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This document contains the complete mathematical framework underlying the Autophage Protocol, including advanced formulas, proofs, and implementation details not covered in the main mathematical reference.
Continuous Decay Differential Equation
$$\frac{dV}{dt} = -\lambda V$$
(A.1)
This fundamental equation describes how token value decreases over time, creating the metabolic property that makes tokens "alive." Unlike traditional currencies that maintain value indefinitely, Autophage tokens must be continuously regenerated through health activities.
Soft Cap Acceleration
Progressive decay acceleration prevents extreme wealth concentration while allowing success:
$$\lambda_{effective} = \begin{cases}
\lambda_{base} \times 1.05 & \text{if } V > 10,000 \\
\lambda_{base} \times 1.10 & \text{if } V > 20,000 \\
\lambda_{base} \times 1.25 & \text{if } V > 50,000 \\
\lambda_{base} \times 1.50 & \text{if } V > 100,000
\end{cases}$$
(A.2)
Someone with 100,000 Rhythm tokens experiences 7.5% daily decay instead of 5%, creating natural redistribution.
Reservoir Inflow Dynamics
$$R_{inflow}(t) = \sum_{i \in \{R,H,F,C\}} \lambda_i \times V_i(t)$$
(A.3)
All decayed tokens flow to the Reservoir for redistribution. This creates a community wealth pool that grows with system activity, funding public health initiatives and rewards.
$$T_{gen} = B \times M \times S \times G \times N \times Sy \times Ge \times V \times A \times \eta(N) \times C(h) \times P(N) \times F_{strategy}$$
(A.4)
Subject to constraints: $T_{gen} \in [0.3B, 20B]$ where $B$ is base reward. Each multiplier component serves a specific behavioral incentive:
Multiplier Components
Consistency Multiplier
$$M_{consistency} = 1 + \log_{10}(\text{consecutive\_days})$$
(A.5)
Group Activity Multiplier
$$M_{group} = 1 + 0.75 \times \min\left(\frac{\text{participants} - 1}{10}, 1\right)$$
(A.6)
Synergy Bonus
$$Sy = \begin{cases}
1.0 & \text{if activities} = 1 \\
1.1 & \text{if activities} = 2 \\
1.2 & \text{if activities} = 3 \\
1.25 & \text{if activities} \geq 4
\end{cases}$$
(A.7)
Complete Price Discovery Formula
$$P(t) = \frac{E_{health}(t) \times (1 + \gamma C_{ratio}(t)) + V_{market}(t) \times (1 - C_{ratio}(t))}{S_{active}(t) \times V(t) \times (1 + M_{activity}(t))}$$
(A.8)
Where:
- $E_{health}(t) = 0.80$ USD average health activity energy cost
- $\gamma \in [1.2, 3.0]$ energy gradient based on system health
- $C_{ratio}(t) = \frac{V_{catalyst}(t)}{\sum_{i \in \{R,H,F,C\}} V_i(t)}$ is the catalyst token ratio (target: 0.25)
- $S_{active}(t)$ active token supply
- $V(t)$ velocity of token circulation
- $M_{activity}(t)$ activity level multiplier
The catalyst ratio $C_{ratio}(t)$ is the key to metabolic price discovery. When catalyst tokens are exactly 25% of total supply, the pricing formula balances energy costs and market valuation equally. When catalyst ratio deviates:
- High catalyst ratio (>0.25): Prices weight more toward energy costs, reducing speculation
- Low catalyst ratio (<0.25): Prices weight more toward market valuation, increasing liquidity
Energy Gradient Calculation
$$\gamma = 1.2 + 1.8 \times \left(1 - \frac{H_{system}}{100}\right)$$
(A.9)
System health $H_{system}$ ranges from 0-100%, with lower health triggering higher energy gradients to incentivize participation during low activity periods.
Inverted Kleiber's Law for Networks
$$C_{individual}(N) = C_{base} \times \left(\frac{N}{N_0}\right)^{-1/4}$$
(A.10)
Individual metabolic capacity decreases with network size, but total network capacity increases. This creates sustainable growth where early adopters have higher individual capacity (64× at N=1) while maintaining system stability at scale (1.002× at N=1B).
Liebig's Law Implementation
$$H_{system} = \min(H_{users}, H_{apps}, H_{reserves}, H_{velocity}, H_{geographic})$$
(A.11)
System health equals its weakest component. If any dimension falls below 50%, targeted multipliers double to restore balance.
Catalyst Token Generation
$$T_{catalyst} = Q \times R \times P_{tier} \times V_{market}$$
(A.12)
Where:
- $Q$ = Quality score (0-1) from verification richness
- $R$ = Rarity multiplier based on activity uniqueness
- $P_{tier}$ = Privacy tier multiplier (1.0, 1.75, 3.5)
- $V_{market}$ = Market velocity factor
Privacy Tier Pricing
$$P_{proof} = P_{base} \times \begin{cases}
1.0 & \text{Anonymous} \\
1.75 & \text{ProfileID revealed} \\
3.5 & \text{Genetic traits displayed}
\end{cases}$$
(A.13)
Contribution Score
$$S_{contribution} = \sum_{t=0}^{T} D_t \times (0.95)^{T-t} + R_{reputation}$$
(A.14)
Lifetime Reservoir contributions decay at 5% annually to weight recent participation while acknowledging history. Reputation score adds non-monetary contributions like successful proposals and community service.
Proposal Success Probability
$$P_{success} = \frac{1}{1 + e^{-k(I-I_0)}}$$
(A.15)
Sigmoid function where $I$ is health improvement percentage, $I_0 = 5\%$ threshold, and $k = 0.5$ steepness. Proposals achieving 5%+ health improvement have >50% chance of stake return plus bonus.
Gini Coefficient Convergence
$$G(t) = G^* + (G_0 - G^*) \times e^{-\alpha t}$$
(A.16)
Where $G^* \in [0.32, 0.38]$ is target equilibrium, $G_0$ is initial inequality, and $\alpha = 0.02$ is convergence rate. System reaches stable wealth distribution within 180 days.
Monte Carlo Confidence Intervals
$$CI_{95\%} = \bar{X} \pm 1.96 \times \frac{\sigma}{\sqrt{n}}$$
(A.17)
With $n = 10,000$ simulation runs, confidence intervals validate theoretical predictions. All key metrics converge within 2% of mathematical expectations.
Vault Balance Calculation
Wellness vaults protect health savings with reduced decay rates:
$$V_{vault}(t) = V_0 \times e^{-\lambda_{vault} \times t} \times (1 + y)^t$$
(A.18)
Where:
- $\lambda_{vault} = 0.001$ (0.1% daily decay vs 5% for Rhythm tokens)
- $y = 0.0002$ (0.02% daily yield)
- Vaulted tokens can only be used for qualified health expenses
Seasonal Health Multipliers
$$M_{seasonal} = 1 + 0.25 \times \sin\left(\frac{2\pi(\text{month} - 3)}{12}\right)$$
(A.19)
Natural health cycles affect token generation. Peak in spring (March), trough in fall (September). Multiplier ranges from 0.75× to 1.25×.
Price Volatility Protection
$$\text{If } |\Delta P_{24h}| > 0.3 \text{ then } \begin{cases}
V_{max} = V_{max} \times 0.5 \\
S_{bounds} = S_{bounds} \times 1.5 \\
T_{delay} = 300 \text{ seconds}
\end{cases}$$
(A.20)
When 24-hour price change exceeds 30%, velocity caps reduce by 50%, slippage bounds increase by 50%, and 5-minute transaction delays activate.
Bank Run Prevention
$$\text{If } W_{1h} > 0.05 \times S_{total} \text{ then } W_{max} = \min(0.01 \times B_{user}, 10000)$$
(A.21)
If hourly withdrawals exceed 5% of total supply, individual withdrawal limits activate: 1% of user balance or 10,000 tokens, whichever is smaller.
Sybil Attack Detection
$$\text{If } N_{new,1h} > 1000 \text{ then } \begin{cases}
V_{new} = 100 \text{ tokens/day max} \\
T_{verification} = 48 \text{ hours} \\
S_{proof} = \text{"enhanced"}
\end{cases}$$
(A.22)
Circulation Velocity
$$V_{circ} = \frac{\sum_{30d} T_{volume}}{S_{active}} \times \frac{30}{t}$$
(A.26)
Token velocity normalized to 30-day periods. Healthy range: 2-10× monthly. Below 2× indicates hoarding, above 10× suggests speculation.
Genetic Trait Cost Function
$$C_{trait} = \sum_{i=1}^{n} 1000 \times (L_{old,i} + 1) \times (L_{new,i} - L_{old,i})$$
(A.27)
Upgrading genetic traits becomes exponentially expensive at higher levels. Level 0→1 costs 1,000 tokens, while level 9→10 costs 10,000 tokens.
Network Health Score
$$H_{network} = \left(\prod_{i=1}^{5} H_i\right)^{1/5} \times \left(1 - \frac{G_{actual} - G_{target}}{G_{target}}\right)$$
(A.28)
Geometric mean of health dimensions (users, apps, reserves, velocity, geography) adjusted by Gini coefficient deviation from target range [0.32, 0.38].
Proof Rarity Multiplier
$$M_{rarity} = 1 + \log_{10}\left(\frac{N_{total}}{N_{activity} + 1}\right)$$
(A.29)
Rare activities earn higher marketplace premiums. If only 100 of 1 million users complete ultramarathons, rarity multiplier = 5×.
Governance Contribution Score
$$C = 0.5 \times \log_{10}(1 + R_{lifetime}) + 0.3 \times V_{reputation} + 0.2 \times A_{participation}$$
(A.30)
Where:
- $R_{lifetime} = \sum$(Tokens decayed + Tokens donated + Fees paid)
- $V_{reputation} = \frac{\text{Successful verifications}}{\text{Total verifications}} \times \log_{10}(1 + \text{Total verifications})$
- $A_{participation} = \min(1, \frac{\text{Days active in 90}}{90}) \times (1 + \text{Consistency bonus})$
Progressive USDC Bonus Formula
$$\text{USDC Bonus} = 100 \times M(P) \times \max(0, P - 1.0) \times \sqrt{\frac{U}{1000}}$$
(A.31)
Performance multiplier $M(P)$ scales with achievement level:
- 0.5× for 100-150% of target
- 1.0× for 150-200% of target
- 2.0× for 200-300% of target
- 3.0× for 300-400% of target
- 4.0× for 400%+ of target
Dynamic Balance Equation
$$R(t+1) = R(t) + I(t) - W_c(t) - W_b(t) - S(t)$$
(A.32)
Where:
- $R(t)$ = Reservoir balance at time $t$
- $I(t)$ = Inflows (fees + circulation)
- $W_c(t)$ = Cash withdrawals
- $W_b(t)$ = Benefit redemptions
- $S(t)$ = Public health spending
Time-Decayed Cash Access Function
$$A_{cash}(u,t) = \sum_{i=0}^{t} C(u,i) \times D(t-i)$$
(A.33)
Decay function $D(y)$ preserves recent contributions:
- 100% value if less than 1 year old
- 75% value if 1-2 years old
- 50% value if 2-5 years old
- 30% value if 5+ years old
Genetic Baseline Modification Cost
$$\text{Cost} = \begin{cases}
1 \text{ point} & \text{to increase trait by +1} \\
2 \text{ points} & \text{to decrease trait by -1}
\end{cases}$$
(A.34)
Returning nutrients to the system requires more energy than allocating them. Maximum trait value: 10.
Specialist and Generalist Bonuses
$$M_{specialization} = \begin{cases}
1.5 & \text{if any trait} \geq 7 \text{ (specialist)} \\
1.25 & \text{if all traits} \geq 3 \text{ (generalist)} \\
1.0 & \text{otherwise (default)}
\end{cases}$$
(A.35)