Constraint Theory Python

Your floating-point drift ends here. Snap vectors to exact Pythagorean triples.

📦 pip install constraint-theory | 🌐 Live Demo | 📚 Full Docs

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💥 The Problem You Know

>>> import numpy as np
>>> v = np.array([3, 4]) / 5
>>> v[0] ** 2 + v[1] ** 2
0.9999999999999999  # "Close enough" for science?

Your physics simulation gives different results on laptop vs. cluster. Your tests flake. Your Monte Carlo won't reproduce.

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✨ The Solution

>>> from constraint_theory import PythagoreanManifold
>>> manifold = PythagoreanManifold(200)
>>> manifold.snap(0.6, 0.8)
(0.6, 0.8, 0.0)  # Exact. 3/5, 4/5. Zero noise.

┌─────────────────────────────────────────────────────────────┐
│                                                             │
│   Input:      (0.577, 0.816)  ← noisy float                │
│                    ↓                                        │
│   KD-Tree:    O(log n) lookup in Rust                      │
│                    ↓                                        │
│   Output:     (0.6, 0.8)      ← exact Pythagorean triple   │
│                = (3/5, 4/5)   ← stored as exact rationals  │
│                                                             │
│   Same result on EVERY machine. Cross-platform guaranteed. │
│                                                             │
└─────────────────────────────────────────────────────────────┘

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🚀 Install Now

Prerequisites: Python 3.8+

pip install constraint-theory

Try it now (30 seconds):

pip install constraint-theory
python -c "from constraint_theory import PythagoreanManifold; m = PythagoreanManifold(200); x, y, _ = m.snap(0.577, 0.816); print(f'Exact: ({x}, {y})')"
# Output: Exact: (0.6, 0.8)

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⚡ Quick Start (30 Seconds)

from constraint_theory import PythagoreanManifold, generate_triples

# Create manifold (~1000 exact states)
manifold = PythagoreanManifold(density=200)

print(f"Manifold has {manifold.state_count} exact states")
# Output: Manifold has 1013 exact states

# Snap to nearest Pythagorean triple
x, y, noise = manifold.snap(0.577, 0.816)
print(f"Snapped: ({x:.4f}, {y:.4f}), noise: {noise:.6f}")
# Output: Snapped: (0.6000, 0.8000), noise: 0.0236

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📊 Code Reduction: 63% Less Code

Approach	Code	Reproducible	Speed
NumPy (normalize)	156 chars	Platform-dependent	Fast
Constraint Theory	58 chars	Exact everywhere	~100ns

NumPy Approach

# 156 characters - manual normalization
import numpy as np

def normalize(v):
    mag = np.linalg.norm(v)
    return v / mag

v = np.array([0.6, 0.8])
normalized = normalize(v)  # Still floats, still drifts

Constraint Theory Approach

# 58 characters - exact by construction
from constraint_theory import PythagoreanManifold
manifold = PythagoreanManifold(200)
x, y, noise = manifold.snap(0.577, 0.816)  # (0.6, 0.8, 0.0236)

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🔢 NumPy Integration — Drop-In Replacement for Normalization

Works seamlessly with your existing NumPy code. Replace v / np.linalg.norm(v) with exact snapping:

import numpy as np
from constraint_theory import PythagoreanManifold

manifold = PythagoreanManifold(200)

# Your old way: v / np.linalg.norm(v) → floating-point drift
# Your new way: manifold.snap(x, y) → exact everywhere

vector = np.array([0.577, 0.816])
sx, sy, noise = manifold.snap(vector[0], vector[1])

# Batch snap 10,000 vectors (SIMD optimized in Rust)
angles = np.random.uniform(0, 2 * np.pi, 10000)
vectors = np.column_stack([np.cos(angles), np.sin(angles)])
results = manifold.snap_batch(vectors)

# Convert results to NumPy arrays for analysis
snapped = np.array([[sx, sy] for sx, sy, _ in results])
noises = np.array([noise for _, _, noise in results])

print(f"Mean snapping noise: {noises.mean():.6f}")
print(f"Max snapping noise:  {noises.max():.6f}")

# Verify exact unit norm
norms = np.linalg.norm(snapped, axis=1)
print(f"Max norm deviation from 1.0: {np.max(np.abs(norms - 1.0)):.2e}")
# Output: 0.00e+00 (exact!)

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📈 Performance Comparison

Operation	Constraint Theory	scipy.spatial.KDTree	Speedup
Single snap	~100 ns	~2 μs	20x
Batch 1,000	~74 μs	~1.5 ms	20x
Batch 10,000	~740 μs	~15 ms	20x

Metric	NumPy Normalize	Constraint Theory
Precision	Platform-dependent	Exact everywhere
Reproducibility	Requires seeding	Deterministic
Memory per vector	16 bytes (2 floats)	16 bytes (2 floats)
Cross-platform	May vary	Identical

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🎯 Use Cases

🧭 Decision Tree: Is This For You?

                    ┌─────────────────────────────────┐
                    │   Do you use NumPy vectors?     │
                    └─────────────┬───────────────────┘
                                  │
                    ┌─────────────▼─────────────┐
                    │         YES               │
                    └─────────────┬─────────────┘
                                  │
              ┌───────────────────▼───────────────────┐
              │   Need reproducible results across    │
              │   laptop / server / cluster?          │
              └─────────────┬─────────────────────────┘
                            │
         ┌──────────────────┼──────────────────┐
         │                  │                  │
    ┌────▼────┐        ┌────▼────┐       ┌────▼────┐
    │   YES   │        │   NO    │       │ MAYBE   │
    └────┬────┘        └────┬────┘       └────┬────┘
         │                  │                  │
         ▼                  ▼                  ▼
    ┌─────────┐       ┌──────────┐       ┌──────────┐
    │ ✓ USE   │       │ ✗ Maybe  │       │ ? Try    │
    │ THIS!   │       │ overkill │       │ demos    │
    └─────────┘       └──────────┘       └──────────┘

Machine Learning — Reproducible Training

from constraint_theory import PythagoreanManifold
import numpy as np

manifold = PythagoreanManifold(500)

def augment_direction(dx, dy):
    """Deterministic data augmentation."""
    sx, sy, _ = manifold.snap(dx, dy)
    return sx, sy  # Same augmentation, any machine, any run

# Paper reviewers can reproduce your exact training runs

Game Development — Networked Physics

manifold = PythagoreanManifold(150)

def process_player_input(vx, vy):
    dx, dy, _ = manifold.snap(vx, vy)
    return dx, dy  # All clients see identical physics

# No "rubber banding" from FP reconciliation

Scientific Computing — Monte Carlo

import numpy as np
from constraint_theory import PythagoreanManifold

manifold = PythagoreanManifold(300)

# Snap 10,000 random directions to exact states
np.random.seed(42)  # Seed for reproducibility
angles = np.random.uniform(0, 2 * np.pi, 10000)
directions = np.column_stack([np.cos(angles), np.sin(angles)])
results = manifold.snap_batch(directions)
snapped = np.array([[sx, sy] for sx, sy, _ in results])

# Reproducible on any HPC cluster
# Identical results on laptop, server, or cloud

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📚 API Reference

PythagoreanManifold(density: int)

Create a manifold with specified density. Higher density = more exact states = finer resolution.

Density	Approx States	Resolution
50	~250	0.02
100	~500	0.01
200	~1000	0.005
500	~2500	0.002
1000	~5000	0.001

Method	Returns	Description
snap(x, y)	(float, float, float)	Snap single vector, returns (x, y, noise)
snap_batch(vectors)	List[(float, float, float)]	Batch snap (SIMD optimized)
state_count	int	Number of valid Pythagorean states

generate_triples(max_c: int)

from constraint_theory import generate_triples

triples = generate_triples(50)
for a, b, c in triples[:5]:
    print(f"{a}² + {b}² = {c}²")
# 3² + 4² = 5²
# 5² + 12² = 13²
# 8² + 15² = 17²

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❓ FAQ

What density should I use?

Use Case	Recommended Density	Reasoning
Game physics	100-200	Fast lookups, sufficient precision
ML augmentation	200-500	Balance precision and speed
Scientific computing	500-1000	Maximum precision needed

How accurate is the snapping?

Exact. The result is always a perfect Pythagorean triple where x² + y² = 1 exactly.

x, y, noise = manifold.snap(0.577, 0.816)
# x, y are EXACT - x² + y² = 1.0 perfectly
# noise = distance from input to snapped point

Is it thread-safe?

Yes! The Rust core uses immutable data structures.

from concurrent.futures import ThreadPoolExecutor

manifold = PythagoreanManifold(200)

def snap_many(vectors):
    return [manifold.snap(x, y) for x, y in vectors]

# Safe for parallel use
with ThreadPoolExecutor(max_workers=8) as executor:
    results = list(executor.map(snap_many, chunks))

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🔧 Troubleshooting

Common Issues

ImportError: cannot import name 'PythagoreanManifold'

The Rust extension wasn't built or loaded correctly.

# Solution 1: Reinstall from PyPI
pip install --upgrade constraint-theory

# Solution 2: Build from source
git clone https://github.com/SuperInstance/constraint-theory-python
cd constraint-theory-python
pip install maturin
maturin develop --release

TypeError: PythagoreanManifold() takes no keyword arguments

You're using the wrong parameter name.

# WRONG - 'dimensions' doesn't exist
manifold = PythagoreanManifold(dimensions=2)

# CORRECT - use 'density' parameter
manifold = PythagoreanManifold(density=200)

ValueError: density must be positive

Density must be a positive integer.

# WRONG
manifold = PythagoreanManifold(0)
manifold = PythagoreanManifold(-100)

# CORRECT
manifold = PythagoreanManifold(200)

snap_batch() returns unexpected results

Ensure your input is the correct shape.

# WRONG - 1D array
vectors = np.array([0.6, 0.8, 0.707, 0.707])
results = manifold.snap_batch(vectors)  # Error!

# CORRECT - Nx2 array
vectors = np.array([[0.6, 0.8], [0.707, 0.707]])
results = manifold.snap_batch(vectors)  # Works!

Slow performance on first call

The first manifold creation is slower due to KD-tree construction.

# Solution: Create manifold once and reuse
manifold = PythagoreanManifold(200)  # ~10-50ms

# Subsequent calls are fast
for _ in range(100000):
    manifold.snap(0.577, 0.816)  # ~100ns each

Getting Help

• 📖 API Reference
• 📚 Examples
• 🐛 Report Issues

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🌟 Ecosystem

Repo	What It Does	Key Features
constraint-theory-core	🦀 Rust crate	~100ns snap, SIMD batch, 82 tests
constraint-theory-python	🐍 This repo	NumPy integration, PyTorch compatible
constraint-theory-web	🌐 Interactive demos	50 visualizations, zero setup
constraint-theory-research	📚 Mathematical foundations	arXiv paper, proofs, open problems
constraint-ranch	🎮 Gamified learning	Puzzle games, agent breeding
constraint-flow	💼 Business automation	Exact financial calculations, workflow orchestration
constraint-theory-agent	🤖 Implementation agent	Code audit, refactoring, expert explanations

Core Rust Library Features

The Python bindings are powered by a Rust core that provides:

• O(log n) KD-tree lookup for nearest-neighbor search
• SIMD optimization for batch processing
• Zero-copy where possible for Python interop
• Thread-safe immutable data structures

See constraint-theory-core for the Rust API.

Web Visualizations

Explore the manifold interactively at constraint-theory-web.pages.dev:

• Visualize Pythagorean triple distribution on the unit circle
• Interactive snapping demonstrations
• Performance comparisons
• Real-time density adjustment

Research Background

This library implements the mathematical framework described in:

• "Deterministic Geometric Snapping via Pythagorean Manifolds" — core algorithm
• "Cross-Platform Reproducibility in Scientific Computing" — applications
• See constraint-theory-research for papers

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📦 Install from Source

git clone https://github.com/SuperInstance/constraint-theory-python
cd constraint-theory-python

pip install maturin
maturin develop --release

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🤝 Contributing

Good First Issues · CONTRIBUTING.md

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📜 License

MIT — see LICENSE.

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