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Mathematical Specification for SBCM (v2.0)

General Theory of Administrative Hydraulics and Entropic Elasticity

Version: 2.0
Author: Hokuto Koyama (SBCM Alliance, github:Melnus)
References: SBCM Core Theory, Notes #6-#8, Case Studies 001-002

1. Fundamental Units & Potential Capacity

SBCM redefines the municipality not as a mere administrative label, but as a physical "Field Unit (Block)" with a finite capacity to retain energy (wealth).

1.1 The Standard Block ($B_{std}$)

The fundamental quantum of governance structure.

$$ B_{std} = \frac{P_{total}}{N_{muni}} \approx 72,176 \quad [\text{persons}] $$

1.2 Potential Entropic Capacity ($C_{pot}$) [Ref: Note #7]

The limit volume of complexity (wealth, information, infrastructure) that a single block can physically maintain.

$$ C_{pot} = \alpha \cdot P_{density} \cdot M_{mesh} $$

  • $P_{density}$: Population Density (Basic interaction potential).
  • $M_{mesh}$: Mesh Connectivity Coefficient (Local Multiplier/LM3). Higher internal circulation increases the capacity to trap entropy.
  • $\alpha$: Dimensional adjustment constant.

2. Field Theory & The Continuity Equation [Ref: Part 4]

Regional economy is described not as discrete points, but as a continuous "Fluid Field of Wealth."

2.1 The SBCM Governing Equation

The time evolution of Economic Density $\rho$ at coordinate $\mathbf{x}$ follows the Equation of Continuity:

$$ \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = \sigma - \delta(\rho) $$

  • $\rho(\mathbf{x}, t)$: Economic Density (Accumulated Wealth/Stock).
  • $\mathbf{J}(\mathbf{x}, t) = \rho \mathbf{v}$: Economic Flux Vector (Flow of Wealth).
  • $\nabla \cdot \mathbf{J}$: Divergence. The rate of inflow/outflow.
    • $\nabla \cdot \mathbf{J} > 0$: Leakage/Extraction (The Straw Effect).
    • $\nabla \cdot \mathbf{J} < 0$: Retention/Accumulation (Gravity Well).
  • $\sigma$: Source Term (Value Creation via Labor/Energy).
  • $\delta(\rho)$: Sink Term (Dissipation via Maintenance Costs/Entropy).

2.2 Vector Definition of the "Straw Effect"

The Straw Effect is defined as a state where flux vectors align towards a central high-gravity point (e.g., Tokyo), maximizing positive divergence in local regions.

$$ \text{Straw Effect} \iff \forall \mathbf{x} \in \text{Local}, \quad \nabla \cdot \mathbf{J}(\mathbf{x}) \gg 0 $$

3. Theory of Entropic Elasticity [Ref: Note #7]

A regional economy behaves as an Elastic Body, not an infinite container.

3.1 Hooke's Law of Economics

When external budget injection $S_{in}$ exceeds the local capacity $C_{pot}$, the system generates a Restoring Force ($F_{eject}$) that expels excess energy.

$$ \mathbf{F}_{eject} = -k (S_{in} - C_{pot}) $$

  • $k$: Institutional Rigidity Constant. A high $k$ (lack of local vendors/bureaucracy) causes violent rejection of capital, forcing it to bounce back to central contractors.

3.2 The Theorem of Leakage

If the injection speed exceeds the adaptation speed of the capacity, the retention rate approaches zero.

$$ \frac{dS_{in}}{dt} \gg \frac{dC_{pot}}{dt} \implies \lambda \to 1.0 \quad (\text{Total Ejection}) $$

4. Thermodynamic Limit of Growth [Ref: Note #6]

As system complexity ($A$) increases, the maintenance cost (Management Entropy) scales non-linearly.

4.1 Modified Growth Equation

$$ \dot{K} = s A^\sigma K^\alpha - \underbrace{\delta_0 A^\gamma K}_{\text{Complexity Cost}} $$

  • $A$: Scale of Intelligence/Urbanization (Complexity).
  • $\gamma$: Complexity Penalty Coefficient.
    • Centralization (Tokyo) and Digital Sprawl increase $\gamma$.
  • Heat Death Condition: When $\gamma \ge \sigma$, growth is thermodynamically cancelled by maintenance costs, leading to stagnation.

5. Extended Metrics for Physical Auditing

5.1 Fund Rotation Ratio ($R_{fund}$) [Ref: Case Study 001]

A metric to measure the "Metabolic Rate" of a government.

$$ R_{fund} = \frac{\text{Total Fund Balance}}{\text{Total Annual Revenue}} $$

  • $R_{fund} \gg 0.4$: Metabolic Stagnation (e.g., Tokyo). Wealth is frozen; liquidity trap.
  • $R_{fund} \approx 0.15$: Healthy Circulation (e.g., Aichi). Wealth is actively working.

5.2 Physical Security Barrier ($Cost_{total}$) [Ref: Note #8]

The firewall mechanism in G-Cart based on "Gravity."

$$ Cost_{total} = P_{bid} + \alpha \cdot (\text{Distance})^2 $$

  • By adding a cost proportional to the square of physical distance, the algorithm prevents predatory dumping by remote central capital, enforcing local thermodynamic optimization.

6. G-Cart Control Algorithm

Based on the physics above, the G-Cart protocol enforces the following inequality for all budget executions:

The Impedance Matching Condition

$$ I(t) \le \frac{dC_{pot}}{dt} $$

  • Function: Throttles the budget injection rate $I(t)$ to match the local capacity growth rate.
  • Objective: Prevents "Elastic Rebound" (Leakage) and induces "Plastic Deformation" (Permanent Industrial Growth).