Mathematical Theory Cross-References

Token Economy Simulation Suite - Mathematical Reference Guide

This document provides cross-references between the unified mathematical framework and the subsystem-specific formulations.

1. Affine System Integration Points

1.1 Token Supply Dynamics

Unified Framework: $S_i(t+1) = S_i(t) + \Delta S^{\text{inflow}}(t)$
Affine Specific: Buy/sell transactions $\Delta T_{j,i}(t)$
Cross-Reference: Chapter 2.2.1, Chapter 3.2.1

1.2 Commission Rate Dynamics

Unified Framework: $C_j(t) \in [C_{\min}, C_{\max}]$
Affine Dynamics: Based on holding average, market performance
Cross-Reference: Chapter 1.4.2, Section A3.2 (Appendix)

2. Airdrop System Integration Points

2.1 Demand Multiplier Evolution

Unified Framework: $\alpha_i(t+1) = \min(1, \alpha_i(t) + \delta)$
Airdrop Specific: Vesting-dependent growth
Cross-Reference: Chapter 4.2.1, Theorem 2

2.2 Market Sentiment Coupling

Unified Framework: $M(t+1) = f(M(t), \epsilon_t)$
Airdrop Application: Price signal modifier
Cross-Reference: Chapter 3.4.1

3. Bonding Curve Laboratory Integration

3.1 Price Memory Representations

Unified Framework: $M_{i,t} \in \mathbb{R}^W$
Bonding Specific: Rolling price windows
Cross-Reference: Chapter 4.3.1, Theorem 2

3.2 Trend Signal Processing

Unified Framework: $\mathbf{P}(signal) = \sigma(\mathbf{w} \cdot \mathbf{z})$
Bonding Application: Buy/sell decision probability
Cross-Reference: Chapter 3.3.1

4. MCP System Integration

4.1 Resource Capacity Management

Unified Framework: $C_j(t+1) = \min(C_{\max}, C_j(t) \cdot g(t))$
MCP Specific: Base regeneration + economic feedback
Cross-Reference: Chapter 2.3.1, Theorem 4

4.2 Allocation Priority Mechanisms

Unified Framework: $\phi: \mathbf{Actions}, \mathcal{E} \to \mathbf{Allocations}$
MCP Implementation: Random-ordered sequential allocation
Cross-Reference: Chapter 5.2

5. Subsystem-Specific Mathematical Extensions

5.1 Affine-Derived Concepts

Whale Agent Classification: Based on commission rate thresholds
Cross-Token Price Correlation: Inter-dependent market dynamics
Dynamic Curve Parameters: Time-varying bonding curve modifications

5.2 Airdrop-Derived Concepts

Tiered Distribution Logic: Eligibility-weighted allocation
Vesting Schedule Mathematics: Time-dependent accessibility functions
Activity Threshold Mechanics: Performance-based qualification
Market Sentiment Feedback: Auto-correlation and erosion factors

5.3 Bonding-Derived Concepts

Memory-Based Trend Analysis: Historical price pattern recognition
Volatility-Driven Trading: Risk-adjusted decision making
Multi-Segment Curve Dynamics: Phase-transition pricing behavior

5.4 MCP-Derived Concepts

Capacity Regeneration Models: Economic activity-dependent recovery
Load Distribution Functions: Resource utilization balancing
Priority Allocation Schemes: Competition resolution mechanisms
Maintenance Cost Functions: Ongoing operational requirements

6. Mathematical Translations Between Systems

6.1 Concept Equivalences

Universal Concept	Affine Terms	Airdrop Terms	Bonding Terms	MCP Terms
Price Signal	$P_i(t)$	$P(t) + M(t)$	$f(S(t))$	$Pr_j(t)$
Supply Control	$\Delta T_{j,i}$	$\Delta S^{\text{burn}}$	$\Delta S$	$L_j(t+1)-L_j(t)$
Agent Utility	$B_j(t)$	$H_i(t)P(t)$	$C_i+T_i P(t)$	$B_i(t)$
Resource Allocation	Commission share	Distribution fraction	Curve dynamics	Capacity extension

6.2 Parameter Mapping Functions

Φ_affine_to_mcp(b_j): Affiliate.balance → MCP.balance = b_j / scale_factor
Φ_mcp_to_affine(l_j): MCP.load → Affine.supply_projection = base_token * (1 + l_j/capacity)
Φ_bonding_to_affine(σ_p): Bonding.volatility → Affine.commission_rate = min(max_rate, σ_p * γ)
Φ_airdrop_to_all(t_t): Airdrop.vesting_period → Universal.time_window = t_t

These cross-references enable seamless translation between subsystems while preserving mathematical rigor and economic intuition. The mappings preserve the essential dynamics while accommodating system-specific implementation details.