MathOS

Mathematical Research Platform

A comprehensive web-based computational platform for modern mathematics, featuring elliptic curve analysis, Galois theory, modular forms, topology, cryptography, and AI pattern recognition.

Overview

MathOS is a sophisticated mathematical research platform designed to provide researchers, students, and enthusiasts with powerful computational tools for exploring advanced mathematical concepts. Built with Python and Flask, it combines a user-friendly web interface with robust mathematical engines to deliver real-time analysis and visualization.

Purpose

Goal	Description
Research Support	Accelerate mathematical research with computational tools
Education	Provide an accessible platform for learning advanced mathematics
Experimentation	Enable rapid prototyping and testing of mathematical hypotheses
Collaboration	Share findings through a web-based interface

Features

Elliptic Curve Analysis

Click to expand details

Feature	Description
Curve Visualization	Analyze curves of the form y^2 = x^3 + ax + b
Rank Computation	Compute Mordell-Weil rank (exact or approximate)
Torsion Points	Identify torsion subgroups
Discriminant	Calculate discriminant and j-invariant
Group Structure	Analyze rational points structure

Example:

POST /api/curve
{
    "a": 1,
    "b": 1,
    "exact_rank": true
}

Galois Theory

Click to expand details

Feature	Description
Galois Groups	Compute Galois groups of polynomials
Field Extensions	Analyze field extensions
Pre-built Examples	Common polynomials with known groups
Degree Analysis	Determine degree of field extensions

Supported Polynomials:

Polynomial	Galois Group
x^5 - 2	F_20 (Frobenius group)
x^4 + 1	V_4 (Klein four-group)
x^3 - 2	S_3 (Symmetric group)
x^5 - 4x + 2	S_5 (Symmetric group)

Example:

POST /api/galois
{
    "polynomial": "x**5 - 2"
}

Modular Forms

Click to expand details

Feature	Description
Weight & Level	Compute forms of specified weight and level
L-Functions	Analyze associated L-functions
Fourier Coefficients	Generate and visualize coefficients
Eigenforms	Identify Hecke eigenforms

Example:

POST /api/modular
{
    "weight": 2,
    "level": 11
}

Topology

Click to expand details

Feature	Description
Homology Groups	Compute homology groups
Euler Characteristic	Calculate topological invariants
F-Vectors	Face vector analysis
Supported Spaces	Sphere, Torus, Projective Plane, Klein Bottle

Example:

POST /api/topology
{
    "space": "torus",
    "dimension": 2
}

Cryptography

Click to expand details

Feature	Description
Post-Quantum	Kyber and Dilithium benchmarks
ECC	Elliptic curve cryptography
Security Analysis	Lattice and ECC security evaluation
Key Generation	ECC key pair generation

Example:

POST /api/crypto
{
    "action": "pqc"
}

AI Pattern Recognition

Click to expand details

Feature	Description
Sequence Identification	Identify Fibonacci, primes, squares
Recurrence Finding	Detect linear recurrences
Pattern Matching	Recognize mathematical patterns
Prediction	Extend sequences

Example:

POST /api/ai
{
    "sequence": "1,1,2,3,5,8,13"
}

Live Demo

Visit the live application: mathos-ogx3.onrender.com

Note: Free tier may spin down after 15 minutes of inactivity. First request may take approximately 30 seconds to wake up.

Installation

Prerequisites

Requirement	Version
Python	3.7+
pip	Latest
git	Latest
Virtual Environment	Optional but recommended

Local Setup

1. Clone the Repository

git clone https://github.com/Soyebsoyeb/MathOS.git
cd MathOS

2. Create Virtual Environment

# Linux/Mac
python -m venv venv
source venv/bin/activate

# Windows
python -m venv venv
venv\Scripts\activate

3. Install Dependencies

pip install -r requirements.txt

4. Set Environment Variables

Create a .env file in the root directory:

SECRET_KEY=your-secret-key-here
FLASK_ENV=development

5. Run the Application

# Development mode
python math_research_platform/interfaces/web_app.py

# Production mode (using Gunicorn)
gunicorn --worker-class gthread --threads 4 --timeout 120 --bind 0.0.0.0:5000 math_research_platform.interfaces.web_app:app

6. Access the Application

Open your browser and navigate to: http://localhost:5000

Docker Deployment

# Dockerfile
FROM python:3.14-slim

WORKDIR /app

COPY requirements.txt .
RUN pip install --no-cache-dir -r requirements.txt

COPY . .

ENV SECRET_KEY=docker-secret-key

EXPOSE 5000

CMD ["gunicorn", "--worker-class", "gthread", "--threads", "4", "--timeout", "120", "--bind", "0.0.0.0:5000", "math_research_platform.interfaces.web_app:app"]

# Build and run
docker build -t mathos .
docker run -p 5000:5000 mathos

Usage Guide

Web Interface

The application provides a clean, dark-themed interface with the following modules:

1. Home Page

Welcome screen with navigation
Quick links to all modules
Platform overview

2. Curves Module

Input parameters a and b
Compute exact or approximate rank
View results in JSON format
Quick examples for common curves

3. Galois Module

Input polynomial expression
Pre-built examples for common polynomials
Detailed Galois group analysis

4. Modular Forms Module

Set weight and level
Compute form properties
L-function analysis

5. Topology Module

Select space type
Set dimension
Compute homology and Euler characteristic

6. Cryptography Module

Select action type
Benchmark post-quantum algorithms
ECC operations

7. AI Module

Input comma-separated sequences
Identify sequence types
Find recurrences

API Documentation

Base URL

https://mathos-ogx3.onrender.com/api

Authentication

No authentication required for public endpoints.

Endpoints

1. Elliptic Curve Analysis

POST /api/curve

Request Body:

{
  "a": 1.0,
  "b": 1.0,
  "exact_rank": false
}

Response:

{
  "status": "success",
  "result": {
    "curve": "y^2 = x^3 + 1.0x + 1.0",
    "discriminant": -331,
    "j_invariant": -3375/31,
    "rank": 1,
    "torsion": "trivial"
  }
}

2. Galois Theory

POST /api/galois

Request Body:

{
  "polynomial": "x**5 - 2"
}

Response:

{
  "status": "success",
  "result": {
    "polynomial": "x**5 - 2",
    "degree": 5,
    "galois_group": "F20",
    "order": 20
  }
}

3. Modular Forms

POST /api/modular

Request Body:

{
  "weight": 2,
  "level": 11
}

Response:

{
  "status": "success",
  "result": {
    "weight": 2,
    "level": 11,
    "dimension": 1,
    "coeffs": [0, 1, -2, -1, 2, 1, 2, -2, -1, 2, -2, -1]
  }
}

4. Topology

POST /api/topology

Request Body:

{
  "space": "torus",
  "dimension": 2
}

Response:

{
  "status": "success",
  "result": {
    "space": "torus",
    "dimension": 2,
    "euler_characteristic": 0,
    "f_vector": [1, 4, 4, 1]
  }
}

5. Cryptography

POST /api/crypto

Request Body:

{
  "action": "benchmark"
}

Response:

{
  "status": "success",
  "result": {
    "kyber": { "security": 128, "n": 256, "q": 3329 },
    "dilithium": { "security": 256, "n": 256, "q": 8380417 }
  }
}

6. AI Pattern Recognition

POST /api/ai

Request Body:

{
  "sequence": "1,1,2,3,5,8,13"
}

Response:

{
  "status": "success",
  "result": {
    "sequence": [1, 1, 2, 3, 5, 8, 13],
    "identifications": ["Fibonacci sequence"],
    "recurrences": {
      "linear": "a(n) = a(n-1) + a(n-2)",
      "order": 2
    }
  }
}

7. Health Check

GET /health

Response:

{
  "status": "healthy",
  "service": "MathOS",
  "version": "1.0.0"
}

Error Handling

All endpoints return consistent error responses:

{
  "status": "error",
  "error": "Error message description"
}

HTTP Status Codes:

Code	Meaning
200	Success
400	Bad Request (invalid input)
500	Internal Server Error

Technology Stack

Backend

Technology	Version	Purpose
Python	3.14	Core language
Flask	3.1.3	Web framework
Gunicorn	26.0.0	Production WSGI server
SymPy	1.14	Symbolic mathematics
NumPy	2.5.1	Numerical computing
SciPy	1.18	Scientific computing
Flask-CORS	6.0.5	Cross-origin support

Frontend

Technology	Version	Purpose
Bootstrap	5.3.0	UI framework
Bootstrap Icons	1.10.0	Icon library
JavaScript	ES6	Client-side interactivity
HTML5	-	Structure
CSS3	-	Styling (custom dark theme)

Deployment

Technology	Purpose
Render.com	Cloud hosting
GitHub	Version control
Gunicorn	Production server

Project Structure

MathOS/
├── math_research_platform/
│   ├── interfaces/
│   │   └── web_app.py          # Flask web interface (main app)
│   ├── main.py                 # Core platform module
│   ├── __init__.py             # Package initializer
│   └── [other modules]         # Future math modules
├── .gitignore                  # Git ignore rules
├── requirements.txt            # Python dependencies
├── README.md                   # This file
├── LICENSE                     # MIT License
└── [configuration files]       # Future config files

Key Files Explained

web_app.py

Flask application entry point
Contains all routes and API endpoints
Implements health checks
Uses main.py for mathematical computations
Serves HTML pages with Bootstrap interface

main.py

Core mathematical engine
Contains all math computation functions
Exposes platform object with all methods
Handles elliptic curves, Galois theory, etc.

requirements.txt

All Python dependencies
Version-locked for stability
Used by Render and local installations

Deployment

Deploy to Render (Recommended)

Create a Render Account
- Visit render.com
- Sign up with GitHub
Create New Web Service
- Click "New +" -> "Web Service"
- Connect your GitHub repository
- Select Soyebsoyeb/MathOS

Configure Service

Name: MathOS
Environment: Production
Language: Python 3
Branch: main
Region: Oregon (US West)

Set Build Settings

Build Command: pip install -r requirements.txt
Start Command: gunicorn --worker-class gthread --threads 4 --timeout 120 --bind 0.0.0.0:$PORT interfaces.web_app:app
Root Directory: math_research_platform

Set Environment Variables
```
SECRET_KEY = (generate random string)
```
Deploy
- Click "Create Web Service"
- Wait for build and deployment
- Access at: https://mathos.onrender.com

Deploy to Heroku

# Install Heroku CLI
curl https://cli-assets.heroku.com/install.sh | sh

# Create App
heroku create mathos
heroku config:set SECRET_KEY=your-secret-key
git push heroku main

# Scale Dynos
heroku ps:scale web=1

Deploy to AWS EC2

# Launch Ubuntu Instance
# SSH into Instance
ssh -i key.pem ubuntu@ec2-ip

# Install Dependencies
sudo apt update
sudo apt install python3-pip nginx -y
pip install -r requirements.txt

# Run with Gunicorn
gunicorn --worker-class gthread --threads 4 --timeout 120 --bind 0.0.0.0:8000 math_research_platform.interfaces.web_app:app

Performance & Scaling

Current Configuration

Parameter	Value	Purpose
Worker Class	gthread	Handles concurrent requests
Threads	4	Concurrent request handling
Timeout	120s	Long-running computations
Memory	512MB (free)	Sufficient for most operations
CPU	0.1 (free)	Single-core for computations

Optimization Tips

Caching

Implement Redis cache for frequent computations
Cache API responses for repeated queries

Asynchronous Processing

Use Celery for long-running tasks
Implement job queues

Database Integration

Store computation results
Cache historical queries

Load Balancing

Use multiple workers
Implement horizontal scaling

Performance Monitoring

# Add monitoring middleware
@app.before_request
def before_request():
    request.start_time = time.time()

@app.after_request
def after_request(response):
    elapsed = time.time() - request.start_time
    print(f"Request took {elapsed:.2f}s")
    return response

Security

Current Security Measures

Environment Variables

SECRET_KEY stored in environment
No hardcoded secrets

CORS Configuration

Controlled cross-origin access
Prevents unauthorized API access

Input Validation

JSON request validation
Type checking
Bounds checking

Error Handling

Proper exception handling
No sensitive information in errors

Recommendations

Rate Limiting

from flask_limiter import Limiter

limiter = Limiter(app, key_func=lambda: request.remote_addr)

@app.route('/api/curve', methods=['POST'])
@limiter.limit("10 per minute")
def api_curve():
    # ...

HTTPS Enforce

from flask_talisman import Talisman
Talisman(app)

Authentication

from flask_jwt_extended import JWTManager, jwt_required
# Add authentication to sensitive endpoints

Contributing

Ways to Contribute

Code Contributions

New mathematical modules
Performance improvements
Bug fixes
Test coverage

Documentation

API documentation
User guides
Tutorials

UI/UX

Responsive design improvements
Accessibility enhancements
Theme customizations

Development Workflow

Fork Repository

git clone https://github.com/your-username/MathOS.git
cd MathOS

Create Branch
```
git checkout -b feature/AmazingFeature
```
Make Changes
- Write code
- Add tests
- Update documentation
Test
```
python -m pytest
```

Commit

git add .
git commit -m "Add some AmazingFeature"

Push
```
git push origin feature/AmazingFeature
```
Create Pull Request
- Open PR on GitHub
- Describe changes
- Wait for review

Code Style

Python: PEP 8
JavaScript: ESLint
HTML/CSS: Bootstrap conventions

Testing

# test_mathos.py
import unittest
from math_research_platform.main import platform

class TestMathOS(unittest.TestCase):
    def test_elliptic_curve(self):
        result = platform.analyze_elliptic_curve(1, 1)
        self.assertEqual(result['rank'], 1)

if __name__ == '__main__':
    unittest.main()

License

This project is licensed under the MIT License - see the LICENSE file for details.

MIT License

Copyright (c) 2026 Soyebsoyeb

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.

Acknowledgments

Libraries & Frameworks

SymPy - Symbolic mathematics engine
Flask - Web framework
Bootstrap - UI components
Gunicorn - Production server
NumPy / SciPy - Scientific computing

Inspiration

Mathematical research community
Open source contributions
Educational platforms

Contributors

Soyebsoyeb - Creator and maintainer

Hosting

Render.com - Free tier hosting

Support

Resources

Resource	Link
Live Demo	mathos-ogx3.onrender.com
GitHub Issues	Issues
Source Code	Repository
API Documentation	API Docs

Getting Help

Check Documentation - Read the README
Search Issues - Look for similar problems
Create Issue - Report bugs/feature requests
Contact - Reach out to maintainers

Common Issues

Issue	Cause	Solution
App Takes Long to Load	Free tier spin-down	Wait ~30 seconds for wake-up
ModuleNotFoundError	Missing dependencies	Run `pip install -r requirements.txt`
Port Already in Use	Another process on port 5000	Use different port or kill process
CORS Errors	Cross-origin restrictions	Use the API from allowed origins

Future Roadmap

Planned Features

Graph Database - Store and query mathematical structures
Jupyter Integration - Interactive notebooks
Visualization - 3D plots and graphs
Collaboration - Share and discuss results
More Math Modules - Number theory, algebraic geometry
Machine Learning - Automated theorem proving
Mobile App - React Native version
Cloud Storage - Save and load computations
Export Options - PDF, LaTeX, Markdown
Batch Processing - Run multiple computations

Version History

Version	Date	Changes
1.0.0	2026-07-19	Initial release
0.9.0	2026-07-15	Beta testing
0.8.0	2026-07-10	Alpha release

Made with passion and precision by Soyebsoyeb

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Latest commit History 10 Commits
math_research_platform		math_research_platform
.gitignore		.gitignore
README.md		README.md
docs.md		docs.md
requirements.txt		requirements.txt

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MathOS

Table of Contents

Overview

Purpose

Features

Elliptic Curve Analysis

Galois Theory

Modular Forms

Topology

Cryptography

AI Pattern Recognition

Live Demo

Installation

Prerequisites

Local Setup

1. Clone the Repository

2. Create Virtual Environment

3. Install Dependencies

4. Set Environment Variables

5. Run the Application

6. Access the Application

Docker Deployment

Usage Guide

Web Interface

1. Home Page

2. Curves Module

3. Galois Module

4. Modular Forms Module

5. Topology Module

6. Cryptography Module

7. AI Module

API Documentation

Base URL

Authentication

Endpoints

1. Elliptic Curve Analysis

2. Galois Theory

3. Modular Forms

4. Topology

5. Cryptography

6. AI Pattern Recognition

7. Health Check

Error Handling

Technology Stack

Backend

Frontend

Deployment

Project Structure

Key Files Explained

Deployment

Deploy to Render (Recommended)

Deploy to Heroku

Deploy to AWS EC2

Performance & Scaling

Current Configuration

Optimization Tips

Performance Monitoring

Security

Current Security Measures

Recommendations

Contributing

Ways to Contribute

Development Workflow

Code Style

Testing

License

Acknowledgments

Libraries & Frameworks

Inspiration

Contributors

Hosting

Support

Resources

Getting Help

Common Issues

Future Roadmap

Packages