> For the complete documentation index, see [llms.txt](https://kix-2.gitbook.io/kix/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://kix-2.gitbook.io/kix/dats-digital-athlete-tokens/dats/bonding-curve-technical.md).

# The Bonding Curve — Technical

This page is for readers who want to understand the precise mechanics of how DAT prices are set on KiX. It is not required reading to use the platform — but if you are doing serious due diligence, this is where the mathematics lives.

***

![DAT Bonding Curve](/files/FZUgMwzl9DBp9WscHT5f)

## The Core Price Formula

Every footballer on KiX has their own independent bonding curve. The price of one DAT at any given supply level is defined by:

```
P(T) = B + A × T^E
```

Where:

| Variable | Name                  | Value         | Description                                                 |
| -------- | --------------------- | ------------- | ----------------------------------------------------------- |
| T        | Tokens in Circulation | 0 → 1,000,000 | The current number of DATs in existence for this footballer |
| B        | Starting Price        | 0.5 $KIX      | The minimum price — no DAT ever trades below this floor     |
| A        | Scaling Constant      | 1 / 10^10.35  | The master parameter — controls the steepness of the curve  |
| E        | Exponent              | 2.0           | Makes this a quadratic (parabolic) curve                    |

***

## The Scaling Constant — A is Everything

The value of A is not arbitrary. It was chosen with a specific outcome in mind:

**A = 1 / 10^10.35 ≈ 0.0000000000447**

This single number determines that the price of the 1,000,000th DAT — the last possible token for any footballer — will be approximately **45 $KIX**. That is intentional. The entire curve is calibrated around this terminal price anchor.

Change A and you change the whole curve. A larger A steepens the curve (higher prices, faster). A smaller A flattens it (cheaper tokens, slower appreciation). Every other parameter flows from this choice.

***

## The Price Table

Because E = 2, the curve is quadratic — price grows with T². The early supply is priced almost flat. The steep appreciation comes later.

| DATs in Circulation | Price per DAT ($KIX) | Total $KIX in Pool |
| ------------------- | -------------------- | ------------------ |
| 0                   | 0.50                 | 0                  |
| 10,000              | 0.50                 | 5,015              |
| 50,000              | 0.61                 | 26,861             |
| 100,000             | 0.95                 | 64,889             |
| 200,000             | 2.29                 | 219,116            |
| 300,000             | 4.52                 | 552,015            |
| 500,000             | 11.67                | 2,111,182          |
| 750,000             | 25.63                | 6,656,488          |
| 1,000,000           | **\~45.17**          | **15,389,453**     |

The first 100,000 DATs barely move the price above the 0.5 $KIX floor. The curve then steepens sharply. By the time 500,000 DATs exist, the price has reached 11.67 $KIX — and the final 500,000 tokens take the price to 45.17 $KIX.

This design rewards early participants. It also means the curve is self-limiting — the deeper the market goes, the more expensive it becomes to accumulate.

***

## The Cost of a Trade — Area Under the Curve

The price formula P(T) gives the **spot price** at a given supply level. But when you buy or sell multiple DATs in a single transaction, you do not pay the current spot price for all of them. You pay the **area under the curve** between your start and end supply points.

This is the integral of the price function:

```
Cost(T₁ → T₂) = ∫[T₁ to T₂] P(T) dT
               = B×(T₂ - T₁) + (A / (E+1)) × (T₂^(E+1) - T₁^(E+1))
               = B×(T₂ - T₁) + (A/3) × (T₂³ - T₁³)
```

**Why this matters in practice:**

Buying 10,000 DATs when supply is at zero (T₁=0, T₂=10,000) costs **5,015 $KIX** — very close to 10,000 × 0.50 floor price, because the curve is nearly flat there.

Buying the same 10,000 DATs when supply is already at 990,000 (T₁=990,000, T₂=1,000,000) costs approximately **447,232 $KIX** — because you are buying on the steepest part of the curve.

Same quantity of DATs. Vastly different cost. This is what makes early participation in a footballer's DAT pool economically meaningful.

***

## Guaranteed Liquidity — The Closed-Loop Invariant

Every $KIX that enters a footballer's pool stays in that pool. When you buy DATs, the $KIX you pay is locked into the pool. When you sell, that $KIX is returned to you. No fraction is ever routed elsewhere — not to the treasury, not to rewards, not anywhere.

**This is the closed-loop invariant and it is absolute.**

The practical consequence: you can always sell your DATs. The liquidity to buy them back from you is always there, because it was deposited when they were originally purchased. KiX does not rely on a future buyer appearing — the protocol itself is always the buyer of last resort.

***

## The Dynamic Bid Spread

The bonding curve formula above determines the **offer price** (what you pay to buy). The **bid price** (what you receive when you sell) is set at a discount to the offer, calculated dynamically:

```
bid_discount = max(k × (sell_size / pool_depth)^α, decayed_recent_sells / current_depth × w_M)
```

Default parameters:

* **k** = 2.5
* **α** = 1.5
* **half-life** = 6 hours
* **w\_M** = 0.4
* **Spread clamped** between 1% and 12.5%

This spread is entirely retained within the per-footballer pool. It never leaves. This means the pool deepens with every trade — the bid/offer spread compounds into liquidity over time.

The spread widens under two conditions: large sell orders relative to pool depth, and recent concentrated selling pressure. It narrows during calm, balanced trading. This is a manipulation-resistant design — panic selling is absorbed but at a cost; organic trading is cheap.

***

## Slippage and Partial Fills

For large orders, the price may move during the transaction. Rather than failing the entire transaction when price moves beyond a user's tolerance, the protocol executes a **partial fill** — buying or selling the quantity achievable within the price tolerance and returning the remainder to the user's wallet.

This means large traders are protected from unexpected price impact, and the pool is not destabilised by oversized single transactions.

***

## Goalkeepers

Goalkeepers do not have DAT pools. Only outfield players — Defenders, Midfielders, and Forwards — are listed on KiX. This is consistent throughout the performance scoring system, the rewards system, and the bonding curve pools.

***

*For the plain-English explanation of how DAT prices work, see* [*Buying & Selling DATs*](/kix/dats-digital-athlete-tokens/dats/buying-and-selling.md)*.*