A Universal Reynolds Number for Self-Organizing Systems

Towards a Unified Physics of Organization

Version: 1.0
Date: 2026-01-16
Status: THEORETICAL PROPOSAL

1. The Core Hypothesis

We propose that Self-Organization in non-equilibrium systems is governed by a universal phase transition, controlled by a dimensionless parameter analogous to the Reynolds Number in fluid dynamics.

$$ Re_{\Omega} = \frac{\text{Driving Force} \cdot \text{Scale}}{\text{Dissipation / Viscosity}} $$

Laminar Phase ($Re_{\Omega} < Re_c$): The system exhibits emergent order, “extra” binding forces, and structural stability.
Turbulent Phase ($Re_{\Omega} > Re_c$): The system exhibits chaos, entropy maximization, and loss of coherent structure.

2. Domain 1: Galactic Dynamics (Validated)

System: Spacetime / Gravity.
Anomaly: “Missing Mass” (Rotation Curves).
Control Parameter: Gravitational Reynolds Number.
$$ Re_G = \frac{V \cdot R}{\nu_G} $$
Laminar Phase: $\alpha \approx 0.35$ (Dark Matter mimicry).
Turbulent Phase: $\alpha \to 0$ (Newtonian Gravity).
Status: VALIDATED (SPARC Database, N-Body Sim).

3. Domain 2: Biological Healing (Proposed)

System: Tissue Repair / Morphogenesis.
Anomaly: “Missing Regeneration” (Scarring).
Control Parameter: Healing Reynolds Number.
$$ Re_H = \frac{V_{repair} \cdot L_{wound}}{D_{eff}} $$
Laminar Phase: Regeneration (Emergent structural order).
Turbulent Phase: Scarring (Entropic gap filling).
Status: FORMULATED (Physics-First Framework). Needs experimental test.

4. Domain 3: Neural Networks (The Test Case)

To prove universality, we apply the framework to a third, distinct domain: Deep Learning.

System: High-dimensional Optimization Landscape (SGD).
Anomaly: “Generalization Gap” (Why do overparameterized networks generalize?).
Control Parameter: Learning Reynolds Number.
$$ Re_L = \frac{\eta \cdot |\nabla \mathcal{L}|}{\sigma_{noise}} $$
- $\eta$: Learning Rate (Velocity).
- $|\nabla \mathcal{L}|$: Gradient Magnitude (Force).
- $\sigma_{noise}$: Gradient Noise / Batch Size effects (Dissipation).
Laminar Phase ($Re_L < Re_c$):
- Behavior: The network settles into “flat minima” (Hochreiter & Schmidhuber).
- Emergent Property: Generalization. The system “hallucinates” a rule that applies to unseen data (analogous to Dark Matter or Regeneration).
Turbulent Phase ($Re_L > Re_c$):
- Behavior: Chaotic oscillation, sharp minima.
- Outcome: Overfitting / Divergence. The system memorizes noise but fails to capture structure.

Prediction:
We predict that the “Edge of Chaos” (critical batch size) in LLM training corresponds exactly to the phase transition $Re_L \approx Re_c$.

5. The Universal Scaling Law

Across all three domains, the “Order Parameter” ($\alpha$) follows the same logistic decay:

$$ \alpha(Re_{\Omega}) = \frac{\alpha_{max}}{1 + (Re_{\Omega}/Re_c)^\gamma} $$

Domain	$\alpha$ Represents	$\alpha_{max}$	$Re_c$
Gravity	Interaction Strength Boost	$\approx 0.35$	Galactic Scale
Healing	Structural Binding Energy	(TBD)	Tissue Scale
AI	Generalization Gap	(TBD)	Batch Scale

6. Falsification Criteria (The “Kill Switch”)

If we cannot fit experimental data from Domain 3 (AI) to this curve without arbitrary parameter tuning, the Universal Hypothesis is FALSE.
* Test: Plot Test Loss vs. $Re_L$ for a Transformer model.
* Prediction: A sharp phase transition in test loss should occur at a universal critical value, regardless of model size.

7. Conclusion

We are moving from “Analogies” to “Universality Classes.” If Gravity, Biology, and Intelligence all obey the same Reynolds Scaling Law, we have discovered a fundamental constraint on how the universe builds complexity.

Part 1: The Universal Reynolds Number