Resource-Bound Democracy Ceiling Mathematical Model

Resource-Bound Democracy Ceiling Model

---

1. System State Definition

The macro-economic environment is represented as a dynamic state vector:

𝑆(𝑡)=⟨𝑅(𝑡),𝑃(𝑡),μ(𝑡),𝐺(𝑡),𝐺𝐷P𝗀(𝑡),𝑈(𝑡)⟩

Where:

• (𝑅(𝑡)) = total national resources / wealth index

• (𝑃(𝑡)) = population

• (μ(𝑡)=𝑅(𝑡)/𝑃(𝑡)) = mean per-capita wealth

• (𝐺(𝑡)) = inequality index, (0 ≤ 𝐺 ≤ 1)

• (𝐺𝐷P𝗀(𝑡)) = GDP growth rate (normalized)

• (𝑈(𝑡)) = social-stress index (unemployment, wage-gap, housing)

Commentary:
The system evolves continuously in time. The wealth ceiling must remain consistent with macro-state changes.

---

2. Boundary Logic — Democratic Stability Conditions

Tyranny suppression condition:

lim₍G(t) → 1₎ Wₘₐₓ(t) = 0

Equality stability condition:

lim₍G(t) → 0₎ Wₘₐₓ(t) = k(t) · μ(t)

Continuity requirement:

Wₘₐₓ(t) ∈ C¹(ℝ⁺)

Commentary:
The model guarantees tightening at extreme inequality and relaxation in equitable states, without discontinuities or shocks.

---

3. Base Wealth Ceiling Function

Wₘₐₓ(t) = k(t) · μ(t) · (1 − G(t))ⁿ⁽ᵗ⁾

Where:

• ((1−𝐺(𝑡))) = inequality attenuation factor

• (𝑛(𝑡)>0) = sensitivity exponent

Range guarantee:

0 ≤ (1 − G(t))ⁿ⁽ᵗ⁾ ≤ 1

Commentary:
As inequality rises, the attenuation factor shrinks and the ceiling tightens smoothly. When inequality falls, economic freedom expands.

---

4. Innovation-Weighted Merit Multiplier

𝑘(𝑡)=𝑐₀+𝑎₁ⱱⱼ(𝑡)+𝑎₂ⱱᵣ𝒹(𝑡)+𝑎₃ⱱₑ(𝑡)

Where:

• (𝑐₀ ≥ 1) = base merit constant (protects average workers)

• ⱱⱼ(𝑡) = normalized employment contribution

• ⱱᵣ𝒹(𝑡) = normalized research / innovation intensity

• ⱱₑ(𝑡) = normalized export contribution

Domain:

0 ≤ ⱱⱼ,ⱱᵣ𝒹,ⱱₑ ≤ 1

Bound:

1 ≤ k(t) ≤ kₘₐₓ

Commentary:
This prevents zero-wealth traps and ensures every citizen may accumulate at least up to the national mean, while superior contributors earn higher limits.

---

5. Adaptive Sensitivity Dynamics (Feedback Control)

dn/dt = σ · U(t) − β · GDPg(t)

With:

• (σ>0) = social-stress tightening gain

• (β>0) = prosperity relaxation factor

Projection bounds:

nₘᵢₙ ≤ n(t) ≤ nₘₐₓ

Commentary:
Instead of forcing the ceiling directly, the strictness of response adapts.
Stress → tighter inequality response
Stability → softer response

This avoids oscillation or policy shock.

---

6. Hard Safety Cap (Choose Implementation Mode)

A) Systemic-Risk Cap (Recommended)

W𝑐𝑟𝑖𝑡(t) = λ · R(t)

Where:

0 < λ ≪ 1

(e.g., λ = 0.01 ⇒ 1% national-resource control limit)

Commentary:
Prevents oligarchic capture without harming individual prosperity.

---

B) Mean-Multiple Ceiling (Alternative)

Wcrit(t) = Λ · μ(t)

Where:

Λ ≫ 1

(e.g., Λ = 10⁴)

Commentary:
Allows extreme success while bounding dynastic dominance.

---

Final Operative Constraint

Wₘₐₓ(t) = min( Wₘₐₓ(t), W𝚌𝚛𝚒𝚝(t) )

---

7. Overflow Conversion (Circulation Operator)

Overflow:

Ω(t) = max(0, W − Wₘₐₓ(t))

Allocation:

Ω(t) = Φᵢ(t) + Φ𝑐(t) + Φₚ(t)

Where:

Φᵢ = θᵢ Ω,     Φ𝑐 = θ𝑐 Ω,     Φₚ = θₚ Ω

θᵢ + θ𝑐 + θₚ = 1,      θᵢ, θ𝑐, θₚ ≥ 0

Commentary:
Excess wealth becomes:

• innovation capital

• citizen dividends

• infrastructure investment

without confiscation or punishment.

---

8. Lyapunov-Style Stability Condition

Define:

L(t) = G(t) + Ω(t)

Stability requirement:

dG/dt ≤ 0,   dΩ/dt ≤ 0,   L(t) → 0⁺

Commentary:
A democracy is stable when:

• inequality trends downward

• overflow remains negligible

• wealth circulation remains active

---

Model Summary Commentary

This mathematical framework:

• allows wealth accumulation

• rewards innovation & social value

• prevents oligarchic concentration

• self-adjusts via inequality feedback

• preserves incentive while protecting democracy

No policymaker discretion.
No lobbying override.
The ceiling “moves with society.”

It tightens only when inequality rises
and relaxes when prosperity is broadly shared.