# Glossary

**Dynamic Token Supply**: It refers to the overall supply of tokens within the bonding curve ecosystem, which adjusts dynamically based on changes in the reserve. It encompasses the cumulative effect of token issuance and redemption activities on the total supply of tokens, reflecting the system's responsiveness to market demand and liquidity conditions.

**Dynamic Token Issuance: It i**s the process within a bonding curve where new tokens are created or removed from circulation in response to changes in the reserve. It primarily focuses on the minting or burning of tokens based on alterations in the reserve, highlighting the mechanism's capability to dynamically adjust the token supply.

> While Dynamic Token Issuance specifically addresses the creation or destruction of tokens in response to reserve changes, Dynamic Token Supply provides a broader perspective by considering how the total token supply evolves over time in tandem with alterations in the reserve

**Continuous Organizations:** A *Continuous Organization* refers to any organization that set up a *Continuous Securities Offering* (*CSO*) to give to every stakeholder the ability to invest in the organization at any single time.

**Continuous Securities Offering (CSO) :** A *Continuous Securities Offering* (*CSO*) is a novel way for organizations to receive financing without releasing any equity or any governance rights. A CSO uses an organization's realized revenues (i.e. revenues for which a payment has been made) as a collateral to back fully digital securities (called *FAIR Securities* or *FAIRs*) that anyone can buy or sell to speculate on the organization's future revenues.

**The Decentralized Autonomous Trust:** Decentralized Autonomous Trusts (DATs) are innovative financial tools designed to operate within `continuous organizations`. The DAT is a smart contract built to incentivize long-term stakeholder participation and discourage speculation. The smart contract is implements a bonding curve that regulates buy/sell token prices and distribute dividends.

A **Continuous Securities Offering** (*CSO*) : It is a novel way for organizations to receive financing without releasing any equity or any governance rights. A CSO uses an organization's realized revenues (i.e. revenues for which a payment has been made) as a collateral to back fully digital securities (called Continuos Tokens) that anyone can buy or sell to speculate on the organization's future revenues.

## Augmented Bonding Curve

**Hatch**: A [hatch ](https://medium.com/commonsstack/whats-in-a-hatch-3799a398243a)is an initial fundraising mechanism used in token projects, particularly those employing bonding curves. During a hatch sale, tokens are sold directly to participants at a predetermined price. Token Engineering Commons used the hatch mechanism to raise $2million

**Hatch Maximum Goal -** the maximum amount of funds that can be sent to the Hatch. [Deep Dive](https://forum.tecommons.org/t/hatch-goals-deep-dive/62)

**Hatch Minimum Goal -** the amount of funds that needs to be raised during the Hatch to initialize the bonding curve. [Deep Dive](https://forum.tecommons.org/t/hatch-goals-deep-dive/62)

**Hatch Time Limit -** the duration the Hatch is open to collect funds. [Deep Dive](https://forum.tecommons.org/t/hatch-time-limit-deep-dive/63)

**Hatch Tribute -** specifies what % of the funds raised by the Hatch go to the Funding Pool. [Deep Dive](https://forum.tecommons.org/t/hatch-tribute-deep-dive/65)

**Initial Raise** $$d\_0$$**:**&#x54;he initial amount of DAI raised during the hatch sale. $$d\_0 \in \[d\_{min},d\_{max}]$$ where $$d\_{min}$$ & $$d\_{max}$$ represent the minimum and maximum bounds of the raise.

**Initial Reserve** $$R\_0$$**:** The initial reserve amount in the bonding curve after the hatch sale, calculated as a fraction of the initial raise. $$R\_0 = (1-\theta)d\_0$$ where 𝜃 is the fraction not included in the reserve

**Hatch Sale Price** $$p\_0$$: The price of tokens during the hatch sale. Determines the initial supply:  $$S\_0 = \frac{d\_0}{p\_0}$$

**Initial Supply** $$S\_0$$: The total number of coins or tokens that are created when a new cryptocurrency first launches, this starting amount is defined in the smart contract code and serves as the genesis supply for that crypto $$S\_0 = \frac{d\_0}{p\_0}$$

$$
S\_0 = \frac{d\_0}{p\_0}
$$

**Power Function Invariant**: The mathematical relationship between reserve and supply that remains constant.

$$
V(R,S)= \frac {S^{\kappa}}{R}
$$

**Reserve Ratio (RR):** The Reserve Ratio (RR) is a key financial metric that is defined as the ratio of the dollar reserves (R) held within a liquidity pool or bonding curve to the market capitalization of the bonding tokens. This ratio establishes a relationship between the reserves and the total market value of the circulating bonding tokens, indicating the extent to which each unit of currency is backed by reserves at the current spot price. providing a clear measure of the financial stability and backing of the tokenized assets within the system. A higher reserve ratio indicates a higher degree of backing for each unit of currency, enhancing the system's stability and trustworthiness.

Mathematically, it can be expressed as:

$$
RR = \frac{R}{S \cdot P(R)}
$$

where:

* $$R$$  represents the dollar value of the reserves in the pool.
* $$S$$ is the total supply of bonding tokens.
* $$P(R)$$ denotes the current spot price of the bonding tokens, which is a function of the reserves.

In an alternative form, the reserve ratio can be expressed as the reciprocal of a constant, $$\kappa$$, illustrating that:

$$
RR = \frac{1}{\kappa}
$$

**Kappa (𝜅):** $$\kappa$$ is a parameter that shapes the bonding curve, determining the relationship between the reserve and the supply of tokens.

* In the power function invariant κ influences how steeply the token price changes with respect to changes in the reserve.
* In the price function **𝜅**  affects the curvature of the price-reserve relationship.
* In the supply function **κ** determines how the token supply scales with the reserve.
* In the reserve function **κ** defines how the reserve scales with the token supply.

A higher **𝜅**  results in a bonding curve where the price of tokens increases more sharply as the reserve grows, leading to more rapid price escalation. Conversely, a lower **𝜅** leads to a more gradual increase in token price with respect to the reserve.

**Mint fee**: Minting is the process of creating new tokens on bonding curve. The fee a user pays to convert reserve assets into bonded tokens is called the mint fee.

**Burn Fee**: Burning is the process of redeeming tokens back to the curve. Burn fee is what the user pays to convert bonding tokens back to reserve tokens﻿.

**Price Function :** The price function, denoted as $$𝑃(𝑅)$$, represents the price of tokens in terms of the reserve. It determines how much one token costs given a certain amount of reserve.&#x20;

$$
P(R) = \frac{\kappa R^{(\kappa-1)/ \kappa}}{V\_0^{1 / \kappa}}
$$

**Supply Function:** The supply function, denoted as $$S(R)$$, represents the total supply of tokens as a function of the reserve. It shows how many tokens are in circulation given a certain amount of reserve.

$$
S(R) = \sqrt\[\kappa]{V\_0 R}
$$

**Reserve Function:** The reserve function, denoted as $$R(S)$$, represents the amount of reserve needed to support a given supply of tokens. It shows how the reserve scales with the token supply.

$$
R(S)= \frac{S^{\kappa}}{V\_0}
$$

**Invariant Coefficient:** The invariant coefficient is a key parameter in bonding curve models, representing a constant value derived from the initial conditions of the curve. It plays a fundamental role in maintaining the mathematical integrity of the bonding curve and ensuring that the curve's invariant is preserved throughout its operation.

$$
V\_0 = V(R\_0, S\_0) = \frac{S\_0^\kappa}{R\_0} = \left(\frac{1}{p\_0(1-\theta)}\right)^\kappa R\_0^{\kappa-1}
$$

**Post-Hatch Price:** The post-hatch price $$(p\_1)$$ refers to the price of tokens after the conclusion of the hatch sale phase in a token project. It represents the token price following the initial fundraising event&#x20;

$$
p\_1=P(R\_0) = \frac{\kappa R\_0^{(\kappa-1)/ \kappa}}{V\_0^{1 / \kappa}} = \kappa R\_0^{(\kappa-1)/ \kappa} \cdot(1-\theta)p\_0\cdot R\_0^{-(\kappa-1)/\kappa} = \kappa(1-\theta) p\_0
$$

**Return Factor:** The return factor, often denoted by $$\frac{p\_1}{p\_0}$$, represents the ratio of the post-hatch sale token price $$p\_1$$ to the initial hatch sale price $$p\_0​$$. It provides insight into how the token price changes after the hatch sale phase, reflecting the return on investment for participants who purchased tokens during the hatch sale.

$$
\frac{p\_1}{p\_0} = {\kappa}(1-\theta)
$$

**Additional Deposit** $${\Delta}R$$:  The amount of additional reserve tokens deposited to mint new tokens.

**Conservation Equation**: This equation ensures a specific relationship between the reserve amount, the total supply and their changes after a deposit event. It essentially states that after a deposit, the system maintains a constant value $$V\_0$$ through the interplay of reserve and total supply. The power $$\kappa$$influences the impact of changes in supply on the required reserve level.&#x20;

$$
(R+ \Delta R, S+\Delta S) = \frac{(S+\Delta S)^\kappa}{R+\Delta R} =V\_0
$$

**Derived Mint Equation**: This equation tells you how many new stablecoins are minted $${\Delta}S$$ based on the amount of additional reserve added to the system $${\Delta}R$$. It considers the initial system value and the current reserve level, with the power $$\kappa$$ affecting how much a deposit influences the number of minted tokens.

$$
\Delta S = mint\big(\Delta R ; (R,S)\big)= \sqrt\[\kappa]{V\_0(R+\Delta R)}-S
$$

<mark style="background-color:red;">**Realized Price**</mark> $$\bar{P}(\Delta R)$$<mark style="background-color:red;">:</mark> <mark style="background-color:red;"></mark><mark style="background-color:red;">**\[needs t be defined]**</mark>

$$
\bar{P}(\Delta R) =\frac{\Delta R}{\Delta S} = \frac{\Delta R}{\sqrt\[\kappa]{V\_0(R+\Delta R)}-\sqrt\[\kappa]{V\_0(R)}} \rightarrow \big(\frac{\partial S(R)}{\partial R} \big)^{-1} as \Delta R \rightarrow 0
$$

<mark style="background-color:red;">**Spot Price**</mark>**&#x20;\[**<mark style="background-color:red;">**review**</mark>**]**: Spot price is the instantaneous price a user would pay when depositing an infinitesimal amount of reserve tokens into the bonding curve. Spot price is not "real" in the sense that this price is not experienced by a user, but rather reflects the current price at which bonding curve tokens can be minted or redeemed, based on the current state of the curve.

$$
\lim\_{\Delta R \rightarrow 0} \bar{P}(\Delta R)=\big(\frac{\partial S(R)}{\partial R}\big)^{-1}= \big(\frac{V\_0^{1/\kappa} \cdot  R^{1/\kappa-1}}{\kappa}\big)^{-1}= \frac{\kappa R^{1-1/\kappa}}{V\_0^{1/\kappa}} = \frac{\kappa R^{(\kappa-1)/\kappa}}{V\_0^{1/\kappa}} =P(R)
$$

***Formal Expression***

The spot price $$𝑃(𝑅)$$ can be defined mathematically as:&#x20;

$$
\lim\_{\Delta R \rightarrow 0} \bar{P}(\Delta R)=\big(\frac{\partial S(R)}{\partial R}\big)^{-1}=\frac{\kappa R^{(\kappa-1)/\kappa}}{V\_0^{1/\kappa}}  = P(R)
$$

Where,

* $$𝑅$$ is the current reserve of tokens.
* $$𝑆(𝑅)$$ is the supply function of the bonding curve tokens.
* $$𝜅$$ is the elasticity parameter, determining the shape of the bonding curve.
* $$𝑉0$$ is the invariant coefficient, ensuring the mathematical consistency of the bonding curve.

1. **Limit Definition**: \
   $$\lim\_{\Delta R \rightarrow 0} \bar{P}(\Delta R)$$ denotes the limit of the realized price $$\bar{P}(\Delta R)$$ as the change in reserve $$Δ𝑅$$approaches zero. This is the spot price.
2. **Inverse Derivative**:\
   $$\big(\frac{\partial S(R)}{\partial R}\big)^{-1}$$ is the inverse of the derivative of the supply function $$𝑆(𝑅)$$ with respect to the reserve $$𝑅$$. This also equals the spot price.
3. **Derivative Calculation**:\
   $$\big(\frac{\partial S(R)}{\partial R}\big)^{}$$ is calculated as $$\big(\frac{V\_0^{1/\kappa} \cdot  R^{1/\kappa-1}}{\kappa}\big)^{-1}$$. This represents how the supply of tokens changes with a small change in the reserve.
4. **Inverse Derivative Simplification**:\
   $$\big(\frac{\partial S(R)}{\partial R}\big)^{-1}$$ simplifies to $$\frac{\kappa R^{1-1/\kappa}}{V\_0^{1/\kappa}}$$.
5. **Spot Price Expression**:\
   $$\frac{\kappa R^{1-1/\kappa}}{V\_0^{1/\kappa}}$$ further simplifies to $$\frac{\kappa R^{(\kappa-1)/\kappa}}{V\_0^{1/\kappa}}$$, which is the final expression for the spot price $$𝑃(𝑅)$$.

**Friction Coefficient** $$\phi$$: It is a parameter used in bonding curve systems to represent the portion of tokens or value extracted from transactions as fees or charges. It introduces a frictional force that reduces the efficiency of token transactions by imposing a fee on participants who interact with the system.
