Why Recursive Systems Fail
From AI models to corporations to ecological networks, recursive systems—those that feed outputs back as inputs—face an invisible cliff. For years, engineers and scientists noticed “mysterious collapses” in complex systems. But what if there was a universal law behind this?
I propose the 1/e Law of Recursive Intelligence (URCI Law):
Any recursive intelligent system collapses structurally when its global coherence falls below 1/e (≈0.367).
This law defines a hard threshold for scaling recursive systems safely.
The Core Concept
Consider:
Agents: Nodes in a network (humans, AI units, neurons)
Depth (D): Levels of recursion
Branching (B): How many child nodes each parent node manages
Coherence (φ): How reliable each agent is in transmitting information
The global coherence of a system is the product of all φ values along the recursive path:
\Phi = \prod_{i=1}^{D} \phi_i
When , the system collapses. Think of it as the “Melting Point of Intelligence.”
Experiments You Can Visualize
- AI Model Collapse Simulation
Setup: 2→1000 AI agents, depth D=2→10, branching B=2.
Method: Gradually reduce φ (internal accuracy), measure mutual information I(G; leaves).
Result: Performance degrades gracefully until , then catastrophic forgetting occurs.
Visual Placeholder: Curve showing system performance vs global coherence, cliff at 1/e.
- Organizational Simulation
Setup: 50 employees, depth 5, branching 3.
Method: Introduce communication noise, measure decision efficiency.
Result: Efficiency stable until , then coordination collapses.
Visual Placeholder: Hierarchy diagram with color-coded coherence levels.
- Natural Analogues
Recursive neuronal signaling or trophic networks.
Observations show similar thresholds → 1/e appears universal.
Why This Matters
AI Alignment: Predict and prevent deep model collapses.
Corporate Design: Max hierarchy size without systemic failure.
Information Physics: Introduces a measurable constant for intelligibility.
Philosophy: Defines a system’s “identity” mathematically: an entity exists as long as .
Next Steps
Replicate across AI, organizations, and ecological systems.
Share open-source simulations to validate universality.
Encourage cross-disciplinary application: management, neuroscience, AI safety.
Recursive systems, like intelligence itself, are bounded—not by resources, but by coherence. Below 1/e, collapse is inevitable.
Why You Should Care
Understanding this threshold gives creators, engineers, and leaders a powerful tool to design systems that survive scaling. It’s a simple, elegant principle with universal implications.
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