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Spatial Definition and Stability Conditions of a P2P Contribution Structure with Six-Directional Tension Vectors (v1.1)

This version (v1.1) includes additional mathematical control terms to model institutional fairness, deflation-resistance, and activity sustainability.

This paper mathematically proves that a P2P contribution structure possesses an unbreakable "six-directional tension-sphere" as its institutional space, ensuring spatial stability.

■ Mathematical Formulation (Six-Directional Space and Institutional Stability)

Definition of the Six-Directional Tension Field

σ_S = {σ₁, σ₂, σ₃, σ₄, σ₅, σ₆}
σᶍ = kᶍ ⋅ Xᶍ (j = 1, 2, …, 6)

The components of each σᶍ (Credit, Circulation, Choice, Boundary, Production, Will) are defined below.

Institutional Stability Condition (Spatial Convergence)

H(fᵢ, gᵢ, zᵢ; εᵢ, βᵢ) ∈ σ_S

Condition for Spatial Integrity

∀ᶍ ∈ {1, …, 6}, σᶍ > 0 ⇒ σ_S is closed and stable

■ Definition of Each Vector σᶍ

Vector Name Formula Explanation
σ₁ Credit σ₁ = k₁ · Sᵢ · φ(Gᵢ) Sᵢ: transparency of financial reports   Gᵢ: total prestige   φ(Gᵢ): fairness of prestige distribution (Gaussian, Lorenz correction, etc.)
σ₂ Circulation σ₂ = k₂ · Gᵢ Represents metabolic flow of tokens and knowledge; shows self-renewal capability.
σ₃ Choice σ₃ = k₃ · (1 − Aᵢ) · Pᵢ Aᵢ: token acceptance rate (coerciveness of rewards)   Pᵢ: number of contribution actions
σ₄ Boundary σ₄ = k₄ · Bᵢ⁻ Bᵢ⁻: external boundary (nationality, birth, cultural inheritance) that shields the system.
σ₅ Production σ₅ = k₅ · (Fᵢ + Wᵢ) Fᵢ: physical production (e.g., farming)   Wᵢ: supply of skills / wisdom sustaining the system.
σ₆ Will σ₆ = k₆ · Nᵢ Nᵢ: count of participatory actions; indicates whether the system is driven by intentional actors.

■ Geometric Properties of the Six-Directional Tension Sphere σ_S

  • σ_S is a tension field composed of six independent, complementary vectors (σ₁–σ₆) that guarantee institutional integrity.
  • Geometrically, it forms a dynamic sphere whose cross-section resembles a rotating “asanoha” (hemp-leaf) pattern, always maintaining a closed shape.
  • Stability conditions:
    1. ∀ᶍ, σᶍ > 0
    2. H(fᵢ, gᵢ, zᵢ; εᵢ, βᵢ) ∈ σ_S
  • The fgz engine grows internally along the time axis, while σ_S provides external spatial support; their coexistence stabilizes the institution.

■ Significance of the Spatial Structure and Stability

  • The six-directional field σ_S formalizes spatial stability for P2P contribution systems.
  • σ_S, a closed sphere of Credit, Circulation, Choice, Boundary, Production, and Will, spatially ensures integrity.
  • Internal structure (f, g, z; ε, β) converges to π over time, while external structure (σ₁–σ₆) supports the system in space, yielding a life-like institution with self-preservation.

■ Mathematical Extensions for Institutional Stability

To further reinforce fairness, sustainability, and non-inflationary properties within the system, we introduce the following control terms:

1. Prestige Suppression Coefficient α

H(f, g, z; ε, β) = ε ⋅ β ⋅ z ⋅ ∛(f ⋅ g^α)
  • α is a coefficient within the range 0 < α < 1 (recommended: 0.6–0.8).
  • This prevents overreliance on prestige and maintains action-based valuation.

2. Burn Structure and Token Supply Dynamics

dS(t)/dt = E(t) − B(t),  B(t) = λ ⋅ S(t)
  • λ: burn rate (recommended: 0.01/day)
  • This model controls circulating tokens and prevents inflation through exponential decay.

3. Endogenizing Institutional Activity z

z = ((p + v + r) + γ ⋅ Σf + δ ⋅ Σg) / (3 + γ + δ)
  • Suggested parameters: γ = δ = 1.0
  • Institutional vitality dynamically reflects both action and trust contributions from participants.

Summary

These coefficients (α, λ, γ, δ) enhance the model’s controllability and predictive power while preserving fairness and regenerative cycles, allowing for adaptive institutional design.

■ Correspondence with Lyapunov Stability

Let H(fᵢ, gᵢ, zᵢ; εᵢ, βᵢ) be non-negative, continuous, and asymptotically convergent; then H can serve as a Lyapunov function:

  1. H(t) ≥ 0 (non-negative)
  2. limₜ→∞ H(t) = π (asymptotic convergence)
  3. H(t) ∈ σ_S (spatial containment)

If σ_S is assumed to be a closed, bounded set, the fact that H(t) remains within σ_S implies convergence to an invariant set, reinforcing structural stability.

■ Toward Mathematical Rigor of Institutional Stability

  1. Parameter Estimation & Measurability of σᶍ

    • Elements such as Sᵢ, Gᵢ, Pᵢ, Bᵢ⁻, Fᵢ, Nᵢ must be measurable.
    • Constants kᶍ require explicit scaling and weighting.
    • A monitoring design is needed to map σᶍ to empirical metrics.

  2. Concrete Lyapunov Modeling of H

    • Analyze ∂H/∂t to show monotonic decrease or quasi-stable trajectories, supporting asymptotic stability.
    • Model conditions driving the system to z = 0 (institutional death) or instability due to loss of any σᶍ.

These extensions position the model at the intersection of institutional theory, control theory, and quantitative social science, enabling both academic study and practical implementation.