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

The Singularity Equation: Understanding P(α) = C / (1-α)

The FLAT Protocol introduces a fascinating concept with its "Singularity Equation," expressed as P(α) = C / (1-α). This equation is a mathematical model designed to illustrate the theoretical price behavior of FLAT tokens as a function of their absorption into the SAVE mechanism. Let's break down what each variable means and explore its implications.

Deconstructing the Equation

• P (Price): This represents the theoretical market price of the FLAT token. It's the output of the equation, showing how the price changes as other factors shift.
• C (Constant / Initial Price): This is a baseline value, representing the initial price of the FLAT token. It acts as a scaling factor for the equation. For simplicity, in many discussions, C might be considered as 1 unit of currency to illustrate the multiplier effect clearly.
• α (Alpha - Absorption Rate): This is the core driver of the equation. Alpha represents the fraction of the total FLAT token supply that has been locked into the SAVE mechanism. It's a number between 0 (no tokens locked) and 1 (all tokens locked).

What Does "Absorption" Mean?

In the context of FLAT Protocol, "absorption" refers to the act of locking FLAT tokens into the SAVE Vault. When tokens are locked, they are removed from the circulating supply, becoming illiquid. The absorption rate (α) is simply the percentage of the total supply that has been committed to SAVE. For example, if there are 1,000,000 FLAT tokens in total, and 500,000 of them are locked in SAVE, then α = 0.5 (or 50%).

Illustrative Examples: The Multiplier Effect

Let's see how the price (P) changes with varying absorption rates (α), assuming C = 1 for simplicity to highlight the multiplier:

• α = 0% (0.0): If no FLAT tokens are locked in SAVE, then P = 1 / (1 - 0) = 1. The price remains at its initial value (1x).
• α = 50% (0.5): When half of the FLAT tokens are locked, P = 1 / (1 - 0.5) = 1 / 0.5 = 2. The price theoretically doubles (2x).
• α = 90% (0.9): With 90% of tokens locked, P = 1 / (1 - 0.9) = 1 / 0.1 = 10. The price theoretically increases tenfold (10x).
• α = 99% (0.99): If 99% of tokens are locked, P = 1 / (1 - 0.99) = 1 / 0.01 = 100. The price theoretically multiplies by 100 (100x).

As α approaches 1 (meaning nearly all tokens are locked), the denominator (1-α) approaches 0, causing P to theoretically approach infinity. This is the "singularity" aspect of the equation.

The "Finite Energy" Corollary: Why Infinite Price Doesn't Require Infinite Money

One of the intriguing corollaries of this equation is that an "infinite" price doesn't necessarily require an infinite amount of money. As α approaches 1, the circulating supply of FLAT tokens approaches zero. If there are very few tokens available on the market, even a small amount of demand (or "energy" in the protocol's terminology) can theoretically drive the price of those scarce tokens to extremely high levels. This is because the market capitalization (total value) would be P * (circulating supply). If circulating supply is tiny, P can be huge while market cap remains manageable.

Acknowledging the Assumptions: Model vs. Reality

It's crucial to understand that the Singularity Equation is a mathematical model and describes a theoretical limit, not a guaranteed market outcome. Several assumptions underpin this model:

1. Requires Continued Locking: The model assumes a continuous and increasing rate of FLAT tokens being locked into SAVE. If tokens are unlocked, α decreases, and the theoretical price would fall.
2. Assumes Constant C: The initial price (C) is treated as a constant. In reality, market dynamics can influence this baseline.
3. Real Markets Have Friction: The equation doesn't account for real-world market factors like liquidity, trading fees, slippage, large sell orders, or changes in overall market sentiment and demand for th