> For the complete documentation index, see [llms.txt](https://jester.gitbook.io/docs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://jester.gitbook.io/docs/tools/castellan/bonding-curve.md).
# Bonding Curve
The bonding curve system **automatically adjusts** token requirements for different membership tiers based on market conditions. Here's how it works:
**The key innovation is that token requirements adjust inversely to price:**
* **When token price increases β required tokens decrease**
* **When token price decreases β required tokens increase**
For example, if the token price doubles, **youβll need fewer tokens** to maintain your tier.
### **π‘οΈ Built-in Protections**
To ensure **fair and stable adjustments**, the system also applies:
β **Rate Limits:**
* **Daily max change: 2%**
* **Biweekly max change: 10%**
β **Grace Periods:**
* Verified users have **a 7-day grace period** before losing a tier.
* Users within **10% of the requirement** retain their tier (if eligible).
β **Dual Qualification:**
* Users can qualify **by either token count or USD value on holdings**.
β **Exponential Decay for Smooth Adjustments:**
* The system uses **exponential decay** to adjust token requirements gradually based on price changes.
* This ensures **no drastic jumps** when token price fluctuates.
β **Scaled Discounts:**
* Fee discounts increase **as your holdings grow**:
* **Basic:** 15-25% discount
* **Standard:** 30-50% discount
* **Premium:** 60-75% discount
* **Role-based discounts override these** if they offer better rates.
### **π― Why This Matters**
This system ensures:
β **Loyal holders are protected**\
β **Tier requirements adjust smoothly with price changes.**\
β **New users can still enter at reasonable token amounts.**
***
#### Graph Breakdown
* **Dashed lines** represent **token requirements decreasing** when price **increases**.
* **Solid lines** represent **token requirements increasing** when price **decreases**.
* **Each color corresponds to a different membership tier** (**BASIC, STANDARD, PREMIUM**).
* The **black dotted line** marks the **target price ($1)** where token requirements stabilize.
***
## Formulas
### **When Token Price Increases (Requirements Decrease):**
* As the token price **rises**, the number of required tokens **decreases** towards the **target minimum threshold** using **exponential decay**:
$$
T\_{new} = T\_{target} + (T\_{previous} - T\_{target}) \times e^{-k \times \left(\frac{P\_{current}}{P\_{target}}\right)}
$$
**Explanation:**
* **Tnewβ** β The updated minimum token requirement
* **Ttarget** β The lowest possible token requirement for the tier
* **Tprevious** β The last recorded minimum requirement before price changes
* **Pcurrent** β The current token price
* **Ptarget**β The price at which the tier requirement stabilizes
* **k** β A decay constant (default: **2**) that controls the rate of adjustment
* **ee**β Eulerβs number (**β 2.718**), ensuring smooth decay
πΉ **Result:** If the **token price doubles**, the requirement moves closer to **the target minimum** (e.g., BASIC tier moves from 2500 tokens toward 250).
### **When Token Price Decreases (Requirements Increase)**
$$
T\_{new} = \min \left( T\_{start}, T\_{previous} + (T\_{start} - T\_{previous}) \times e^{-k \times \left(1 - \frac{P\_{current}}{P\_{target}}\right)} \right)
$$
**Explanation:**
* **Tstart** β The initial token requirement before any price-based adjustments
* Other variables remain the same as in the first formula.
πΉ **Result:** If the **token price drops**, the system **increases token requirements gradually** but prevents extreme jumps.
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