** ## WORK IN PROGRESS, NOT IMPLEMENTED ## **

We begin with a system that is primarily driven by environmental factors (temperature) and then apply a small, sentiment-based adjustment in a manner inspired by a constant product market maker (CPMM).

  1. Environmental Baseline Emissions

    Let \( T \) be the daily temperature, and define the parameters:

    • \( d \) (death threshold temperature)
    • \( e \) (Endcoin emission rate coefficient)
    • \( g \) (Gaiacoin emission rate coefficient)

    The environment-driven daily emissions for Endcoin \( E_{\text{env}}(T) \) and Gaiacoin \( G_{\text{env}}(T) \) are given by:

    \[ E_{\text{env}}(T) = \exp(e \cdot (d - T)) - 1 \] \[ G_{\text{env}}(T) = \exp(g \cdot T) - 1 \]

    Let the total environment-driven emission be:

    \[ E_{\text{total-env}}(T) = E_{\text{env}}(T) + G_{\text{env}}(T). \]

    In the absence of sentiment, these are the final emissions: \[ E_{\text{final}} = E_{\text{env}}(T), \quad G_{\text{final}} = G_{\text{env}}(T). \]

  2. Introducing Sentiment

    Define a sentiment parameter \( S \in [-1, 1] \), which represents the market’s bias. For \( S=0 \), there is no sentiment-driven preference. Positive \( S \) tilts toward Gaiacoin, and negative \( S \) tilts toward Endcoin.

    Convert \( S \) to a linear ratio: \[ R_{\text{linear}} = \frac{S + 1}{2}. \]

    Thus, when \( S=0 \), \( R_{\text{linear}}=0.5 \). Values above 0.5 lean toward Gaiacoin; values below 0.5 lean toward Endcoin.

    To limit sentiment’s impact, we allow \( S \) to be scaled down so that it moves within a narrower range (e.g., \([-0.1, 0.1]\)) to ensure that temperature remains the dominant factor.

  3. CPMM-Inspired Adjustment

    In a constant product market maker (CPMM), the relationship between two token quantities \( x \) and \( y \) is \( x \cdot y = K \), a constant. Here, we use a similar idea to slightly adjust the emission distribution.

    First, establish the baseline product from the environment scenario: \[ K_{\text{env}} = E_{\text{env}}(T) \times G_{\text{env}}(T). \]

    At zero sentiment (\( R_{\text{linear}}=0.5 \)), we must have the final values match the environment: \[ E_{\text{final}} = E_{\text{env}}, \quad G_{\text{final}} = G_{\text{env}}. \]

    This condition is naturally satisfied by: \[ E_{\text{final}} + G_{\text{final}} = E_{\text{total-env}}, \] \[ E_{\text{final}} \cdot G_{\text{final}} = K_{\text{env}}. \]

    To incorporate sentiment softly, let: \[ \Delta = R_{\text{linear}} - 0.5. \]

    We introduce a small parameter \(\delta\) to control sensitivity. Adjust \( K \) slightly based on \(\Delta\): \[ K_{\text{dynamic}} = K_{\text{env}} \times (1 - \delta \Delta). \]

    • If \(\Delta > 0\) (favor Gaiacoin), \( K_{\text{dynamic}} < K_{\text{env}} \) slightly, nudging the solution toward higher Gaiacoin emissions.
    • If \(\Delta < 0\) (favor Endcoin), \( K_{\text{dynamic}} > K_{\text{env}} \) slightly, nudging the solution toward higher Endcoin emissions.
    • If \(\Delta = 0\), \( K_{\text{dynamic}} = K_{\text{env}} \), leaving the distribution unchanged from the environment baseline.

  4. Solving the Adjusted System

    We now solve: \[ E_{\text{final}} + G_{\text{final}} = E_{\text{total-env}}, \] \[ E_{\text{final}} \cdot G_{\text{final}} = K_{\text{dynamic}}. \]

    Substitute \( G_{\text{final}} = E_{\text{total-env}} - E_{\text{final}} \): \[ E_{\text{final}}(E_{\text{total-env}} - E_{\text{final}}) = K_{\text{dynamic}}, \] \[ -E_{\text{final}}^2 + E_{\text{total-env}}E_{\text{final}} - K_{\text{dynamic}} = 0. \]

    Solve the quadratic equation for \( E_{\text{final}} \). There will be two solutions; choose the one that yields a final ratio \( G_{\text{final}} / E_{\text{total-env}} \) consistent with the slight sentiment-driven direction. Because we only made a small adjustment to \( K \), the final solution will closely approximate the environment values at zero sentiment and smoothly tilt toward one token or the other as sentiment changes.

Summary:

This revised formula ensures that temperature-driven emissions remain the primary factor determining the daily distribution of Endcoin and Gaiacoin. The sentiment introduces only minor adjustments around the environment-defined baseline, achieved by slightly shifting the product \( K \) of the CPMM-like relation. As a result, at neutral sentiment, the final emissions match the environment scenario exactly. Small positive or negative sentiment causes proportionally small, smooth deviations in the emission ratio, resisting extreme swings and maintaining overall stability.