Status: pre-launch. This page describes target design; this document does not establish a deployed service.
L(x) = x/(1-x): The Mathematical Regulator
Current reading — 2026-09-27
For 0 ≤ x < 1, L(x) = x/(1−x); it is undefined at x = 1 and diverges as x approaches 1 from below. A candidate price throttle does not prove liquidity, repayment, bounded distress or non-gameability. Utilization definitions, rates, hard limits and authorized failure controls are model-specific. See the Lombard Bridge risk boundary. [I]
This is an editorial boundary, not a new runtime contract, upstream repeal or proof of adoption. See current status and the product roles. Original revision dates remain as recorded below.
Historical research draft — retained on 2026-09-27; not current terms or implementation instructions
The stereographic projection function L(x) = x/(1-x) governs all rate-sensitive mechanisms in SoResFi. It replaces the need for discrete circuit breakers, governance-controlled rate caps, and emergency interventions. The math IS the regulator.
The Function
L(x) = x / (1 - x)
Where:
x = utilization ratio (0 <= x < 1)
L(x) = cost multiplier (0 to infinity)
| x (utilization) | L(x) (multiplier) | Interpretation |
|---|---|---|
| 0.00 | 0.00 | No utilization. Zero cost. |
| 0.10 | 0.11 | Light use. Negligible cost. |
| 0.25 | 0.33 | Moderate. Below base. |
| 0.50 | 1.00 | Equilibrium. Base rate. |
| 0.75 | 3.00 | Heavy use. 3x base. |
| 0.90 | 9.00 | Stressed. 9x base. |
| 0.95 | 19.00 | Critical. 19x base. |
| 0.99 | 99.00 | Near-failure. 99x base. |
| 1.00 | infinity | Impossible. Cannot reach. |
Key Property: Asymptotic Barrier
As x approaches 1, L(x) approaches infinity. This means:
- You can never fully utilize the system. The cost of the last unit approaches infinite.
- No discrete threshold is needed. Traditional circuit breakers trigger at an arbitrary point (e.g., "halt trading at 10% drop"). L(x) is continuous -- cost rises smoothly, with no discontinuity.
- No governance decision required. The barrier is mathematical, not political. No committee decides when to intervene.
Application: ZAI -> SKY Minting
When minting SKY from ZAI collateral:
minting_cost = base_cost * L(x)
Where x = total_sky_minted / (total_zai_collateral * max_ratio)
As the system approaches its collateralization limit:
- Minting becomes exponentially expensive.
- The last few percent of capacity cost orders of magnitude more than the first.
- Full utilization (x=1) requires infinite cost -- it is mathematically unreachable.
Application: Lombard Bridge Interest (Target Design)
When borrowing SKY via the Lombard Bridge:
interest_rate = 0.05 * L(x)
Where x = pool_utilized / pool_total
At 50% utilization, the rate is 5% (base). At 90%, it is 45%. At 99%, it is 495%. The lending pool cannot be fully drained because the cost of the last unit is infinite.
Application: PRISM Faucet
The Faucet dilution rate is modulated by L(x) applied to the ratio of new issuance to outstanding supply:
effective_dilution_cost = base_dilution_cost * L(dilution_ratio / cap)
As dilution approaches the 3% cap, the cost of further dilution climbs asymptotically.
Why Not a Circuit Breaker?
Traditional circuit breakers have three problems:
- Arbitrary thresholds: Who decides the trigger point? A committee. This introduces discretion, politics, and capture risk.
- Cliff effects: When the breaker triggers, the market goes from fully operational to fully halted. This discontinuity causes panic and front-running.
- Gaming: If the threshold is known, actors can position to profit from the halt.
L(x) solves all three:
- No threshold: The function is continuous. There is no trigger point to debate.
- No cliff: Cost increases smoothly. There is no discontinuity to panic around.
- No gaming: The cost at any point is determined by the current state. There is nothing to front-run.
The Equator: L(x=0.5) = 1.0
The point x = 0.5, where L(x) = 1.0, is the equator -- the natural operating point. Below the equator, the system operates below base cost (incentivizing use). Above the equator, costs escalate (discouraging over-use). The equator is the basin of attraction. Well-functioning markets oscillate around it.
L(x) is not a feature of SoResFi. It is the governing equation. Every rate, every cost, every capacity limit in the protocol is a specific instantiation of the stereographic projection. One function. Universal application. No discretion.