TNFRResonant Fractal Nature Theory
A mathematical framework for modeling coherent patterns on graph-coupled networks through the nodal equation .
A shift in network modeling
TNFR models coherent dynamic patterns that persist through resonance, rather than discrete objects.
Model Coherence
Capture process, not state. Do not model objects; instead, model the continuous structural coherence that binds them.
Measure Resonance
Think structure, not substance. A pattern (like a vortex or neural assembly) dissolves when its resonant coupling fails.
Embrace Emergence
Focus on autopoietic (self-generated) and multiscalar patterns that emerge naturally from local nodal reorganization.
What TNFR defines
TNFR is a framework for describing the dynamics of coherent patterns on graph-coupled networks. It is defined by four formal components:
- 01
The Structural Triad
Every node carries three core attributes:
EPI: Coherent structural form νf: Reorganization capacity φ: Synchronization phase
- 02
The Nodal Equation
Governing the evolution of every node: ∂t/∂EPI =ν f⋅ΔNFR(t)
- 03
13 Canonical Operators
The exclusive mechanism by which a node's state may change. Form (EPI) is never mutated directly.
- 04
Unified Grammar (U1–U6)
Constrains valid operator sequences to ensure integral-convergence and structural stability.
- 05
Fractal-Resonant Nodes
NFRs are regions of structural coherence that are multiscalar (fractal), autopoietic, relational, and temporal.
- 06
Structural-field Tetrad
(Φs∣∇φ∣,Kφ,ξCΦ s,∣∇φ∣,K φ,ξ C) spanning the minimal discrete derivative tower to characterize the entire network.
| Field | Order | Meaning |
|---|---|---|
| \Phi_s | 0th — global aggregation | Structural potential (global stability) |
| |\nabla\varphi| | 1st — local derivative | Phase gradient (local stress) |
| K_\varphi | 2nd — discrete Laplacian | Phase curvature (geometric confinement) |
| \xi_C | non-local-correlation | Coherence length (spatial correlations) |
What you can do with it
Quick start
Mirrors README.md in the repository — verbatim, no paraphrasing.
from tnfr.sdk import TNFR
# Create, connect, evolve
net = TNFR.create(20).ring().evolve(5)
print(net.results().summary())
# -> C=0.987, Si=0.912, N=20, E=20, rho=0.105
# Inspect the four structural fields
tetrad = net.tetrad()
print(tetrad.summary())
# -> Phi_s=0.0312, |grad_phi|=0.0841, |K_phi|=0.1523, xi_C=2.3147 (N=20)