staking-reward-streamer/docs/MathSpec.md

Table of Contents

Mathematical Specification of Staking Protocol

Important

All values in this document are expressed as unsigned integers.

Summary

Constants

Symbol	Source	Value	Unit	Description
SCALE_{FACTOR}		\pu{1 \times 10^{18}}	(1)	Scaling factor to maintain precision in calculations.
M_{MAX}		\pu{4 \mathrm{(1)}}	(1)	Maximum multiplier of annual percentage yield.
\mathtt{APY}		100	percent	Annual percentage yield for multiplier points.
\mathsf{MPY}	M_{MAX} \times \mathtt{APY}	400	percent	Multiplier points accrued maximum percentage yield.
\mathsf{MPY}^\mathit{abs}	100 + (2 \times M_{\text{MAX}} \times \mathtt{APY})	900	percent	Multiplier points absolute maximum percentage yield.
T_{RATE}	(minimal blocktime)	2 (chain dependent)	seconds	The accrue rate period of time over which multiplier points are calculated.
T_{DAY}		86400	seconds	One day.
T_{YEAR}	\lfloor365.242190 \times T_{DAY}\rfloor	31556925	seconds	One (mean) tropical year.
A_{MIN}	\lceil\tfrac{T_{YEAR} \times 100}{T_{RATE} \times \mathtt{APY}}\rceil	2629744	tokens per period	Minimal value to generate 1 multiplier point in the accrue rate period (T_{RATE}). (A_{MIN} \propto T_{RATE})
A_{MAX}	\frac{2^{256} - 1}{\text{APY} \times T_{\text{RATE}}}		tokens per period	Maximum value to not overflow unsigned integer of 256 bits.
T_{MIN}	90 \times T_{DAY}	7776000	seconds	Minimum lockup period, equivalent to 90 days.
T_{MAX}	M_{MAX} \times T_{YEAR}	126227700	seconds	Maximum of lockup period.

Variables

System and User Parameters

\Delta a\rightarrow Amount Difference

Difference in amount, can be either reduced or increased depending on context.

---

\Delta t\rightarrow Time Difference of Last Accrual

---

The time difference defined as:

\Delta t = t_{now} - t_{last}, \quad \text{where}  \Delta t > T_{RATE}

---

t_{lock}\rightarrow Time Lock Duration

A user-defined duration for which a_{bal} remains locked.

---

t_{now}\rightarrow Time Now

The current timestamp seconds since the Unix epoch (January 1, 1970).

---

t_{lock, \Delta}\rightarrow Time Lock Remaining Duration

Seconds a_{bal} remains locked, expressed as:

\begin{align} &t_{lock, \Delta} = max(t_{lock,end},t_{now}) - t_{now}  \\
\text{ where: }\quad & t_{lock, \Delta} = 0\text{ or }T_{MIN} \le t_{lock, \Delta} \le (M_{MAX} \times T_{YEAR})\end{align}

---

State Related

a_{bal}\rightarrow Amount of Balance

Amount of tokens in balance, where a_{bal} \ge A_{MIN}.

---

t_{lock,end}\rightarrow Time Lock End

Timestamp marking the end of the lock period, its state can be defined as:

t_{lock,end} = \max(t_{now}, t_{lock,end}) + t_{lock}

The value of t_{lock,end} can be updated only within the functions:

• \mathcal{f}^{stake}(\mathbb{Acc}, \Delta a, \Delta t_{lock});
• \mathcal{f}^{lock}(\mathbb{Acc}, \Delta t_{lock});

---

t_{last}\rightarrow Time of Accrual

Timestamp of the last accrued time, its state can be defined as:

t_{last} = t_{now}

The value of t_{last} is updated by all functions that change state:

• f^{accrue}(\mathbb{Acc}, a_{bal},\Delta t),
• \mathcal{f}^{stake}(\mathbb{Acc}, \Delta a, \Delta t_{lock});
• \mathcal{f}^{lock}(\mathbb{Acc}, \Delta t_{lock});
• \mathcal{f}^{unstake}(\mathbb{Acc}, \Delta a);

---

mp_\mathcal{M}\rightarrow Maximum Multiplier Points

Maximum value that mp_\Sigma can reach.

Relates as mp_\mathcal{M} \propto a_{bal} \cdot (t_{lock} + \mathsf{MPY}).

Altered by functions that change the account state:

• \mathcal{f}^{stake}(\mathbb{Acc}, \Delta a, \Delta t_{lock});
• \mathcal{f}^{lock}(\mathbb{Acc}, \Delta t_{lock});
• \mathcal{f}^{unstake}(\mathbb{Acc}, \Delta a).

It's state can be expressed as the following state changes:

Increase in Balance and Lock

\begin{aligned}
mp_\mathcal{M} &= mp_\mathcal{M} + mp_\mathcal{A}(\Delta a, M_{MAX} \times T_{YEAR}) \\
&\quad + mp_\mathcal{B}(\Delta a, t_{lock,\Delta} + t_{lock}) \\
&\quad + mp_\mathcal{B}(a_{bal}, t_{lock}) \\
&\quad + mp_\mathcal{I}(\Delta a)
\end{aligned}

Increase in Balance only

\begin{aligned}
mp_\mathcal{M} &= mp_\mathcal{M} + mp_\mathcal{A}(\Delta a, M_{MAX} \times T_{YEAR}) \\
&\quad + mp_\mathcal{B}(\Delta a, t_{lock,\Delta}) \\
&\quad + mp_\mathcal{I}(\Delta a)
\end{aligned}

Increase in Lock only

mp_\mathcal{M} = mp_\mathcal{M} + mp_\mathcal{B}(a_{bal}, t_{lock})

Decrease in Balance

mp_\mathcal{M} = mp_\mathcal{M} - mp_\mathcal{R}(mp_\mathcal{M}, a_{bal}, \Delta a)

---

mp_{\Sigma}\rightarrow Total Multiplier Points

Altered by all functions that change state:

• \mathcal{f}^{stake}( \mathbb{Acc}, Delta a, t_{lock}) \longrightarrow Stake Amount With Lock
• \mathcal{f}^{lock}(\mathbb{Acc}, t_{lock}) \longrightarrow Increase Lock;
• \mathcal{f}^{unstake}( \mathbb{Acc}, Delta a)\longrightarrow Unstake Amount Unlocked;
• \mathcal{f}^{accrue}(\mathbb{Acc}) \longrightarrow Accrue Multiplier Points.

The state can be expressed as the following state changes:

For every T_{RATE}

mp_{\Sigma} = min(\mathcal{f}mp_\mathcal{A}(a_{bal},\Delta t), mp_\mathcal{M} -  mp_\Sigma)

Increase in Balance and Lock

\begin{aligned}
mp_{\Sigma} &= mp_{\Sigma} + mp_\mathcal{B}(\Delta a, t_{lock, \Delta} + t_{lock}) \\
&\quad + mp_\mathcal{B}(a_{bal}, t_{lock}) \\
&\quad + mp_\mathcal{I}(\Delta a)
\end{aligned}

Increase in Balance only

mp_{\Sigma} = mp_{\Sigma} + mp_\mathcal{B}(\Delta a, t_{lock, \Delta}) + mp_\mathcal{I}(\Delta a)

Increase in Lock only

mp_{\Sigma} = mp_{\Sigma} + mp_\mathcal{B}(a_{bal}, t_{lock})

Decrease in Balance

mp_{\Sigma} = mp_{\Sigma} - mp_\mathcal{R}(mp_{\Sigma}, a_{bal}, \Delta a)

---

\mathbb{Acc}\rightarrow Account Storage Schema

Defined as following:

\begin{gather}
	\mathbb{Acc}  \\
	\overbrace{
		\begin{align}
			a_{bal} & : \text{balance},  \\
			t_{lock,end} & : \text{lock end}, \\
			t_{last} & : \text{last accrual}, \\
			mp_\Sigma & : \text{total MPs}, \\
			mp_\mathcal{M} & : \text{maximum MPs}
		\end{align}
	}
\end{gather}

---

\mathbb{Acc} \cdot W \rightarrow Account Weight

The account weight for an individual account j combines its staked balance and accumulated Multiplier Points. This weight determines the proportion of rewards the account is entitled to.

\mathbb{Acc} \cdot W = \mathbb{Acc} \cdot a_{\text{bal}} + \mathbb{Acc} \cdot mp_{\Sigma}

Where:

• \mathbb{Acc} \cdot a_{\text{bal}}: Staked balance of account j.
• \mathbb{Acc} \cdot mp_{\Sigma}: Total Multiplier Points of account j.

\mathbb{System}\rightarrow System Storage Schema

Defined as following:

\begin{gather}
	 \mathbb{Sys}  \\
	\overbrace{
		\begin{align}
			\mathbb{Acc}\mathrm{[]} & : \text{accounts}, \\
			a_{bal} & : \text{total staked}, \\
			mp_\Sigma & : \text{MP supply}, \\
			mp_\mathcal{M} & : \text{MP supply max}
		\end{align}
	}
\end{gather}

---

\mathbb{Sys} \cdot W \rightarrow Total Weight

The total weight of the system is the aggregate of all staked tokens and Multiplier Points (MP) across all accounts. It serves as the denominator in reward distribution calculations.

\mathbb{Sys} \cdot W = \mathbb{Sys} \cdot a_{\text{bal}} + \mathbb{Sys} \cdot mp_{\Sigma}

Where:

• \mathbb{Sys} \cdot a_{\text{bal}}: Total tokens staked in the system.
• \mathbb{Sys} \cdot mp_{\Sigma}: Total Multiplier Points accumulated in the system.

---

R_i \rightarrow Cumulative Reward Index

The reward index represents the cumulative rewards distributed per unit of total weight (staked balance plus Multiplier Points) in the system. It is a crucial component for calculating individual rewards.

R_i = R_i + \left( \frac{R_{\text{new}} \times SCALE_{FACTOR}}{\mathbb{Sys} \cdot W} \right)

Where:

• R_{\text{new}}: The amount of new rewards added to the system.
• \mathbb{Sys} \cdot W: The total weight in the system, calculated as the sum of all staked balances and total Multiplier Points.
• SCALE_{FACTOR}: Scaling factor to maintain precision.

---

Pure Mathematical Functions

Note

This function definitions represent direct mathematical input -> output methods, which don't change state.

\mathcal{f}{mp_\mathcal{I}}(\Delta a) \longrightarrow Initial Multiplier Points

Calculates the initial multiplier points based on the balance change \Delta a. The result is equal to the amount of balance added.

\boxed{
	\begin{equation}
		\mathcal{f}{mp_\mathcal{I}}(\Delta a) = \Delta a
	\end{equation}
}

Where

• $\Delta a$: Represents the change in balance.

---

\mathcal{f}{mp_\mathcal{A}}(a_{bal}, \Delta t) \longrightarrow Accrue Multiplier Points

Calculates the accrued multiplier points over a time period $\Delta t$, based on the account balance $a_{bal}$ and the annual percentage yield \mathtt{APY}.

\boxed{
	\begin{equation}
		\mathcal{f}mp_\mathcal{A}(a_{bal}, \Delta t) = \dfrac{a_{bal} \times \Delta t \times \mathtt{APY}}{100 \times T_{YEAR}}
	\end{equation}
}

Where

• $a_{bal}$: Represents the current account balance.
• $\Delta t$: The time difference or the duration over which the multiplier points are accrued, expressed in the same time units as the year (typically days or months).
• $T_{YEAR}$: A constant representing the duration of a full year, used to normalize the time difference $\Delta t$.
• $\mathtt{APY}$: The Annual Percentage Yield (APY) expressed as a percentage, which determines how much the balance grows over a year.

---

\mathcal{f}{mp_\mathcal{B}}(\Delta a, t_{lock}) \longrightarrow Bonus Multiplier Points

Calculates the bonus multiplier points earned when a balance $\Delta a$ is locked for a specified duration t_{lock}. It is equivalent to the \mathcal{f}{mp_\mathcal{A}}(a_{bal}, \Delta t) \longrightarrow Accrue Multiplier Points but specifically applied in the context of a locked balance, using \Delta t as t_{lock}.

\begin{aligned}
	&\mathcal{f}mp_\mathcal{B}(\Delta a, t_{lock})  = \mathcal{f}mp_\mathcal{A}(\Delta a, t_{lock}) \\
	&\boxed{
		\begin{equation}
			\mathcal{f}mp_\mathcal{B}(\Delta a, t_{lock})  = \dfrac{\Delta a \times t_{lock} \times \mathtt{APY}}{100 \times T_{YEAR}}
		\end{equation}
	}
\end{aligned}

Where:

• $\Delta a$: Represents the amount of the balance that is locked.
• $t_{lock}$: The duration for which the balance $\Delta a$ is locked, measured in units of seconds.
• $T_{YEAR}$: A constant representing the length of a year, used to normalize the lock period $t_{lock}$ as a fraction of a full year.
• $\mathtt{APY}$: The Annual Percentage Yield (APY), expressed as a percentage, which indicates the yearly interest rate applied to the locked balance.

---

\mathcal{f}{mp_\mathcal{R}}(mp, a_{bal}, \Delta a) \longrightarrow Reduce Multiplier Points

Calculates the reduction in multiplier points when a portion of the balance $\Delta a$ is removed from the total balance $a_{bal}$. The reduction is proportional to the ratio of the removed balance to the total balance, applied to the current multiplier points $mp$.

\boxed{
	\begin{equation}
		\mathcal{f}{mp_\mathcal{R}}(mp, a_{bal}, \Delta a) = \dfrac{mp \times \Delta a}{ a_{bal}}
	\end{equation}
}

Where:

• $mp$: Represents the current multiplier points.
• $a_{bal}$: The total account balance before the removal of $\Delta a$.
• $\Delta a$: The amount of balance being removed or deducted.

---

State Functions

These function definitions represent methods that modify the state of both $\mathbb{System}$ and $\mathbb{Acc}$. They perform various pure mathematical operations to implement the specified state changes, affecting either the system as a whole and the individual account states.

\mathcal{f}^{stake}(\mathbb{Acc},\Delta a, t_{lock}) \longrightarrow Stake Amount With Lock

Purpose: Allows a user to stake an amount \Delta a with an optional lock duration t_{lock}.

---
title: Stake Storage Access Flowchart
---
flowchart LR
	BonusMP{{Bonus MP}}
	InitialMP{{Initial MP}}
	Balance
	LockEnd[Lock End]
	TotalMP[Total MPs]
	MaxMP[Maximum MPs]
	FBonusMP{Calc Bonus MP}
	FMaxMP{Calc Max Accrue MP}
	M_MAX([MAX MULTIPLIER])
    Balance --> InitialMP
    Balance --> FMaxMP
    M_MAX --> FMaxMP
    InitialMP --> TotalMP
    InitialMP --> MaxMP
    BonusMP --> TotalMP
    BonusMP --> MaxMP
    FMaxMP --> MaxMP
    LockEnd --> FBonusMP
    Balance --> FBonusMP
    FBonusMP --> BonusMP

Steps

Accrue Existing Multiplier Points (MPs)

Call the \mathcal{f}^{accrue}(\mathbb{Account}) \longrightarrow Accrue Multiplier Points function to update MPs and last accrual time.

Calculate the New Remaining Lock Period (\Delta t_{lock})

\Delta t_{lock} = max(\mathbb{Acc} \cdot t_{lock,end}, t_{now}) + t_{lock} - t_{now}

Verify Constraints

Ensure new balance (a_{bal} + \Delta a) meets the minimum amount (A_{MIN}):

\mathbb{Acc} \cdot a_{bal} + \Delta a > A_{MIN}

Ensure the New Remaining Lock Period (\Delta t_{lock}) is within Allowed Limits

\Delta t_{lock} = 0 \lor T_{MIN} \le \Delta t_{lock} \le T_{MAX}

Calculate Increased Bonus MPs

For the new amount (\Delta a) with the New Remaining Lock Period (\Delta t_{lock}):

\Delta \hat{mp}^\mathcal{B} = \mathcal{f}mp_\mathcal{B}(\Delta a, \Delta t_{lock})

For extending the lock (t_{lock}) on the existing balance (\mathbb{Acc} \cdot a_{bal}):

\Delta \hat{mp}^\mathcal{B} = \Delta \hat{mp}^\mathcal{B} + \mathcal{f}mp_\mathcal{B}(\mathbb{Acc} \cdot a_{bal}, t_{lock})

Calculate Increased Maximum MPs (\Delta mp_\mathcal{M})

\Delta mp_\mathcal{M} = \mathcal{f}mp_\mathcal{I}(\Delta a) + \Delta \hat{mp}^\mathcal{B} + \mathcal{f}mp_\mathcal{A}(\Delta a, M_{MAX} \times T_{YEAR})

Calculate Increased Total MPs (\Delta mp_\Sigma)

\Delta mp_\Sigma = \mathcal{f}mp_\mathcal{I}(\Delta a) + \Delta \hat{mp}^\mathcal{B}

Verify Constraints

Ensure the New Maximum MPs (\mathbb{Acc} \cdot mp_\mathcal{M} + \Delta mp_\mathcal{M}) is within the Absolute Maximum MPs:

\mathbb{Acc} \cdot mp_\mathcal{M} + \Delta mp_\mathcal{M} \le \frac{a_{bal} \times \mathsf{MPY}^\mathit{abs}}{100}

Update account State

Maximum MPs:

\mathbb{Acc} \cdot mp_\mathcal{M} = \mathbb{Acc}\cdot mp_\mathcal{M} + \Delta mp_\mathcal{M}

Total MPs:

\mathbb{Acc} \cdot mp_\Sigma = \mathbb{Acc} \cdot mp_\Sigma + \Delta mp_\Sigma

Balance:

\mathbb{Acc} \cdot a_{bal} = \mathbb{Acc} \cdot a_{bal} + \Delta a

Lock end time:

\mathbb{Acc} \cdot t_{lock,end} = max(\mathbb{Acc} \cdot t_{lock,end}, t_{now}) + t_{lock}

Update System State

Maximum MPs:

\mathbb{Sys} \cdot mp_\mathcal{M} = \mathbb{Sys} \cdot mp_\mathcal{M} + \Delta mp_\mathcal{M}

Total MPs:

\mathbb{Sys} \cdot mp_\Sigma = \mathbb{Sys} \cdot mp_\Sigma + \Delta mp_\Sigma

Total staked amount:

\mathbb{Sys} \cdot a_{bal} = \mathbb{Sys} \cdot a_{bal} + \Delta a

---

\mathcal{f}^{lock}(\mathbb{Acc}, t_{lock}) \longrightarrow Increase Lock

Note

Equivalent to:

\mathcal{f}_{stake}(\mathbb{Acc},0, t_{lock}) 

Purpose: Allows a user to lock the \mathbb{Acc} \cdot a_{bal} with a lock duration t_{lock}.

---
title: Lock Storage Access Flowchart
---
flowchart LR
	BonusMP{{Bonus MP}}
	LockEnd[Lock End]
	TotalMP[Total MPs]
	MaxMP[Maximum MPs]
	FBonusMP{Calc Bonus MP}
    BonusMP --> TotalMP
    BonusMP --> MaxMP
    LockEnd --> FBonusMP
    Balance --> FBonusMP
    FBonusMP --> BonusMP

Steps

Accrue Existing Multiplier Points (MPs)

Call the \mathcal{f}^{accrue}(\mathbb{Account}) \longrightarrow Accrue Multiplier Points function to update MPs and last accrual time.

Calculate the New Remaining Lock Period (\Delta t_{lock})

\Delta t_{lock} = max(\mathbb{Acc} \cdot t_{lock,end}, t_{now}) + t_{lock} - t_{now}

Verify Constraints

Ensure the New Remaining Lock Period (\Delta t_{lock}) is within allowed limits:

\Delta t_{lock} = 0 \lor T_{MIN} \le \Delta t_{lock} \le T_{MAX}

Calculate Bonus MPs for the Increased Lock Period

\Delta \hat{mp}^\mathcal{B} = mp_\mathcal{B}(\mathbb{Acc} \cdot a_{bal}, t_{lock})

Verify Constraints

Ensure the New Maximum MPs (\mathbb{Acc} \cdot mp_\mathcal{M} + \Delta \hat{mp}^\mathcal{B}) is within the Absolute Maximum MPs:

\mathbb{Acc} \cdot mp_\mathcal{M} + \Delta \hat{mp}^\mathcal{B} \le \frac{a_{bal} \times \mathsf{MPY}^\mathit{abs}}{100}

Update account State

Maximum MPs:

\mathbb{Acc} \cdot mp_\mathcal{M} = \mathbb{Acc} \cdot mp_\mathcal{M} + \Delta \hat{mp}^\mathcal{B}

Total MPs:

\mathbb{Acc} \cdot mp_\Sigma = \mathbb{Acc} \cdot mp_\Sigma + \Delta \hat{mp}^\mathcal{B}

Lock end time:

\mathbb{Acc} \cdot t_{lock,end} = max(\mathbb{Acc} \cdot t_{lock,end}, t_{now}) + t_{lock}

Update System State

Maximum MPs:

\mathbb{Sys} \cdot mp_\mathcal{M} = \mathbb{Sys} \cdot mp_\mathcal{M} + \Delta mp_\mathcal{B}

Total MPs:

\mathbb{Sys} \cdot mp_\Sigma = \mathbb{Sys} \cdot mp_\Sigma + \Delta mp_\mathcal{B}

---

\mathcal{f}^{unstake}(\mathbb{Acc}, \Delta a) \longrightarrow Unstake Amount Unlocked

Purpose: Allows a user to unstake an amount \Delta a.

---
title: Unstake Storage Access Flowchart
---
flowchart LR
	Balance
	TotalMP[Total MPs]
	MaxMP[Maximum MPs]
	FReduceMP{Calc Reduced MP}
    TotalMP --> FReduceMP
    MaxMP --> FReduceMP
    Balance --> FReduceMP
    FReduceMP --> Balance
    FReduceMP --> TotalMP
    FReduceMP --> MaxMP

Steps

Accrue Existing Multiplier Points (MPs)

Call the \mathcal{f}^{accrue}(\mathbb{Account}) \longrightarrow Accrue Multiplier Points function to update MPs and last accrual time.

Verify Constraints

Ensure the account is not locked:

\mathbb{Acc} \cdot t_{lock,end} < t_{now}

Ensure that account have enough balance:

\mathbb{Acc} \cdot a_{bal} > \Delta a

Ensure that new balance (\mathbb{Acc} \cdot a_{bal} - \Delta a) will be zero or more than minimum allowed:

\mathbb{Acc} \cdot a_{bal} - \Delta a = 0 \lor \mathbb{Acc} \cdot a_{bal} - \Delta a > A_{MIN}

Calculate Reduced Amounts

Maximum MPs:

\Delta mp_\mathcal{M} =\mathcal{f}mp_\mathcal{R}(\mathbb{Acc} \cdot mp_\mathcal{M}, \mathbb{Acc} \cdot a_{bal}, \Delta a)

Total MPs:

\Delta  mp_\Sigma = \mathcal{f}mp_\mathcal{R}(\mathbb{Acc} \cdot mp_\Sigma, \mathbb{Acc} \cdot a_{bal}, \Delta a)

Update account State

Maximum MPs:

\mathbb{Acc} \cdot mp_\mathcal{M} = \mathbb{Acc} \cdot mp_\mathcal{M} - \Delta mp_\mathcal{M}

Total MPs:

\mathbb{Acc} \cdot mp_\Sigma = \mathbb{Acc} \cdot mp_\Sigma - \Delta mp_\Sigma

Balance:

\mathbb{Acc} \cdot a_{bal} = \mathbb{Acc} \cdot a_{bal} - \Delta a

Update System State

Maximum MPs:

\mathbb{Sys} \cdot mp_\mathcal{M} = \mathbb{Sys} \cdot mp_\mathcal{M} - \Delta mp_\mathcal{M}

Total MPs:

\mathbb{Sys} \cdot mp_\Sigma = \mathbb{Sys} \cdot mp_\Sigma - \Delta mp_\Sigma

Total staked amount:

\mathbb{Sys} \cdot a_{bal} = \mathbb{Sys} \cdot a_{bal} - \Delta a

---

\mathcal{f}^{accrue}(\mathbb{Acc}) \longrightarrow Accrue Multiplier Points

Purpose: Accrue multiplier points (MPs) for the account based on the elapsed time since the last accrual.

---
title: Accrue Storage Access Flowchart
---
flowchart LR
	AccruedMP{{Accrued MP}}
	Balance
	LastMint[Last Mint]
	TotalMP[Total MPs] --> MAX{max}
	MaxMP[Maximum MPs] --> MAX
	FAccruedMP{Calc Accrued MP}
	NOW((NOW)) --> FAccruedMP
	FAccruedMP --> LastMint
	LastMint --> FAccruedMP
    Balance --> FAccruedMP
    FAccruedMP --> AccruedMP
    AccruedMP --> MAX
    MAX --> TotalMP

Steps

Calculate the time Period since Last Accrual

\Delta t = t_{now} - \mathbb{Acc} \cdot t_{last}

Verify Constraints

Ensure the accrual period is greater than the minimum rate period:

\Delta t > T_{RATE}

Calculate Accrued MP for the Accrual Period

\Delta \hat{mp}^\mathcal{A} = min(\mathcal{f}mp_\mathcal{A}(\mathbb{Acc} \cdot a_{bal},\Delta t) ,\mathbb{Acc} \cdot mp_\mathcal{M} - \mathbb{Acc} \cdot mp_\Sigma)

Update account State

Total MPs:

\mathbb{Acc} \cdot mp_\Sigma = \mathbb{Acc} \cdot mp_\Sigma + \Delta \hat{mp}^\mathcal{A}

Last accrual time:

\mathbb{Acc} \cdot t_{last} = t_{now}

Update System State

Total MPs:

\mathbb{Sys} \cdot mp_\Sigma = \mathbb{Sys} \cdot mp_\Sigma + \Delta \hat{mp}^\mathcal{A}

---

Reward Index Update

The reward index is updated whenever new rewards are added to the system. This update ensures that rewards are accurately tracked and distributed based on the current total weight.

1. Calculate New Rewards:

R_{\text{new}} = R_{\text{bal}} - R_{\text{accounted}}

Where:

• R_{\text{bal}}: Current balance of reward tokens in the contract.
• R_{\text{accounted}}: Total rewards that have already been accounted for.

2. Update Reward Index:

R_i = R_i + \left( \frac{R_\text{new} \times SCALE_{FACTOR}}{\mathbb{Sys} \cdot W} \right)

3. Account for Distributed Rewards:

R_\text{accounted} = R_\text{accounted} + R_\text{new}

---

Reward Calculation for Accounts

Each account's rewards are calculated based on the difference between the current reward index and the account's last recorded reward index. This ensures that rewards are distributed proportionally and accurately.

1. Calculate Reward Index Difference:

\Delta \mathbb{Acc} \cdot R_i = \mathbb{Sys} \cdot R_i - \mathbb{Acc} \cdot R_i

2. Calculate Reward for Account j:

\text{reward}^j = \frac{\mathbb{Acc} \cdot W \times \Delta \mathbb{Acc} \cdot R_i}{SCALE_{FACTOR}}

3. Update Account Reward Index:

\mathbb{Acc} \cdot R_i = R_i

---

Distribute Rewards

When distributing rewards to an account, ensure that the reward does not exceed the contract's available balance. Adjust the accounted rewards accordingly to maintain consistency.

1. Determine Transfer Amount:

\text{amount} = \min(\text{reward}^j, R_{\text{bal}})

1. Adjust Accounted Rewards:

R_{\text{accounted}} = R_{\text{accounted}} - \text{amount}

3. Transfer Reward Tokens:

\text{RewardToken.transfer}(j, \text{amount})

Reward Calculation

At any given point, the total reward accumulated by an account j is calculated as follows:

\text{reward}^j = \frac{(\mathbb{Acc} \cdot a_{\text{bal}} + \mathbb{Acc} \cdot mp_{\Sigma}) \times (R_i - \mathbb{Acc} \cdot R_i)}{SCALE_{FACTOR}}

This formula ensures that rewards are distributed proportionally based on both the staked tokens and the accrued Multiplier Points, adjusted by the changes in the global reward index since the last reward calculation for the account.

---

\mathcal{f}^{\text{updateRewardIndex}}(\mathbb{Sys}) \longrightarrow Update Reward Index

Calculates and updates the global reward index based on newly added rewards and the current total weight in the system.

\boxed{
\begin{equation}
\mathcal{f}^{\text{updateRewardIndex}}(\mathbb{Sys}) = R_i + \left( \frac{R_{\text{new}} \times SCALE_{FACTOR}}{\mathbb{Sys} \cdot W} \right)
\end{equation}
}

Where:

• R_{\text{new}}: Calculated as R_{\text{bal}} - R_{\text{accounted}}.
• \mathbb{Sys} \cdot W: Defined as \mathbb{Sys} \cdot a_{\text{bal}} + \mathbb{Sys} \cdot mp_{\Sigma}.

---

\mathcal{f}^{\text{calculateReward}}(\mathbb{Acc} \cdot a_{\text{bal}}, \mathbb{Acc} \cdot mp_{\Sigma}, R_i, \mathbb{Acc} \cdot R_i) \longrightarrow Calculate Account Reward

Calculates the reward for an account j based on its staked balance, Multiplier Points, and the change in the global reward index.

\boxed{
\begin{equation}
\mathcal{f}^{\text{calculateReward}}(\mathbb{Acc} \cdot a_{\text{bal}}, \mathbb{Acc} \cdot mp_{\Sigma}, R_i, \mathbb{Acc} \cdot R_i) = \frac{(\mathbb{Acc} \cdot a_{\text{bal}} + \mathbb{Acc} \cdot mp_{\Sigma}) \times (R_i - \mathbb{Acc} \cdot R_i)}{SCALE_{FACTOR}}
\end{equation}
}

Where:

• \mathbb{Acc} \cdot a_{\text{bal}}: Staked balance of account j.
• \mathbb{Acc} \cdot mp_{\Sigma}: Total Multiplier Points of account j.
• R_i: Current cumulative reward index.
• \mathbb{Acc} \cdot R_i: Reward index at the last update for account j.
• SCALE_{FACTOR}: Scaling factor to maintain precision.

---

Support Functions

Maximum Total Multiplier Points

The maximum total multiplier points that can be generated for a determined amount of balance and lock duration.

\boxed{
	\begin{equation}
		\hat{\mathcal{f}}mp_{\mathcal{M}}(a_{bal}, t_{\text{lock}}) = a_{bal} + \frac{a_{bal} \times \mathtt{APY} \times \left( T_{\text{MAX}} + t_{\text{lock}} \right)}{100 \times T_{\text{YEAR}}}
	\end{equation}
}

Maximum Accrued Multiplier Points

The maximum multiplier points that can be accrued over time for a determined amount of balance. It's \mathcal{f}{mp_\mathcal{A}}(a_{bal}, Delta t) \longrightarrow Accrue Multiplier Points using \Delta t = M_{MAX} \times T_{YEAR}

\boxed{
	\begin{equation}
		\hat{\mathcal{f}}mp_{A}^{max}(a_{bal}) = \frac{a_{bal} \times \mathsf{MPY}}{100}
	\end{equation}
}

Maximum Absolute Multiplier Points

The absolute maximum multiplier points that some balance could have, which is the sum of the maximum lockup time bonus and the maximum accrued multiplier points.

\boxed{
	\begin{equation}
		\hat{\mathcal{f}}mp_\mathcal{M}^\mathit{abs}(a_{bal}) = \frac{a_{bal} \times \mathsf{MPY}^\mathit{abs}}{100}
	\end{equation}
}

Retrieve Bonus Multiplier Points

Returns the Bonus Multiplier Points from the Maximum Multiplier Points and Balance.

\boxed{
\begin{equation}
\hat{\mathcal{f}}\hat{mp}^\mathcal{B}(mp_\mathcal{M}, a_{bal}) = mp_\mathcal{M} - \left(a_{\text{bal}} + \hat{\mathcal{f}}mp_{A}^{max}(a_{bal}) \right)
\end{equation}
}

Retrieve Accrued Multiplier Points

Returns the accrued multiplier points from Total Multiplier Points, Maximum Multiplier Points and Balance.

\boxed{
\begin{equation}
\hat{\mathcal{f}}\hat{mp}^\mathcal{A}(mp_\Sigma, mp_\mathcal{M}, a_{bal}) =  mp_\Sigma + \hat{\mathcal{f}}mp_{A}^{max}(a_{bal})  - mp_\mathcal{M}
\end{equation}
}

Time to Accrue Multiplier Points

Retrieves how much seconds to a certain a_{bal} would reach a certain mp

\boxed{
	\begin{equation}
		t_{rem}(a_{bal},mp_ {target}) = \frac{mp_{target} \times 100 \times T_{YEAR}}{a_{bal} \times \mathtt{APY}}
	\end{equation}
}

Locked Time (t_{lock})

Caution

Use for reference only. If implemented with integers, for a_{bal} < T_{YEAR}, due precision loss, the result may be an approximation.

Estimates the time an account set as locked time.

\boxed{
	\begin{equation}
		\hat{\mathcal{f}}\tilde{t}_ {lock}(mp_{\mathcal{M}}, a_{bal}) \approx \left\lceil \frac{(mp_{\mathcal{M}} - a_{bal}) \times 100 \times T_{YEAR}}{a_{bal} \times \mathtt{APY}}\right\rceil - T_{\text{MAX}}
	\end{equation}
}

Where:

• mp_{\mathcal{M}}: Maximum multiplier points calculated the a_{bal}
• a_{bal}: Account balance used to calculate the mp_{\mathcal{M}}

Remaining Time Lock Available to Increase

Caution

Use for reference only. If implemented with integers, for a_{bal} < T_{YEAR}, due precision loss, the result may be an approximation.

Retrieves how much time lock can be increased for an account.

\boxed{
	\begin{equation}
		t_{rem}^{lock}(a_{bal},mp_\mathcal{M}) \approx \frac{\left(\hat{\mathcal{f}}mp_\mathcal{M}^\mathit{abs}(a_{bal}) - mp_\mathcal{M}\right)\times T_{YEAR}}{a_{bal}}
	\end{equation}
}