**Mitigation**: Implement a minimum reserve pool balance threshold. If this threshold is approached, gradually reduce rewards or mint a small amount of new tokens to prevent reserve pool depletion.

2. **Assumption**: The performance scoring system is resistant to manipulation.
**Mitigation**: Implement additional checks and balances, such as peer review of performance scores or using external data sources to validate performance claims.

3. **Assumption**: The supply adjustment mechanism will maintain price stability.
**Mitigation**: Monitor price volatility and implement circuit breakers or more granular supply adjustments if needed.

4. **Assumption**: Validators will always have an incentive to perform well.
**Mitigation**: Regularly review and adjust the reward structure to ensure it remains competitive and encourages desired behaviors.

By following this detailed workflow and considering the potential faulty assumptions, we can create a robust and adaptive system for dynamic supply adjustment and performance-based rewards in the indemnification protocol.