# MOR emission logic

The `LinearDistributionIntervalDecrease` library implements a reward calculation strategy designed for protocols that emit a fixed token amount per interval, with a predictable linear decrease in the emission rate over time. This pattern is used for long-term token emission schemes (e.g., MOR) that reward staking participants based on when they joined and how long they stayed.

## Overview

The key concept is *linear emission with interval-based reduction*. Instead of updating user rewards every second, the protocol defines a base reward amount that decreases linearly every fixed time interval (e.g., s). This model balances emission precision and computational efficiency, making it ideal for large-scale staking systems.

## **Emission scheme params**

* `initialAmount` — the starting reward amount per interval (e.g., 1000 tokens per day).
* `decreaseAmount` — how much the reward decreases *per interval* (e.g., 10 tokens less per day).
* `interval` — time in seconds (e.g., 86400 for daily).
* `payoutStart` — timestamp when emissions start.
* `startTime` / `endTime` — the specific time window to compute rewards for.

Internally, reward calculation over a range (e.g., user staking from day 4 to day 10) is split into three parts:

1. **Partial First Interval**

   If the time window begins *within* an interval (not at its start), a proportional reward is computed based on how much of that interval is included.
2. **Full Intervals in the Middle**

   If the period spans multiple full intervals, the total reward for these is computed efficiently using an arithmetic sum of decreasing values:

```
        initialReward * ip_ - (decreaseAmount_ * (ip_ * (ip_ - 1))) / 2
```

        This handles the geometric drop without looping over each interval.

3. **Partial Last Interval**

   If the time window ends *within* an interval, a proportional reward for the trailing part is added, similar to the first part.

If the input `endTime_` exceeds this cutoff, it’s clamped to avoid negative rewards.<br>

## Emission cutoff

The system prevents reward underflow by computing a maximum emission end time. This is the timestamp when the emissions would naturally reduce to zero:

```
maxIntervals = ceil(initialAmount / decreaseAmount)
maxEndTime = payoutStart + maxIntervals * interval
```


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