Our Science | Sparse Supernova

Fundamental Research Published: Dec 2025

The Universal Saturation Principle

We have proven that hyperbolic saturation is not a coincidence, but a mathematical necessity in any system with finite capacity. This unifies biology, physics, and information theory into a single framework.

f(x) = x / (x + K)

The Discovery

Our research proves that the hyperbolic saturation form emerges universally in any system satisfying three axioms:

Finite Capacity: The system has a limit (e.g., enzyme sites, channel bandwidth, or agreement states).
Reversible Competition: Agents compete for this capacity (binding/unbinding).
Steady-State Equilibrium: The system reaches a balance where net flux is zero.

This unifies Michaelis-Menten kinetics, Langmuir isotherms, Monod growth, and Shannon channel capacity into a single master equation.

Applications & Impact

By understanding the "Geometric Concentration" of disagreement states (1/dim), we can apply this principle to solve complex problems:

AI Alignment: Deriving observer agreement from first principles using concentration of measure.
Social Consensus: Predicting agreement dynamics in large groups based on heterogeneity.
Distributed Systems: Optimizing Byzantine fault tolerance in blockchain ledgers.
Neural Binding: Modeling cortical synchronization and noise.

Read Paper on Zenodo View Code on GitHub

Applied Research Published: Dec 2025

How Sparse Supernova Uses USL in AI

The Universal Saturation Law (USL) is the foundation of Sparse Supernova's AI systems. It provides a control law for determining when extra capacity stops paying back, enabling us to balance performance, efficiency, and sustainability in real-time.

USL as a Control Law

USL defines the relationship between capacity (dimensionality) and drift (novelty or instability) in AI systems:

FRAI(dim, D) = D / (D + 1/dim)

- dim: The "capacity knob"—embedding dimensions, retrieved facts, or context window size.
- D: Drift, measuring how unstable or novel the world is for a specific user or project.

This formula allows Sparse Supernova to dynamically adjust routing, memory, and compute resources based on real-time conditions, ensuring optimal performance without waste.

Applications in Sparse Supernova Systems

USL is integrated across four critical areas of Sparse Supernova's architecture:

Memory Retrieval: Dynamically adjusts the number of facts retrieved based on drift, reducing token usage and improving efficiency.
Memory Promotion: Decides when to promote short-term memory (STM) to long-term memory (LTM) based on stability.
Model Selection: Determines when to escalate to larger models or deeper reasoning chains, avoiding unnecessary compute.
RAG Chunking: Optimizes chunk size and retrieval breadth in Retrieval-Augmented Generation (RAG).