Derivation of the Self-Stake Requirement

1. Design goals

Let q be the total stake an orchestrator wants to qualify, including its self-stake, and let B(q) be the required operator bond.

We want a requirement that:

• Starts at 50% for up to 1,000 LPT of qualified stake.
• Declines to 1% at 1,000,000 LPT, with a 1% floor thereafter.
• Decreases continuously in percentage between those endpoints.
• Requires an increasing absolute amount of self-stake as qualified stake grows.
• Uses a simple, predictable formula without intermediate thresholds.

2. Choosing and deriving the power law

A power law provides constant proportional scaling: within the declining interval, each 10× increase in qualified stake produces the same proportional reduction in the required fraction. This is a design choice, not something uniquely implied by the endpoints.

For endpoints (q₀, p₀) and (q₁, p₁), write:

$$ p(q)=p_0\left(\frac{q}{q_0}\right)^{-\alpha} $$

Here p(q) is the required self-stake fraction, so 0.50 means 50%. Applying the second endpoint gives:

$$ \frac{p_1}{p_0}=\left(\frac{q_1}{q_0}\right)^{-\alpha} \qquad\Longrightarrow\qquad \alpha=\frac{\ln(p_0/p_1)}{\ln(q_1/q_0)} $$

Using q₀ = 1,000, p₀ = 0.50, q₁ = 1,000,000, and p₁ = 0.01:

$$ \boxed{\alpha=\frac{\ln 50}{\ln 1000}\approx0.566323335} $$

Use the exact logarithmic expression in the formulas; the decimal is an approximation.

3. Required self-stake fraction

The complete requirement is: