CRUX: How Breakthrough AI Solves Complex Math Problems Autonomously

When an AI system independently generates 9,000+ lines of mathematical reasoning, solves USAMO’s most challenging problem, and validates scientific hypotheses, we’re witnessing a historic shift in artificial intelligence research.

What Does This Mean?

Imagine an AI that doesn’t just solve high school math problems but independently tackles Olympiad-level challenges and conducts original mathematical research. This is CRUX’s groundbreaking capability – redefining AI reasoning boundaries through its innovative IC-RL (In-Context Reinforcement Learning) architecture.

Developed by Tooliense, CRUX achieves:

  • 🧠 Fully autonomous complex problem-solving
  • 📚 Independent hypothesis validation and theorem derivation
  • ⚡ Multi-layered intelligent agent collaboration
  • 🔬 Self-optimization without model weight adjustments

Let’s explore this game-changing technological advancement.

The Core Innovation: IC-RL Learning Framework

Traditional AI Training vs CRUX’s IC-RL Approach

Method How It Works Advantages/Limitations
Standard Model Training Adjusts neural network weights Requires massive data, high update costs
IC-RL Optimizes prompt context as policy parameters Instant optimization, no weight updates needed
Conventional Reinforcement Learning Adjusts behavior via reward signals Long training cycles, low sample efficiency
IC-RL Feedback Mechanism Uses natural language feedback as reward signals Real-time reasoning optimization

How IC-RRL Operates

CRUX transforms prompt engineering into optimizable strategy parameters:

graph LR
A[Initial Prompt] --> B(Execute Reasoning)
B --> C{Receive Feedback}
C -->|Optimize Prompt| D[New Prompt Version]
D --> B

This mechanism enables CRUX to refine its “thinking framework” mid-problem, mirroring how researchers optimize derivations on scratch paper.

The Multi-Agent Architecture: AI’s “Research Institute”

Hierarchical Agent Structure

CRUX’s revolutionary Professor-Specialist framework:

🎓 Professor Agent (Command Center)
├── 🔬 Math Specialist (Number Theory/Algebra)
├── 🔬 Logic Specialist (Proof Derivation)
└── 🔬 Domain Specialist (Problem Context)
    └── 🧑🔬 Sub-Specialists (Dynamically Created)

Dynamic Workflow

For complex challenges:

  1. Professor decomposes problem into sub-tasks
  2. Dispatches appropriate specialists
  3. Specialists can recursively spawn sub-teams
  4. Results aggregate at professor level
  5. Final integrated solution emerges

Real-World Case: Solving 2025 USAMO Problem #6 required 8 specialist layers and 127 cross-domain collaborations.

Demonstrated Breakthroughs

Mathematical Olympiad Problem Solving

CRUX completely solved the 2025 USAMO (United States of America Mathematical Olympiad) final problem:

  • ⏱️ Continuous solving duration: 1+ hour
  • 📝 Internal reasoning: 9,000+ lines
  • ✅ Output: Complete mathematical proof
  • 📄 ./2025USAMO/2025_USAMO_p6.pdf

Autonomous Mathematical Research

More remarkably, CRUX conducted independent mathematical discovery:

  • Starting from only the TTRL hypothesis, it autonomously derived:

    • 9 systematic lemmas with complete proofs
    • Full convergence proofs for theoretical frameworks
    • Practical δ-bookkeeping methodology
  • 📄 ./arXiv/TTRL-paper.pdf

Performance Benchmarks

Capability Conventional AI CRUX System
Problem Complexity High School Level USAMO Competition Level
Reasoning Depth 10-100 step chains 9,000+ line rigorous derivations
Research Capability Pattern Recognition Original Mathematical Discovery
Architecture Scalability Single Model Recursive Multi-Layer Agents

Technical Implementation: From Theory to Practice

Core System Components

CRUX comprises two fundamental modules:

🧠 ./self-evolve/

  • IC-RL algorithm implementation
  • Professor-Specialist architecture
  • Dynamic function calling
  • ./self-evolve/ReadMe.md

🌐 ./crux-agent/

  • Production-ready FastAPI + Next.js implementation
  • Real-time reasoning tracking
  • Multi-provider support (OpenAI/DeepSeek)
  • ./crux-agent/README.md

Quick Setup Guide

Method 1: Core Engine Implementation

# Clone repository
git clone https://github.com/your-org/crux.git
cd crux/self-evolve

# Install dependencies
pip install -r requirements.txt

# Configure API keys
export OPENAI_API_KEY="your-key-here"

# Run basic demonstration
python -m self-evolve.examples.example_usage

# Execute Professor-Specialist framework
python -m self-evolve.examples.professor_graduate_example

Method 2: Full Web Application Deployment

# Backend setup
cd crux/crux-agent
pip install -r requirements.txt
cp .env.example .env # Configure API keys

# Launch services (three terminals required)
redis-server              # Terminal 1
python worker.py          # Terminal 2
uvicorn app.main:app --reload # Terminal 3

# Frontend launch
cd crux-mvp
pnpm install
pnpm dev

Access complete functionality at http://localhost:3000

Technical Q&A: Addressing Key Questions

❓ How does CRUX fundamentally differ from conventional AI?

CRUX advances through context optimization rather than weight updates. Like human researchers, it improves by refining its “thinking framework” rather than altering its “brain structure,” enabling real-time optimization.

❓ Why is the multi-agent architecture revolutionary?

By simulating academic research hierarchies:

  1. Professor agents act as principal investigators
  2. Domain specialists serve as subject experts
  3. Sub-specialists form execution teams
    This structure handles complexity exceeding single-model capacity.

❓ What’s IC-RL’s learning efficiency?

In USAMO problem-solving:

  • Average 3.7 optimizations per reasoning step
  • Final prompts 18x more efficient than initial versions
  • Critical breakthroughs occurred at 43rd major context refinement

❓ Can developers utilize this technology?

Absolutely! The open-source system includes:

  • Support for mainstream AI APIs (OpenAI/DeepSeek)
  • Production-ready web application
  • Commercial-friendly MIT license
# Minimal test implementation
from self_evolve import ProfessorAgent

prof = ProfessorAgent()
solution = prof.solve("Prove that √2 is irrational")
print(solution.proof)

Future Research & Applications

Current Research Trajectory

  1. Cross-Domain Transfer: Extending mathematical reasoning to physics theorem proving
  2. Dynamic Specialist Discovery: AI-generated expert types for novel problems
  3. Resource Optimization: Intelligent computation allocation for critical steps

Practical Implementation Areas

Field Application
Education Olympiad mathematical coaching
Research Automated conjecture validation
Engineering Formal verification of complex systems
Algorithmics Novel algorithm correctness proofs

Conclusion: A New Research Paradigm

CRUX represents more than technical innovation – it pioneers autonomous AI research. When a system independently accomplishes:

  • 9,000-line mathematical derivations
  • Original lemma discovery
  • Theoretical framework construction
    We witness AI’s evolution from “pattern recognition tool” to “research partner.”

As the project manifesto declares:

✨ “The LLM already possesses knowledge; we orchestrate the right specialists asking precise questions through dynamic intelligence hierarchies.” ✨