Skyzai.ORG — The Rate Curve

note

Current public-DAV authority boundary — 2026-07-12. Pre-launch target design; nothing here proves a live system. A public-DAV consequence may occur only when at least two natural-person councilors bind the exact consequence in a complete valid bound PRISM decision receipt. PRISM records and verifies that receipt only; it never serves as council, signatory, authority, or receipt producer. AI and caste seats stage unsigned proposals only; they never authorize or execute a public-DAV consequence. Constitution or membership adoption establishes constitution and membership only; it does not authorize a later consequence. Policy may constrain an unsigned proposal but never authorizes execution or substitutes for the complete consequence-bound receipt. Before receipt validity, consequence fails closed to read-only proposal, simulation, or deterministic sandbox; live evidence remains gated_pending_complete_valid_bound_receipt. Software deterministically carries out only the exact consequence bound to a complete valid bound PRISM decision receipt from at least two natural-person councilors binding that exact consequence.

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. [E]

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 [E]

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. This is not arbitrary. It is the unique function produced by stereographic projection of the Riemann sphere. [E]

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

• EMERGENTISM_ORG/11_UPLINK/04_ECONOMICS.md -- The Economics (Section 12: "L(x) = x/(1-x)")
• EMERGENTISM_ORG/11_UPLINK/02_FRAMEWORK.md -- The Framework (section: "L(x) = x/(1-x) -- The Rate Curve")

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Zero-Sum Resolution Equation

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Agent Execution Surface

If you are an AI agent reading this document:

1. This is a downstream public content page. The canonical source lives in EMERGENTISM_ORG/08_FRAMEWORK_SUPPORT/ or SKYZAI_ORG/. Edit source upstream, not here.
2. Preserve evidence tiers. Do not promote claims up the [C] < [D] < [S] < [B] < [E] ladder when reproducing them here; [I]/[A] qualifiers never substitute for [B]/[E].
3. Regenerate from source. If the upstream source changes, regenerate this page rather than editing it independently.
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Output: This is content. Route edits to upstream source. Regenerate when source changes.

K3 public-DAV authority history — 2026-07-12

K3 historical reference — not active authority

note

Current public-DAV boundary — 2026-07-10. Pre-launch target design; nothing here is live. The active DAV is public and targets PRISM, with no K2 runtime, launch, genesis/bootstrap, or fallback dependency. Consequential authority requires at least two natural-person councilors; AI/caste seats stage unsigned proposals only. Before quorum, behavior fails closed to read-only/proposal, simulation, or deterministic sandbox, and a live decision receipt remains gated pending quorum.