Math_Specification/2_Invariants.ipynb

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    "# Cross applicability of invariants\n",
    "\n",
    "*This document is a mathematical analysis of the invariants on the bonding curve & the prediction market, and proves that the invariants will compose.*\n",
    "\n",
    "![img](https://i.imgur.com/lweSU59.png)\n",
    "\n",
    "The invariant $V = \\frac{S^\\kappa}{R}$ is conserved under the bond-to-mint and burn-to-withdraw mechanism. However, an attestation action may alter the value of $V_0$.\n",
    "\n",
    "Likewise, the invariant $I = \\frac{C\\alpha}{1-\\rho} = R + C\\alpha$ is conserved under the attestation mechanism, but its value may change upon a bond-to-mint or burn-to-withdraw action.  \n",
    "\n",
    "| | Bond/Burn | Attest Positive/Negative |\n",
    "| :-: | :-: | :-: |\n",
    "| $V(R, S)$ | Conserved | ? |\n",
    "| $I(\\alpha, R)$ | ? | Conserved |\n",
    "\n",
    "\n",
    "The constant values of $V_0$ upon bond/burn and $I_0$ upon attestation can be calculated from the initial values of corresponding state variables. \n",
    "\n",
    "| | Bond/Burn | Attest Positive/Negative |\n",
    "| :-: | :-: | :-: |\n",
    "| $V(R, S)$ | $V = \\frac{S^\\kappa}{R}$ | ? |\n",
    "| $I(\\alpha, R)$ | ? | $I = \\frac{C\\alpha}{1-\\rho} = R + C\\alpha$ |\n",
    "\n",
    "### Change in $V$ upon Attestation\n",
    "Consider that an agent has made a positive attestation action. The changes in the state variables corresponding to the attestation are\n",
    "\n",
    "$$S_{f, t+1} = S_{f,t} - \\Delta s$$\n",
    "$$Q_{1,t+1} = Q_{1,t} + \\Delta q_1$$\n",
    "$$\\alpha_{t+1} = \\varepsilon(\\alpha) + (1-\\varepsilon)\\left(\\frac{S_1 + S_0}{S_1+S_0+\\Delta s}\\right)(\\alpha)+\\left(\\frac{\\Delta s}{S_1+S_0+\\Delta s}\\right)(\\bar\\alpha)$$ \n",
    "\n",
    "where $\\bar\\alpha = \\frac{\\Delta s R}{A(C+R)-\\Delta s (C)}$ and $A = \\frac{1}{Q_1(Q_1+\\Delta q_1)} \\left[q_1(Q_1\\Delta s - \\Delta q_1 s)+\\Delta q_1(Q_1s_1+Q_1\\Delta s)\\right]$\n",
    "\n",
    "<br/>$V$ depends on the state variables $S, R, \\kappa$. Upon a positive attestation, \n",
    "* $S$ does not change, since $S = S_1+S_0+S_{free}$\n",
    "* $R$ does not change\n",
    "* $\\kappa$ changes due to its relationship with $\\alpha$. $\\kappa_{t}$ &rarr; $\\kappa_{t+1}$\n",
    "\n",
    "During Attestation, $I$ is conserved through the relationship\n",
    "$$I = \\frac{C \\alpha}{1-\\rho}$$ where $\\rho=1/\\kappa$. Solving for $\\kappa$, we have $$\\kappa = \\frac{I}{I-C\\alpha}$$\n",
    "\n",
    "Upon an attestation, $\\alpha_t \\rightarrow \\alpha_{t+1}$, so $\\kappa_t \\rightarrow \\kappa_{t+1}$ given by \n",
    "$$\\kappa_{t+1}=\\frac{I}{I-C\\alpha_{t+1}}\n",
    "$$\n",
    "\n",
    "This gives $V_t \\rightarrow V_{t+1}$ \n",
    "$$V_{t+1} = \\frac{S^{\\kappa_{t+1}}}{R}$$\n",
    "\n",
    "### Change in $I$ upon Bond/Burn\n",
    "Consider that an agent has made a bond action. The changes in the state variables corresponding to the bonding are\n",
    "$$R_{t+1} = R_t + \\Delta R$$\n",
    "$$S_{t+1} = \\sqrt[\\kappa]{V(R_t + \\Delta R_t)}$$\n",
    "$$P_{t+1} = \\frac{R_t + \\Delta R_t}{\\sqrt[\\kappa]{V(R_t + \\Delta R_t)}}$$\n",
    "\n",
    "$I$ depends on the state variables $C, \\alpha, R, \\rho$\n",
    "* $C$ does not change since it's a constant \n",
    "* $\\alpha$ does not change\n",
    "* $R$ changes. $R_{t} \\rightarrow R_{t+1}$\n",
    "* $\\rho$ does not change\n",
    "\n",
    "During Bond/Burn, $V$ is conserved under the relationship\n",
    "$$V = \\frac{S^\\kappa}{R}$$ \n",
    "\n",
    "Since $I = R + C\\alpha$ and $R_t \\rightarrow R_{t+1}$, we have\n",
    "$$ I_{t+1} - I_{t} = (R_{t+1} + C\\alpha) - (R_{t} + C\\alpha) $$\n",
    "$$ I_{t+1} - I_{t} = (R_{t+1} - R_{t})$$\n",
    "$$ I_{t+1} = I_{t} + \\Delta R$$\n",
    "\n",
    "The changed state variable $I_{t+1}$ can also be expressed as\n",
    "$$I_{t+1} = R_{t+1} + C\\alpha$$\n",
    "\n",
    "## Composing Results\n",
    "The changes in invariants $V$ and $I$ upon attestation and bond/burn are shown in the table below.  \n",
    "\n",
    "\n",
    "| | Bond/Burn | Attest Positive/Negative |\n",
    "| :-: | :-: | :-: |\n",
    "| $V(R, S)$ | $V =  \\frac{S^\\kappa}{R}$| $V_{t+1} = \\frac{S^{\\kappa_{t+1}}}{R}$|\n",
    "| $I(\\alpha, R)$ | $I_{t+1} = R_{t+1}+C\\alpha$ | $I = \\frac{C\\alpha}{1-\\rho} = R + C\\alpha$ |\n",
    "\n"
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    "## Effects of Attestations on Price\n",
    "\n",
    "W have identified that $\\kappa$ changes upon an attestation action. The price of a token $P_t$ at time $t$ is defined by \n",
    "$$P_t = \\kappa_t \\frac{R}{S}$$\n",
    "\n",
    "Given that $R$ and $S$ don't change during an attestation, the price $P_{t+1}$ after an attestion action is\n",
    "$$P_{t+1} = \\kappa_{t+1} \\frac{R}{S}$$\n",
    "\n",
    "\n",
    "Which gives \n",
    "$$\\Delta P = P_{t+1} - P_t = (\\kappa_{t+1} - \\kappa_t) \\frac{R}{S} = \\Delta \\kappa \\frac{R}{S}$$\n",
    "\n",
    "During an attestation, $I$ does not change and is given by \n",
    "$$I = \\frac{C\\alpha}{1-(1/\\kappa)}$$\n",
    "\n",
    "Solving for $\\kappa$\n",
    "$$\\kappa = \\frac{I}{I-C\\alpha}$$\n",
    "\n",
    "Thus $$\\Delta P = \\frac{R}{S} \\left(\\frac{I}{I-C\\alpha_{t+1}} - \\frac{I}{I-C\\alpha_{t}} \\right)$$\n",
    "\n",
    "Simplifying, $$\\Delta P = \\frac{IRC}{S} \\left(\\frac{\\Delta \\alpha}{(I-C\\alpha_{t+1})(I-C\\alpha_t)} \\right)$$\n",
    "\n",
    "Thus the price $P$ changes upon attestation through the relationship $\\Delta P = \\Delta \\kappa \\frac{R}{S}$"
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