tnfr 0.0.3.5

TNFR (Resonant Fractal Nature Theory): a mathematical framework for modeling coherent patterns in complex systems through resonance-based dynamics on networks.

TNFR: Resonant Fractal Nature Theory

A mathematical framework for modeling coherent patterns in complex systems through resonance-based dynamics on networks.

A single nodal equation drives every node. From it, a complete transport and geometric structure emerges — measured by the engine, verified to machine precision, and anchored to classical, experimentally-established physics. The graph is only the substrate; the dynamics generates its own geometry.

pip install tnfr

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Core Ideas

All systems evolve via the nodal equation:

$$\frac{\partial \text{EPI}}{\partial t} = \nu_f \cdot \Delta\text{NFR}(t)$$

Structural changes occur exclusively through 13 canonical operators (AL, EN, IL, OZ, UM, RA, SHA, VAL, NUL, THOL, ZHIR, NAV, REMESH) governed by unified grammar rules U1-U6. Each operator carries a canonical contract: it acts on exactly one channel of the nodal equation — the form EPI, the capacity νf, the phase θ, or the pressure ΔNFR — at node or network scale.

System state is characterized by four structural fields — the four orders of the discrete structural-derivative tower (the tetrad). Each is associated with a constant as a notational label; only π is a genuine structural scale (the phase-wrap bound of the phase sector). γ, e, φ are notational, not derived structural scales.

Field	Tower order	Meaning & bound
Φ_s	0th (aggregation)	Structural potential (global stability); empirical bound, no closed form
|∇φ|	1st (local)	Phase gradient (local stress); bound |∇φ| ≤ π (phase wrap)
K_φ	2nd (local)	Phase curvature; bound |K_φ| < 0.9π; K_φ = L_rw·φ
ξ_C	correlation	Coherence length; scale set by spectral gap, ξ_C ∝ 1/√λ₂

Complete theory: AGENTS.md

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From One Equation, a Geometry Emerges

TNFR is more than dynamics on a graph. The graph is only the substrate; the nodal equation generates its own geometry, which the engine measures rather than postulates. Every structure below is verified to machine precision and anchored to classical, experimentally-established phenomena:

• Transport layer (empirically anchored) — channel by channel, the nodal equation is a graph-Laplacian diffusion. From it emerge diffusion, synchronization (Kuramoto), random walks, effective resistance (Ohm/Kirchhoff), and standing-wave modes — all textbook phenomena.
• Emergent symplectic substrate (TNFR-native) — the same dynamics carries a phase space with conserved charges (Noether), a Hamiltonian equal to the energy functional, complete integrability, and a polarization structure (Stokes/Poincaré).
• Orthogonal structure — the dissipative (transport) and conservative (symplectic) parts are the two orthogonal Helmholtz–Hodge components of one flow.

Honest scope: this reorganizes known mathematics and physics inside a single framework, verified in code. It is a characterization of structure the nodal equation already contains — not a claim of new physics.

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Research Status

Two clearly-separated layers:

Solid and verified. The engine, the tetrad, grammar U1–U6, conservation laws, and the emergent transport + symplectic geometry are implemented, anchored to experimentally-established phenomena, and covered by 2,041 tests.

Open research programs. TNFR is also used to probe famous open problems. These are honest, in-progress programs that do not claim proofs:

Program	Done	Open
TNFR–Riemann (P1–P49)	discrete operator σ_c → 1/2; ζ↔L attack surface	Riemann Hypothesis (gap G4) — paused at T-HP
TNFR–Navier–Stokes (N1–N17)	NS-G5 closed at discrete-operator level	continuum limit / Clay (NS-G1..G4) — open
TNFR–Yang–Mills (Y1–Y5)	finite U(1) structural diagnostics	non-Abelian mass gap — open (Branch B)
TNFR–P vs NP (PNP-1)	coherence verification O(|E|) vs synthesis trapping	worst-case separation — open (Branch B)
TNFR–BSD (BSD-1)	rank separation via structural-pressure accumulation	rank ↔ order of vanishing — open (Branch B)
TNFR–Hodge (HC-1)	discrete Hodge = homology exactly (Eckmann)	(p,p) bigrading + algebraicity — structurally blind (Branch B3-leaning)

See AGENTS.md and the theory/ research notes for the full, audited status.

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Quick Start

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

# Structural Field Tetrad — four canonical 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)
print(tetrad.is_safe())  # canonical threshold checks

# Conservation laws — Noether charge, Lyapunov stability
cons = net.conservation()
print(cons.summary())
# -> Q=1.2340, E=0.5678, dE/dt=-0.0012 (STABLE), quality=0.998

# Emergent symplectic substrate — the geometry the dynamics generates
sub = net.symplectic_substrate()
print(sub.summary())
# -> dim=80, H_sub=0.0000, U=0.0000, div(X_H)=0.00e+00 (VALID)
# dim = 4N phase space; div(X_H)=0 => Liouville (volume-preserving)

# One-shot comprehensive analysis
analysis = TNFR.analyze(net)
# Returns: coherence, tetrad, conservation, tensor_invariants,
#          emergent_fields, integrity, features

# Grammar-aware evolution (proactive U1-U6 enforcement)
net.evolve_grammar_aware(steps=10)

# Direct operator usage
import networkx as nx
from tnfr.operators.definitions import Emission, Coherence, Silence
from tnfr.metrics.coherence import compute_coherence

G = nx.erdos_renyi_graph(20, 0.2)
for node in G.nodes():
    Emission()(G, node)
    Coherence()(G, node)
    Silence()(G, node)

print(f"Coherence: {compute_coherence(G):.3f}")

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Installation

pip install tnfr                       # stable release
pip install -e ".[dev-minimal]"        # development
pip install -e ".[test-all]"           # full test suite
pip install -e ".[compute-jax]"        # JAX backend
pip install -e ".[compute-torch]"      # PyTorch backend

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Project Structure

src/tnfr/
├── operators/         # 13 canonical operators + grammar U1–U6 (62 modules)
├── physics/           # Tetrad, conservation, emergent symplectic substrate, structural diffusion (29 modules)
├── engines/           # Self-optimization, pattern discovery, GPU/FFT (8 modules across 5 subpackages)
├── dynamics/           # Nodal equation integration
├── riemann/           # TNFR–Riemann program (61 modules, P1–P49; paused at T-HP, RH open)
├── navier_stokes/     # TNFR–Navier–Stokes program (N1–N17; NS-G5 closed at discrete level, Clay open)
├── yang_mills/        # TNFR–Yang–Mills diagnostics (Y1–Y5; Branch B, mass gap open)
├── sdk/               # Simplified & Fluent API (7 modules)
│   └── simple.py      # Tetrad, conservation, symplectic substrate, grammar-aware dynamics
├── mathematics/       # Number theory, backends
├── constants/         # Canonical constants (mpmath 35-digit precision)
├── metrics/           # Coherence, Si, phase sync, telemetry
├── validation/        # Structural health monitoring
└── factorization/     # Spectral factorization workflow

examples/              # 162 examples in 10 thematic subfolders (see examples/README.md)
tests/                 # 2,041 tests
theory/                # Theoretical derivations
benchmarks/            # 50 performance & structural-validation scripts

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Documentation

Resource	Description
AGENTS.md	Primary reference — complete TNFR theory, operators, grammar, fields
theory/UNIFIED_GRAMMAR_RULES.md	U1-U6 grammar derivations from physics
theory/FUNDAMENTAL_THEORY.md	Structural-field tetrad & field scales
docs/STRUCTURAL_FIELDS_TETRAD.md	Field implementation specifications
docs/STRUCTURAL_INTERFACE_THEORY.md	Structural-interface programme: pipelines, fair benchmarks, validated results, limitations
theory/TNFR_RIEMANN_RESEARCH_NOTES.md	TNFR-Riemann program
theory/TNFR_NAVIER_STOKES_RESEARCH_NOTES.md	TNFR-Navier–Stokes program
theory/TNFR_YANG_MILLS_RESEARCH_NOTES.md	TNFR–Yang–Mills structural gap programme (Y1–Y5; Branch B classified)
theory/TNFR_P_VS_NP_RESEARCH_NOTES.md	TNFR–P vs NP synthesis-vs-verification programme (PNP-1; Branch B, not a proof)
theory/TNFR_BSD_RESEARCH_NOTES.md	TNFR–Birch–Swinnerton-Dyer structural-pressure programme (BSD-1; Branch B, not a proof)
theory/TNFR_HODGE_RESEARCH_NOTES.md	TNFR–Hodge discrete cochain programme (HC-1; Branch B3-leaning strong negative, not a proof)
theory/GLOSSARY.md	Terminology and definitions
examples/	Sequential tutorials
ARCHITECTURE.md	System design
CONTRIBUTING.md	Development guidelines

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Testing

pytest                             # all tests (2,041 under tests/)
pytest tests/sdk/                  # SDK tests (tetrad, conservation, grammar)
pytest tests/unit/                 # unit tests
.\make.cmd smoke-tests             # smoke tests (Windows)
make smoke-tests                   # smoke tests (Unix)

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Citation

@software{tnfr_python_engine,
  author = {Martinez Gamo, F. F.},
  title = {TNFR-Python-Engine: Resonant Fractal Nature Theory Implementation},
  year = {2026},
  version = {0.0.3.5},
  doi = {10.5281/zenodo.17602860},
  url = {https://github.com/fermga/TNFR-Python-Engine}
}

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License

MIT — see LICENSE.md.

Links

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