coordinate-system 7.0.0

High-performance 3D coordinate system library with unified differential geometry, quantum frame algebra, and Christmas Equation (CFUT)

Coordinate System Library

High-performance 3D coordinate system and differential geometry library for Python

Author: PanGuoJun Version: 7.0.0-alpha License: MIT DOI: 10.5281/zenodo.14435613

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What's New in v7.0.0-alpha (2026-01-14)

• 🎄 Christmas Equation: New complex_geometric_physics module implementing unified field theory
• Complex Frame Unified Theory (CFUT): Geometry + Topology = Complex Matter + Topological Force
• Einstein Tensor: Compute Ĝ_μν from complex frame field U(x)
• Chern-Simons Current: Topological current K̄_μ for gauge field analysis
• Energy-Momentum Tensor: Real-imaginary decomposition for matter and topology
• Complete English Translation: All code documentation now in English for worldwide use

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Module Structure

coordinate_system/
├── coordinate_system.pyd/.so      # C++ core (vec3, quat, coord3)
├── complex_geometric_physics.py   # 🎄 Christmas Equation, CFUT unified theory
├── spectral_geometry.py           # FourierFrame [GL(1,C)], spectral analysis
├── u3_frame.py                    # U3Frame [U(3)], gauge field theory
├── differential_geometry.py       # Surface curvature calculation
├── visualization.py               # 3D visualization
└── curve_interpolation.py         # C2-continuous interpolation

Group Correspondence

Class	Group	DOF	Use Case
coord3	Sim(3) = R³ ⋊ (SO(3) × R⁺)	10	3D coordinate transform
ComplexFrame	U(3) complex field	18	🎄 Unified field theory (CFUT)
FourierFrame	GL(1,C) = U(1) × R⁺	2	Spectral geometry, heat kernel
U3Frame	U(3) = SU(3) × U(1)	9	Gauge field theory

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Installation

pip install coordinate-system

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Quick Start

Basic Coordinate System

from coordinate_system import vec3, quat, coord3

# Vector operations
v1 = vec3(1, 2, 3)
v2 = vec3(4, 5, 6)
dot = v1.dot(v2)
cross = v1.cross(v2)

# Quaternion rotation
q = quat(1.5708, vec3(0, 0, 1))  # 90° around Z
rotated = q * v1

# Coordinate transform
frame = coord3.from_angle(1.57, vec3(0, 0, 1))
world_pos = v1 * frame    # Local -> World
local_pos = world_pos / frame  # World -> Local

Differential Geometry

from coordinate_system import Sphere, compute_gaussian_curvature

sphere = Sphere(radius=1.0)
K = compute_gaussian_curvature(sphere, u=0.5, v=0.5)  # K = 1.0

Gaussian Curvature via Lie Bracket: $$K = -\frac{\langle [G_u, G_v] e_v, e_u \rangle}{\sqrt{\det(g)}}$$

Spectral Geometry (FourierFrame)

from coordinate_system import (
    FourierFrame, IntrinsicGradient, BerryPhase, ChernNumber, HeatKernel
)

# Create frame field
frame_field = [[FourierFrame(q_factor=1.0 + 0.1j*(i+j))
                for j in range(16)] for i in range(16)]

# Intrinsic gradient: G_μ = d/dx^μ log Q
grad_op = IntrinsicGradient(frame_field)

# Berry phase: γ = ∮ G_μ dx^μ
berry = BerryPhase(grad_op)
path = [(4, 4), (4, 12), (12, 12), (12, 4), (4, 4)]
gamma = berry.compute_along_path(path, closed=True)

# Heat kernel trace
heat = HeatKernel(frame_field)
trace = heat.trace(t=0.1)

Gauge Field Theory (U3Frame)

from coordinate_system import U3Frame, GaugeConnection, FieldStrength
import numpy as np

# Create U(3) frame
frame = U3Frame()

# Symmetry decomposition: U(3) = SU(3) × U(1)
su3_comp, u1_phase = frame.to_su3_u1()

# Gauge transforms
frame_u1 = frame.gauge_transform_u1(np.pi/4)
frame_su3 = frame.gauge_transform_su3(np.random.randn(8) * 0.1)

# Gauge connection and field strength
conn_x = GaugeConnection(su3_component=np.random.randn(8) * 0.1)
conn_y = GaugeConnection(su3_component=np.random.randn(8) * 0.1)
F_xy = conn_x.field_strength(conn_y)

# Yang-Mills action
S_YM = F_xy.yang_mills_action()

🎄 Complex Geometric Physics (Christmas Equation)

from coordinate_system import (
    ComplexFrame,
    EnergyMomentumTensor,
    ChristmasEquation,
    create_flat_spacetime_frame,
    create_curved_spacetime_frame,
    create_gauge_field_frame,
    M_PLANCK,
    LAMBDA_TOPO
)
import numpy as np

# Create complex frames
flat_frame = create_flat_spacetime_frame()
curved_frame = create_curved_spacetime_frame(curvature=0.1)
gauge_frame = create_gauge_field_frame(field_strength=0.1)

# Initialize Christmas Equation solver
solver = ChristmasEquation()
print(f"Planck mass: {solver.M_P:.3e} GeV")
print(f"Topological coupling: {solver.lambda_topo:.4f}")

# Compute geometric quantities
G_tensor = solver.einstein_tensor(curved_frame)
K_current = solver.chern_simons_current(gauge_frame)

# Create matter energy-momentum tensor
matter_real = np.diag([1.0, 0.1, 0.1, 0.1])
matter_imag = np.zeros((4, 4))
T_matter = EnergyMomentumTensor(matter_real, matter_imag)

# Solve the Christmas Equation
# M_P²/2 Ĝ_μν[U] + λ/(32π²) ∇̂_(μ K̄_ν)[U] = T̂_μν^(top)[U] + T̂_μν^(mat)
solution = solver.solve_christmas_equation(gauge_frame, T_matter)
print(f"Geometric term norm: {np.linalg.norm(solution['geometric_term']):.6e}")
print(f"Topological term norm: {np.linalg.norm(solution['topological_term']):.6e}")
print(f"Equation balanced: {solution['balanced']}")

The Christmas Equation unifies geometry and topology:

• Left side: Geometry (Einstein tensor) + Topology (Chern-Simons current)
• Right side: Topological source + Matter source
• Real part U^(R): Geometric properties (metric, curvature, spacetime)
• Imaginary part U^(I): Topological properties (phase winding, gauge symmetry)

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Key Formulas

Concept	Formula	Code
🎄 Christmas Equation	$\frac{M_P^2}{2} \hat{G}{\mu\nu}[U] + \frac{\lambda}{32\pi^2} \hat{\nabla}{(\mu} \bar{K}{\nu)}[U] = \hat{T}{\mu\nu}^{(\text{top})}[U] + \hat{T}_{\mu\nu}^{(\text{mat})}$	ChristmasEquation.solve_christmas_equation()
Einstein Tensor	$\hat{G}{\mu\nu} = R{\mu\nu} - \frac{1}{2}g_{\mu\nu} R$	ChristmasEquation.einstein_tensor()
Chern-Simons Current	$\bar{K}\mu = \varepsilon{\mu\nu\rho\sigma} \text{Tr}(A^\nu F^{\rho\sigma})$	ChristmasEquation.chern_simons_current()
Intrinsic Gradient	$G_\mu = \frac{d}{dx^\mu} \log C(x)$	IntrinsicGradient
Curvature Tensor	$R_{\mu\nu} = [G_\mu, G_\nu]$	CurvatureFromFrame
Gaussian Curvature	$K = -\langle [G_u, G_v] e_v, e_u \rangle / \sqrt{\det g}$	compute_gaussian_curvature
Berry Phase	$\gamma = \oint G_\mu dx^\mu$	BerryPhase
Chern Number	$c_1 = \frac{1}{2\pi} \iint R_{\mu\nu} dS$	ChernNumber
Heat Kernel	$\text{Tr}(e^{t\Delta}) \sim (4\pi t)^{-d/2} \sum_k a_k t^k$	HeatKernel
Yang-Mills Action	$S = -\frac{1}{4g^2} \text{Tr}(F_{\mu\nu} F^{\mu\nu})$	FieldStrength.yang_mills_action()

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FourierFrame vs U3Frame

Property	FourierFrame	U3Frame
Group	GL(1,C) = C×	U(3)
DOF	2 (phase + magnitude)	9 (unitary matrix)
Use Case	Spectral analysis, heat kernel	Gauge field theory
Fourier Transform	fourier_transform(θ)	gauge_transform_u1(θ)
Conformal Transform	conformal_transform(λ)	Not supported
SU(3) Transform	Not supported	gauge_transform_su3(...)

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Performance

Operation	Ops/second
Vector addition	5,200,000
Quaternion rotation	1,800,000
Gaussian curvature	85,000
Spectral transform (GPU)	12,000

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Changelog

v7.0.0-alpha (2026-01-14)

• 🎄 Christmas Equation: New complex_geometric_physics.py module
• Complex Frame Unified Theory (CFUT): Unified field equation implementation
• ComplexFrame: U(x) = U^(R)(x) + iU^(I)(x) decomposition
• EnergyMomentumTensor: Real-imaginary tensor decomposition
• ChristmasEquation: Einstein tensor, Chern-Simons current, topological energy-momentum
• Complete English Translation: All documentation and code comments in English
• DOI: Added Zenodo DOI 10.5281/zenodo.14435613

v6.0.4 (2025-12-08)

• frames.py → spectral_geometry.py
• Removed fourier_spectral.py
• Unified theory documentation

v6.0.3 (2025-12-04)

• U3Frame: U(3) unitary frame
• GaugeConnection, FieldStrength

v6.0.1 (2025-12-04)

• FourierFrame spectral geometry
• Berry phase, Chern number, heat kernel

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License

MIT License - Copyright (c) 2024-2025 PanGuoJun

Links

• PyPI: https://pypi.org/project/coordinate-system/
• GitHub: https://github.com/panguojun/Coordinate-System
• Email: [e-mail]