Explainer: The Singularity Equation — what P(α) = C/(1-α) means

The FLAT Protocol introduces a fascinating mechanism for price discovery and stability governed by what we call the "Singularity Equation": P(α) = C / (1 - α). This equation describes a theoretical relationship between the price of the FLAT token and its absorption rate within the protocol. Let's break down what each component means and explore its implications.

Understanding the Variables

• P (Price): This represents the price of the FLAT token.
• C (Constant): This is a constant value, representing the initial or base price of FLAT. It acts as a foundational anchor for the price calculation.
• α (Alpha - Absorption Rate): This variable, expressed as a fraction between 0 and 1, signifies the "absorption rate" of FLAT.

What is the Absorption Rate (α)?

The absorption rate (α) quantifies the fraction of the total FLAT supply that is locked within the protocol's SAVE module. When users "SAVE" their FLAT, they are essentially removing it from the circulating supply and locking it to earn yield. This locking mechanism is crucial as it directly influences the absorption rate. An increase in the amount of FLAT locked in SAVE leads to a higher α.

Walking Through Examples

Let's illustrate how the absorption rate impacts the price according to the Singularity Equation:

• α = 0% (0): If no FLAT is locked in SAVE, the equation becomes P = C / (1 - 0) = C. The price is simply the initial constant, C. This represents the baseline value when there is no absorption.
• α = 50% (0.5): If half of the total FLAT supply is locked, P = C / (1 - 0.5) = C / 0.5 = 2C. The price of FLAT doubles from its initial constant value.
• α = 90% (0.9): With 90% of the supply locked, P = C / (1 - 0.9) = C / 0.1 = 10C. The price increases tenfold.
• α = 99% (0.99): As the absorption rate approaches 100%, P = C / (1 - 0.99) = C / 0.01 = 100C. The price experiences a dramatic hundredfold increase.

As α approaches 1 (or 100% absorption), the denominator (1 - α) approaches 0, causing the price P to theoretically approach infinity. This mathematical characteristic is why it's referred to as a "singularity."

The "Finite Energy" Corollary

The concept of a "finite energy" corollary addresses a common misconception: does an infinite price require infinite money? The answer is no. This corollary highlights that as the absorption rate approaches 1, the circulating supply of FLAT approaches 0. Therefore, while the price per token becomes extremely high, the total market capitalization (price multiplied by circulating supply) remains finite. This is because the amount of available tokens for purchase diminishes as more are locked, meaning fewer tokens are needed to reach a high per-token price.

Acknowledging the Assumptions

It's crucial to acknowledge the assumptions underlying this mathematical model:

1. Requires Continued Locking: The model assumes that users will continue to lock FLAT in the SAVE module. If locked tokens are consistently withdrawn, the absorption rate decreases, and the price would react accordingly.
2. Assumes Constant C: The equation treats 'C' as a constant initial price. In reality, market dynamics can influence the perceived base value.
3. Real Markets Have Friction: The Singularity Equation is a theoretical ideal. Real-world markets are subject to various frictions, including liquidity constraints, transaction costs, and emotional trading, which are not captured in this simplified model.

Mathematical Model vs. Market Reality

The Singularity Equation provides a powerful mathematical framework for understanding the potential price behavior of FLAT under specific conditions of absorption. It describes a theoretical limit and the mechanics of a supply-constrained asset. However, it is essential to distinguish this mathematical model from market reality. Real-world prices are influenced by a multi