Skyzai.ORG — The Rate Curve
Status: pre-launch. This page describes target design; this document does not establish a deployed service.
Current reading — 2026-09-28
For 0 ≤ x < 1, x/(1−x) grows without bound as x approaches 1. Divergence alone does not prevent over-leverage or catastrophe: enforced limits, execution controls and adversarial evidence are separate requirements. Here x must be defined for each model; the normalized viability fraction is not automatically the debt-to-collateral ratio. The website's monetary illustration uses its explicitly labelled ratio and scale, not the predecessor's universal-rate claim. [I]
No adopted interest rate, revenue split, yield, instrument right or risk policy is established here. See SKY, LP-100 and current status.
Historical rate and financial-design proposal — not current terms or a safety guarantee
L(x) = x/(1-x). The one curve that governs ALL rates. Self-limiting. The circuit breaker.
The Function
L(x) = x / (1 - x)
Where x = nu / (phi + nu) is the normalized viability fraction (a value between 0 and 1). L(x) maps the unit interval to [0, infinity). It is the stereographic projection from the Riemann sphere. It is the odds form of any probability-like fraction. [D][A]
Every rate in the organism derives from this one function: interest rates, flow decay, risk multipliers, leverage limits, position sizing, fee schedules. Everything. [S]
The Curve
L(x)
|
20+ /
| /
10+ /
| /
5+ /
| /
3+ /
2+ /
1+- - - - - - - * - - - - - <-- EQUATOR (x=0.5, L=1.0)
| /
0.5+ /
0.2+ /
0+---+---+---+---+---+---+---+---+---+---+-> x
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
STAGNATION | SAFE ZONE | DANGER | CATASTROPHE
(0 - 0.2) | (0.3 - 0.7) | (0.8-0.9) | (> 0.95)
The Table
| x | L(x) | Meaning |
|---|---|---|
| 0.00 | 0.00 | No viability. Dead. |
| 0.25 | 0.33 | Low leverage. Conservative. |
| 0.50 | 1.00 | The equator. phi = nu. Maximum balance. Operating point. |
| 0.618 | 1.62 | The golden ratio. Phi-split threshold (61.8/38.2). |
| 0.80 | 4.00 | Danger zone. 4x leverage. One shock kills you. |
| 0.90 | 9.00 | Critical. 9x leverage. Liquidation imminent. |
| 0.95 | 19.0 | Catastrophe. System cannot survive perturbation. |
| 1.00 | infinity | Singularity. South pole. Pure viability, no coherence. |
Key Property: The Circuit Breaker
As x approaches 1, L(x) approaches infinity. The system CANNOT over-lever because the math forbids it. L(x) IS the circuit breaker. No governance vote, no human judgment, no override. The curve itself prevents catastrophe. [S]
This is why the equator is the operating point. Target design: demurrage would push idle ZAI away from 0. The Faucet closes at low transparency (prevents x approaching 1). Grace Exit is a target design (escape before catastrophe). The curve is the organism's immune system expressed as mathematics. [S]
Applications Across the Organism
SKY Interest Rate
Interest rate = 5% base x L(x)
Where x is the system utilization ratio.
| Utilization (x) | L(x) | Interest Rate |
|---|---|---|
| 50% | 1.00 | 5% |
| 61.8% | 1.62 | 8.1% |
| 80% | 4.00 | 20% |
| 95% | 19.0 | 95% |
High demand raises cost, which reduces demand, which lowers cost. Homeostasis without central control. [S]
Position Sizing
The Kelly criterion for optimal bet sizing maps directly to this curve. At the equator (x = 0.5), you bet half. As edge decreases, the curve pulls the bet size down faster than linearly. [S]
Risk Multipliers
Confidence thresholds use L(x). The Three Gates require 90% confidence (x = 0.9, L = 9) to open the Faucet. Below that threshold, the uncertainty premium is too high. [S]
Phi-Split
The golden ratio threshold (x = 0.618, L = 1.62) determines the constitutional split: 61.8% to LP treasury, 38.2% to the buy wall. Changing this split requires Lane B (supermajority) because it is encoded at the phi-split threshold. [S]
Why This Curve and Not a Linear Function
| Property | Linear | L(x) = x/(1-x) |
|---|---|---|
| Safe zone behavior | Constant slope | Approximately linear (easy to reason about) |
| Danger zone behavior | No warning | Superlinear amplification (impossible to ignore) |
| Boundary | No boundary | Infinite cost at x = 1 (impossible to cross) |
| Origin | Arbitrary | Stereographic projection from S² [D][A] |
A linear fee schedule (constant marginal cost) gives no warning before failure. L(x) provides near-linear behavior in the safe zone, superlinear warning in the danger zone, and infinite cost at the boundary. Within this specific derivation, it is the function produced by stereographic projection of the Riemann sphere onto this axis. [D][A]
Derivation
On S², the ratio phi/nu = cot²(theta/2). When x = nu/(phi+nu), the normalized viability fraction, L(x) = x/(1-x) maps the unit interval to [0, infinity). It is the odds form of any probability-like fraction. The geometry of the sphere generates the circuit breaker. No parameter tuning required. [S]
Sources
- The Economics (Section 12: "L(x) = x/(1-x)") (private source)
- The Framework (section: "L(x) = x/(1-x) -- The Rate Curve") (private source)