# Scale-Invariant Yield Mechanics

> ‼️ Nexus is materially different from custodial yield products, bridges, wrapped-Bitcoin structures, and validator-trust models because BTC remains under owner-controlled custody while the application coordinates DeFi activity around it.

A prevalent misconception in traditional decentralized finance is that as Total Value Locked (TVL) scales, the yield per dollar invested inherently compresses. While true for traditional order books and directional market-making, this axiom does not apply to the [Constant Product Market Maker (CPMM)](/yield-economics-and-amm-design/amm-models.md) architecture utilized by the Nexus Protocol.

Within Nexus, arbitrage yields are non-dilutable.

#### The Scale-Invariance Theorem

The foundation of CPMM is the invariant curve: $$x \times y = k$$

Because this curve is homogeneous of degree 2, scaling liquidity $$L \equiv \sqrt{k}$$ strictly causes both capital allocation and arbitrage flows to stretch proportionally.

Mechanism Explained

When the external market price shifts from $$P\_{0}$$ to $$P\_{1}$$, an arbitrageur executes a trade against the Nexus pool to capture the spread. To move the AMM price to match the external CEX price, they must push a specific volume $$V\_{\text{in}}$$ against the curve.

Because of the geometric properties of $$x \times y = k$$, if the pool doubles in size ($$TVL\times 2$$), the volume required to move the price from $$P\_{0}$$ to $$P\_{1}$$ also exactly doubles ($$V\_{\text{in}}\times 2$$).

Since liquidity providers collect a fixed fee percentage ($$\gamma$$) on incoming volume, their absolute fee revenue doubles alongside the TVL. Therefore, the relative yield (Fees / TVL) cancels out the size parameter completely:

$$\text{Yield} = \frac{\gamma|\Delta \sqrt{P}|}{(1-\gamma) 2 \sqrt{P\_{0}}}$$

The yield depends entirely on external price volatility ($$|\Delta \sqrt{P}|$$) and the fee rate ($$\gamma$$), completely independent of how large the liquidity pool grows.

#### The Position Profile

Institutions providing liquidity to Nexus must understand their exposure mathematically: Liquidity Providers are long volatility, short drift.

Instead of being diluted by competitors depositing into the same pool, Nexus LPs benefit immensely from deep aggregate liquidity. Massive pools suffer lower slippage on routine trades, attracting larger tier-1 market arbitrage flows, which generate reliable, steady fee accretion during sideways, volatile market action.


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