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This document provides a comprehensive mathematical reference for all equations, formulas, and theoretical foundations of the Autophage Protocol. Each equation is presented with rigorous notation, followed by intuitive explanations and practical examples.

1Fundamental Token Dynamics

1.1Balance Evolution Equation

$$V_i^{(u)}(t+1) = V_i^{(u)}(t)(1 - \delta_i) + G_i^{(u)}(t)$$ (1)
Definition 1.1. The balance $V_i^{(u)}(t)$ represents the quantity of token species $i \in \{$Rhythm, Healing, Foundation, Catalyst$\}$ held by user $u$ at discrete time $t$.

This fundamental equation governs token balance evolution. The balance at time $t+1$ equals the previous balance reduced by decay factor $(1-\delta_i)$ plus newly generated tokens $G_i^{(u)}(t)$.

Example 1.1. Consider a user with 1000 Rhythm tokens ($\delta_{Rhythm} = 0.05$) who earns 50 new tokens. Their balance evolves as: $$V_{Rhythm}^{(u)}(t+1) = 1000(1 - 0.05) + 50 = 950 + 50 = 1000$$ The decay exactly balances generation, achieving equilibrium.

1.2Continuous-Time Formulation

For theoretical analysis, the continuous-time differential equation is employed:

$$\frac{dV}{dt} = -\lambda V + g(t)$$ (2)

where $\lambda = -\ln(1-\delta)$ transforms the discrete decay rate to a continuous decay constant.

Theorem 1.1. The solution to equation (2) with initial condition $V(0) = V_0$ and constant generation rate $g$ is: $$V(t) = V_0 e^{-\lambda t} + \frac{g}{\lambda}(1 - e^{-\lambda t})$$
Proof. Using the integrating factor method with $\mu(t) = e^{\lambda t}$, the general solution is obtained. As $t \to \infty$, $V(t) \to g/\lambda$, confirming the equilibrium balance.

1.3Half-Life Analysis

The half-life $t_{1/2}$ determines the time required for tokens to decay to half their initial value in absence of generation:

$$t_{1/2} = \frac{\ln(2)}{\lambda} = \frac{\ln(2)}{-\ln(1-\delta)}$$ (3)
Token Species Daily Decay $\delta$ Half-life $t_{1/2}$ Economic Purpose
Rhythm 0.05 13.51 days Daily activity incentive
Healing 0.0075 92.42 days Quarterly health cycles
Foundation 0.001 693.15 days Long-term value storage
Catalyst 0.02–0.10 34.66–6.93 days Dynamic stabilization
Table 1: Token species decay parameters and economic functions

2Token Generation Mechanics

2.1Multi-Factor Reward System

Token generation incorporates multiple factors to incentivize sustained healthy behavior:

$$G_i^{(u)}(t) = \sum_{j \in A_i} B_{i,j} \times M_{total}^{(u,j)}(t) \times \mathbb{1}[\text{activity } j \text{ verified}]$$ (4)

where $A_i$ denotes activities generating token species $i$, $B_{i,j}$ represents base rewards, and $\mathbb{1}[\cdot]$ is the indicator function ensuring verification.

2.2Multiplier Composition

$$M_{total} = M_{streak} \times M_{group} \times M_{time} \times M_{genetic} \times M_{synergy} \times M_{quality}$$ (5)

Subject to the constraint: $M_{total} \in [0.3, 20]$ to prevent exploitation while rewarding excellence.

2.2.1Streak Multiplier

$$M_{streak} = 1 + \log_{10}(\text{consecutive days})$$ (6)
Example 2.1. Streak progression:

2.2.2Circadian Rhythm Multiplier

$$M_{time} = 1 + 0.3\sin\left(\frac{2\pi(h - 6)}{24}\right)$$ (7)

This models natural biological rhythms, with peak rewards at 12:00 (noon) and minimum at 00:00 (midnight).

3Advanced Decay Mechanics

3.1Progressive Decay for Large Balances

To prevent excessive accumulation, decay rates increase progressively with balance:

$$\delta_i^{eff}(V) = \delta_i \times \left(1 + \beta \times \max\left(0, \frac{V - \tau}{\tau}\right)^\gamma\right)$$ (8)

where $\tau$ is the soft cap threshold, $\beta \in [0.5, 2.0]$ controls sensitivity, and $\gamma \in [1.0, 1.5]$ shapes the curve.

Example 3.1. For Foundation tokens with $\tau = 10,000$, $\beta = 1$, $\gamma = 1$:

3.2Catalyst Token Dynamic Decay

Catalyst tokens uniquely adjust their decay rate to maintain system equilibrium:

$$\delta_{catalyst}(t) = \delta_{base} \times \left(1 + \beta \times \left|\frac{V_{catalyst}(t)}{\sum_{i\in S} V_i(t)} - \rho_{target}\right|^\gamma\right)$$ (9)

This mechanism targets $\rho_{target} = 0.25$ (25% of total supply), increasing decay when the ratio deviates.

4System Economics and Equilibrium

4.1Gini Coefficient

The Gini coefficient measures wealth inequality:

$$G = \frac{\sum_{i=1}^n \sum_{j=1}^n |x_i - x_j|}{2n^2\bar{x}}$$ (10)

Equivalently, for sorted wealth $x_1 \leq x_2 \leq ... \leq x_n$:

$$G = \frac{2\sum_{i=1}^n i \cdot x_i}{n\sum_{i=1}^n x_i} - \frac{n+1}{n}$$ (11)
Theorem 4.1. Under the Autophage Protocol with parameters specified in Table 1, the Gini coefficient converges to $G^* \in [0.32, 0.38]$ for populations $n \geq 100$.

4.2Metabolic Price Discovery

Token prices emerge from energy costs and market dynamics:

$$P(t) = \frac{E_{health}(t) \times (1 + \gamma C_{ratio}(t)) + V_{market}(t) \times (1 - C_{ratio}(t))}{S_{active}(t) \times V(t) \times (1 + M_{activity}(t))}$$ (12)

This endogenous pricing mechanism ensures tokens reflect both creation effort and market valuation.

5Biological Scaling Laws

5.1Kleiber's Law for Networks

System efficiency scales with network size following biological principles:

$$\eta(N) = 1 + \alpha \times \left(\frac{N}{N_0}\right)^{1/4}$$ (13)

where $\alpha = 0.3$ and $N_0 = 1000$ normalize the scaling relationship.

Remark. The $1/4$ power law derives from fractal distribution networks in biological systems, optimizing resource distribution while minimizing transport costs.

5.2Liebig's Law of the Minimum

System health equals its weakest component:

$$H_{system} = \min(H_{users}, H_{reserves}, H_{apps}, H_{geographic}, H_{velocity})$$ (14)

This ensures balanced growth across all system dimensions, preventing unsustainable expansion in any single area.

For additional mathematical formulas including wellness vaults, circuit breakers, token dynamics, and governance mathematics, see the extended mathematics reference.