AlphaProof Nexus: AI Mathematical Agent Solves 9 Erdős Centenary Problems in One Night

Tuesday, May 26, 2026

Introduction: The Historic Leap from “Computational Tool” to “Original Research Partner”

On May 21, 2026, Google DeepMind released a groundbreaking paper (arXiv:2605.22763v1) introducing AlphaProof Nexus, a novel AI mathematical agent system. This system successfully solved 9 open Erdős problems that had remained unsolved for decades—in one single night—with the oldest problem existing for 56 years!

This breakthrough’s significance extends far beyond technology itself. Fields Medal laureate Tim Gowers remarked: “If this paper were submitted to the Annals of Mathematics by a human, I would毫不犹豫 recommend its acceptance without hesitation.” This marks AI’s formal evolution from a mere “computational assistant tool” into a true partner in original mathematical research.

This article provides an in-depth analysis of AlphaProof Nexus’s technical architecture, core algorithmic principles, and demonstrates key implementations through complete Python/Go code examples. We will also explore this technology’s profound implications for mathematical research, AI Agent development, and broader scientific domains.

1. Background: Why Are Erdős Problems So Important?

1.1 Paul Erdős and Century-Old Challenges in Discrete Mathematics

Paul Erdős (1913-1996) was one of the greatest mathematicians of the 20th century, proposing over 3,000 mathematical problems throughout his life—many of which remain unsolved today. These “Erdős problems” span combinatorics, number theory, graph theory, and other fields, representing “pearls on the crown of mathematics.”

Key characteristics of Erdős problems:

Simple statements: Often describable in just a few sentences
Extremely difficult proofs: May require hundreds of pages of rigorous reasoning
Profound impact: Solving one often opens new mathematical branches

1.2 The 9 Erdős Problems Solved This Time

According to the AlphaProof Nexus paper, here are the problems solved:

Problem #	Year Proposed	Problem Type	Duration
Erdős #12	1970	Set Theory/Combinatorics	56 years
Erdős #125	1996	Additive Combinatorics	30 years
Erdős #138 variant	1981	van der Waerden Theory	45 years
Erdős #846	-	Plane Geometry/Graph Theory	-
…	…	…	…

1.3 Key Statistics

Experiment Scale:
- Total attempted: 353 Erdős problems
- Successfully solved: 9 problems
- Cost per problem: a few hundred dollars
- Maximum iterations: 3000 per problem

Other Achievements:
- OEIS Conjectures: 44 proven out of 492
- Application Domains: Combinatorics, Optimization, Graph Theory, Algebraic Geometry, Quantum Optics

2. System Architecture: Four-Layer Progressive Agent Design

2.1 Architecture Overview

AlphaProof Nexus employs a four-layer progressive Agent architecture, progressively enhancing proof capabilities from simple to complex:

┌─────────────────────────────────────────────────────────────┐
│                   AlphaProof Nexus Architecture              │
├─────────────────────────────────────────────────────────────┤
│                                                             │
│   Problem Input → [Agent A] → [Agent B] → [Agent C] → [Agent D] │
│                           ↓           ↓          ↓         │
│                      +AlphaProof  +Evolution  Complete     │
│                           ↓           ↓          ↓         │
│                      ←←←← Iterative Loop (max 3000) ←←←←        │
│                           ↓           ↓          ↓         │
│                      ←←←← Lean Compiler Verification ←←←←←         │
│                                                             │
│   Output: Proved Theorems (Lean Formalized) + NL Proof     │
└─────────────────────────────────────────────────────────────┘

2.2 Agent A: Basic Version—LLM + Lean Feedback Loop

Agent A is the most basic version, consisting of multiple parallel LLM sub-agents, each interacting with Gemini 3.1 Pro through multi-turn conversations to generate proof drafts, then verified by the Lean compiler.

Python Code Example: Agent A Core Implementation

import asyncio
from dataclasses import dataclass
from typing import List, Optional, Dict
import anthropic

@dataclass
class ProofAttempt:
    """Proof attempt record"""
    problem: str
    lean_code: str
    error_message: Optional[str]
    iteration: int

class AgentA:
    """Agent A: Basic LLM + Lean Verification Loop"""
    
    def __init__(self, model_name: str = "claude-sonnet-4-20250514"):
        self.client = anthropic.Anthropic()
        self.model_name = model_name
        self.max_iterations = 3000
        self.lean_verifier = LeanVerifier()
    
    async def solve_problem(
        self, 
        problem_statement: str,
        lean_template: str
    ) -> ProofAttempt:
        """
        Core loop for solving mathematical problems
        
        Args:
            problem_statement: Natural language description of the problem
            lean_template: Lean proof template
            
        Returns:
            ProofAttempt: Proof attempt record
        """
        lean_code = lean_template
        iteration = 0
        
        while iteration < self.max_iterations:
            # Step 1: LLM generates proof
            response = await self._generate_proof(
                problem_statement, 
                lean_code
            )
            
            # Step 2: Lean compiler verification
            verification_result = self.lean_verifier.verify(lean_code)
            
            if verification_result.is_valid:
                return ProofAttempt(
                    problem=problem_statement,
                    lean_code=lean_code,
                    error_message=None,
                    iteration=iteration
                )
            
            # Step 3: Fix based on error feedback
            lean_code = await self._fix_proof(
                lean_code,
                verification_result.error_message
            )
            iteration += 1
        
        return ProofAttempt(
            problem=problem_statement,
            lean_code=lean_code,
            error_message="Max iterations reached",
            iteration=iteration
        )
    
    async def _generate_proof(
        self, 
        problem: str, 
        current_lean: str
    ) -> str:
        """Call LLM to generate Lean proof code"""
        message = self.client.messages.create(
            model=self.model_name,
            max_tokens=4096,
            messages=[
                {
                    "role": "user",
                    "content": f"""Given the following math problem:
{problem}

Current Lean code (with errors):
```lean
{current_lean}

Please provide the corrected Lean proof code. Focus on fixing any syntax errors and improving the proof strategy.""" } ] ) return message.content[0].text

async def _fix_proof(
    self, 
    lean_code: str, 
    error: str
) -> str:
    """Fix proof based on Lean error message"""
    return lean_code

class LeanVerifier: “““Lean compiler verifier”””

def __init__(self, lean_path: str = "/usr/local/bin/lean"):
    self.lean_path = lean_path

def verify(self, lean_code: str) -> VerificationResult:
    """Verify Lean proof correctness"""
    import subprocess
    import tempfile
    
    with tempfile.NamedTemporaryFile(
        mode='w', 
        suffix='.lean', 
        delete=False
    ) as f:
        f.write(lean_code)
        temp_path = f.name
    
    try:
        result = subprocess.run(
            [self.lean_path, temp_path],
            capture_output=True,
            text=True,
            timeout=30
        )
        
        if result.returncode == 0:
            return VerificationResult(is_valid=True)
        else:
            return VerificationResult(
                is_valid=False,
                error_message=result.stderr
            )
    finally:
        import os
        os.unlink(temp_path)

@dataclass class VerificationResult: “““Verification result””” is_valid: bool error_message: Optional[str] = None


### 2.3 Agent B: Integrating AlphaProof Reinforcement Learning

Agent B integrates AlphaProof—a reinforcement learning system specifically designed for mathematical proofs—on top of Agent A. When sub-agents get stuck on sub-goals, they can invoke AlphaProof for tree search to tackle local difficulties.

**Go Code Example: AlphaProof Reinforcement Learning Module**

```go
package alphaproof

import (
	"context"
	"math"
	"math/rand"
)

// ProofState represents the state during proof process
type ProofState struct {
	LeanCode     string
	Goals        []ProofGoal    // Goals to prove
	ProvenGoals  []ProofGoal    // Proven goals
	Tactics      []string       // Sequence of tactics used
	Score        float64        // Evaluation score of current state
}

// ProofGoal represents a mathematical goal to prove
type ProofGoal struct {
	Type    string // Goal type: "theorem", "lemma", "corollary"
	Name    string // Goal name
	Statement string // Mathematical statement
}

// AlphaProof is a reinforcement learning-driven proof search system
type AlphaProof struct {
	policyNetwork   *PolicyNetwork
	valueNetwork    *ValueNetwork
	temperature     float64
	numSimulations  int
	maxDepth        int
}

// PolicyNetwork: Policy network for selecting next proof tactic
type PolicyNetwork struct {
	hiddenSize int
	outputSize int
	weights    [][][]float64
}

// ValueNetwork: Value network for evaluating state value
type ValueNetwork struct {
	hiddenSize int
	weights    [][]float64
}

// NewAlphaProof creates a new AlphaProof instance
func NewAlphaProof(hiddenSize, outputSize int) *AlphaProof {
	return &AlphaProof{
		policyNetwork:  NewPolicyNetwork(hiddenSize, outputSize),
		valueNetwork:   NewValueNetwork(hiddenSize),
		temperature:    1.0,
		numSimulations: 800,
		maxDepth:       50,
	}
}

// MCTS uses Monte Carlo Tree Search to find optimal proof tactics
func (ap *AlphaProof) MCTS(ctx context.Context, state *ProofState) (string, error) {
	root := NewMonteCarloTree(state)
	
	for i := 0; i < ap.numSimulations; i++ {
		select {
		case <-ctx.Done():
			return "", ctx.Err()
		default:
		}
		
		// Selection
		node := root.Select()
		
		// Expansion
		if !node.IsTerminal() {
			action := ap.policyNetwork.SelectAction(node.State, ap.temperature)
			node = node.Expand(action)
		}
		
		// Simulation
		reward := ap.simulate(node.State)
		
		// Backpropagation
		node.Backpropagate(reward)
	}
	
	// Select best action
	bestChild := root.BestChild()
	return bestChild.Action, nil
}

// simulate performs random simulation on state, returns final reward
func (ap *AlphaProof) simulate(state *ProofState) float64 {
	currentState := state.Copy()
	depth := 0
	
	for !currentState.IsComplete() && depth < ap.maxDepth {
		tactics := ap.getAvailableTactics(currentState)
		if len(tactics) == 0 {
			break
		}
		
		// Select based on policy probabilities
		probs := ap.policyNetwork.GetActionProbabilities(currentState, tactics)
		selectedIdx := ap.sampleFromDistribution(probs)
		selectedTactic := tactics[selectedIdx]
		
		// Apply tactic
		currentState.Apply(selectedTactic)
		depth++
	}
	
	// Calculate reward
	return ap.calculateReward(currentState)
}

// calculateReward calculates reward for state
func (ap *AlphaProof) calculateReward(state *ProofState) float64 {
	if state.IsComplete() {
		return 1.0 // Completely proved
	}
	
	// Value network evaluation
	value := ap.valueNetwork.Evaluate(state)
	
	// Progress reward
	progressReward := float64(len(state.ProvenGoals)) / 
		float64(len(state.ProvenGoals)+len(state.Goals))
	
	// Combined reward
	return 0.7*value + 0.3*progressReward
}

// MonteCarloTree represents Monte Carlo tree node
type MonteCarloTree struct {
	state    *ProofState
	parent   *MonteCarloTree
	children []*MonteCarloTree
	action   string
	visits   int
	wins     float64
	uct      float64
}

// NewMonteCarloTree creates a new MCT root node
func NewMonteCarloTree(state *ProofState) *MonteCarloTree {
	return &MonteCarloTree{
		state:  state,
		visits: 1,
		wins:   0,
	}
}

// Select uses UCT algorithm to select child node
func (mct *MonteCarloTree) Select() *MonteCarloTree {
	if mct.IsFullyExpanded() {
		bestChild := mct.children[0]
		bestUCT := mct.children[0].uct
		
		for _, child := range mct.children[1:] {
			if child.uct > bestUCT {
				bestChild = child
				bestUCT = child.uct
			}
		}
		return bestChild.Select()
	}
	return mct
}

// Expand expands tree node
func (mct *MonteCarloTree) Expand(action string) *MonteCarloTree {
	newState := mct.state.Copy()
	newState.Apply(action)
	
	child := &MonteCarloTree{
		state:  newState,
		parent: mct,
		action: action,
		visits: 1,
	}
	
	mct.children = append(mct.children, child)
	return child
}

// Backpropagate updates statistics through backpropagation
func (mct *MonteCarloTree) Backpropagate(reward float64) {
	mct.visits++
	mct.wins += reward
	
	if mct.parent != nil {
		exploration := math.Sqrt(math.Log(float64(mct.parent.visits)) / float64(mct.visits))
		mct.uct = (mct.wins / float64(mct.visits)) + 0.5*exploration
		
		mct.parent.Backpropagate(reward)
	}
}

// BestChild returns the best child node
func (mct *MonteCarloTree) BestChild() *MonteCarloTree {
	maxVisits := 0
	bestChild := mct.children[0]
	
	for _, child := range mct.children {
		if child.visits > maxVisits {
			maxVisits = child.visits
			bestChild = child
		}
	}
	
	return bestChild
}

// IsTerminal checks if it's a terminal state
func (mct *MonteCarloTree) IsTerminal() bool {
	return mct.state.IsComplete() || len(mct.children) == 0
}

// IsFullyExpanded checks if fully expanded
func (mct *MonteCarloTree) IsFullyExpanded() bool {
	tactics := getAvailableTacticsStatic(mct.state)
	return len(mct.children) >= len(tactics)
}

// Helper functions
func (ap *AlphaProof) sampleFromDistribution(probs []float64) int {
	r := rand.Float64()
	cumulative := 0.0
	
	for i, p := range probs {
		cumulative += p
		if r < cumulative {
			return i
		}
	}
	return len(probs) - 1
}

func (ap *AlphaProof) getAvailableTactics(state *ProofState) []string {
	return []string{
		"rw",       // Rewrite
		"simp",     // Simplify
		"intro",    // Introduce variables
		"apply",    // Apply lemma
		"induction", // Mathematical induction
		"cases",   // Case analysis
		"split",    // Split
		"use",      // Use hypothesis
	}
}

func getAvailableTacticsStatic(state *ProofState) []string {
	return []string{
		"rw", "simp", "intro", "apply", 
		"induction", "cases", "split", "use",
	}
}

// NewPolicyNetwork creates a policy network
func NewPolicyNetwork(hiddenSize, outputSize int) *PolicyNetwork {
	return &PolicyNetwork{
		hiddenSize: hiddenSize,
		outputSize: outputSize,
	}
}

// SelectAction selects action
func (pn *PolicyNetwork) SelectAction(state *ProofState, temp float64) string {
	tactics := []string{"rw", "simp", "intro", "apply"}
	idx := rand.Intn(len(tactics))
	return tactics[idx]
}

// GetActionProbabilities gets action probability distribution
func (pn *PolicyNetwork) GetActionProbabilities(state *ProofState, tactics []string) []float64 {
	prob := 1.0 / float64(len(tactics))
	return make([]float64, len(tactics))
}

// NewValueNetwork creates a value network
func NewValueNetwork(hiddenSize int) *ValueNetwork {
	return &ValueNetwork{
		hiddenSize: hiddenSize,
	}
}

// Evaluate evaluates state value
func (vn *ValueNetwork) Evaluate(state *ProofState) float64 {
	progress := float64(len(state.ProvenGoals)) / 
		float64(len(state.ProvenGoals)+len(state.Goals)+1)
	return progress
}

2.4 Agent C: Introducing Evolutionary Algorithm

Agent C introduces evolutionary algorithms. Multiple sub-agents no longer work independently but share a population database. Each proof draft is scored by an LLM reviewer (using ELO rating system), with high-scoring drafts preferentially sampled, mutated, and evolved.

Python Code Example: Evolutionary Algorithm Core

import numpy as np
from dataclasses import dataclass, field
from typing import List, Optional, Callable
import random

@dataclass
class ProofIndividual:
    """Proof individual in evolutionary algorithm"""
    id: str
    lean_code: str
    fitness: float = 0.0
    elo_rating: float = 1500.0
    wins: int = 0
    losses: int = 0
    
    def __hash__(self):
        return hash(self.id)

class PopulationDatabase:
    """Population database - stores and manages proof individuals"""
    
    def __init__(self, max_size: int = 1000):
        self.individuals: List[ProofIndividual] = []
        self.max_size = max_size
        self.generation = 0
    
    def add(self, individual: ProofIndividual) -> None:
        """Add new individual"""
        if len(self.individuals) >= self.max_size:
            self.individuals.sort(key=lambda x: x.fitness)
            self.individuals.pop(0)
        
        self.individuals.append(individual)
    
    def get_top_n(self, n: int) -> List[ProofIndividual]:
        """Get top N individuals"""
        sorted_ind = sorted(
            self.individuals, 
            key=lambda x: x.fitness, 
            reverse=True
        )
        return sorted_ind[:n]
    
    def sample(self, k: int, selection_pressure: float = 0.7) -> List[ProofIndividual]:
        """
        Fitness-based weighted sampling
        
        Args:
            k: Sample count
            selection_pressure: Selection pressure (0-1)
        """
        if not self.individuals:
            return []
        
        fitnesses = np.array([ind.fitness for ind in self.individuals])
        exp_fitness = np.exp(fitnesses * selection_pressure)
        probs = exp_fitness / exp_fitness.sum()
        
        indices = np.random.choice(
            len(self.individuals),
            size=min(k, len(self.individuals)),
            p=probs,
            replace=False
        )
        
        return [self.individuals[i] for i in indices]

class EvolutionEngine:
    """Evolution engine - implements proof evolution optimization"""
    
    def __init__(
        self,
        population_db: PopulationDatabase,
        mutation_rate: float = 0.1,
        crossover_rate: float = 0.7,
        elite_ratio: float = 0.1
    ):
        self.population_db = population_db
        self.mutation_rate = mutation_rate
        self.crossover_rate = crossover_rate
        self.elite_ratio = elite_ratio
        self.llm_judge = LLMJudge()
    
    def evolve_generation(self) -> List[ProofIndividual]:
        """Execute one generation of evolution"""
        current_pop = self.population_db.individuals.copy()
        
        # Elite preservation
        elite_count = int(len(current_pop) * self.elite_ratio)
        elites = self.population_db.get_top_n(elite_count)
        
        # Select parents
        parents = self.population_db.sample(k=len(current_pop) * 2)
        
        # Produce next generation
        next_generation = []
        next_generation.extend(elites)
        
        # Crossover and mutation
        while len(next_generation) < len(current_pop):
            parent1, parent2 = random.sample(parents, 2)
            
            if random.random() < self.crossover_rate:
                child = self._crossover(parent1, parent2)
            else:
                child = self._copy_individual(parent1)
            
            if random.random() < self.mutation_rate:
                child = self._mutate(child)
            
            child.fitness = self.llm_judge.evaluate(child.lean_code)
            
            next_generation.append(child)
            self.population_db.add(child)
        
        self.population_db.generation += 1
        return next_generation
    
    def _crossover(
        self, 
        parent1: ProofIndividual, 
        parent2: ProofIndividual
    ) -> ProofIndividual:
        """Crossover operation"""
        if random.random() < 0.5:
            lean_code = parent1.lean_code
        else:
            lean_code = parent2.lean_code
        
        return ProofIndividual(
            id=f"{parent1.id}_{parent2.id}_crossover",
            lean_code=lean_code
        )
    
    def _mutate(self, individual: ProofIndividual) -> ProofIndividual:
        """Mutation operation"""
        lean_code = individual.lean_code
        
        mutations = [
            lambda c: c + "\n-- mutated",
            lambda c: "-- mutated\n" + c,
            lambda c: c.replace(".", "._"),
        ]
        
        mutation = random.choice(mutations)
        mutated_code = mutation(lean_code)
        
        return ProofIndividual(
            id=f"{individual.id}_mutated_{random.randint(1000,9999)}",
            lean_code=mutated_code
        )
    
    def _copy_individual(self, individual: ProofIndividual) -> ProofIndividual:
        """Copy individual"""
        return ProofIndividual(
            id=f"{individual.id}_copy",
            lean_code=individual.lean_code
        )

class LLMJudge:
    """LLM Reviewer - evaluates proof quality using ELO system"""
    
    def __init__(self, k_factor: float = 32):
        self.k_factor = k_factor
    
    def evaluate(self, lean_code: str) -> float:
        """Evaluate proof fitness"""
        score = 0.5
        
        if "sorry" not in lean_code:
            score += 0.2
        
        if 10 < lean_code.count("\n") < 500:
            score += 0.15
        
        if lean_code.startswith("theorem") or lean_code.startswith("lemma"):
            score += 0.15
        
        return min(1.0, max(0.0, score))
    
    def update_elo(
        self, 
        winner: ProofIndividual, 
        loser: ProofIndividual
    ) -> tuple[float, float]:
        """Update ELO ratings"""
        expected_winner = 1 / (1 + 10 ** (
            (loser.elo_rating - winner.elo_rating) / 400
        ))
        expected_loser = 1 - expected_winner
        
        winner.elo_rating += self.k_factor * (1 - expected_winner)
        loser.elo_rating += self.k_factor * (0 - expected_loser)
        
        winner.wins += 1
        loser.losses += 1
        
        winner.fitness = winner.elo_rating / 2000
        loser.fitness = loser.elo_rating / 2000
        
        return winner.elo_rating, loser.elo_rating

2.5 Agent D: Complete Synergy System

Agent D is the ultimate combination, integrating evolutionary algorithm, AlphaProof, and Gemini 3.1 Pro in coordinated combat, unified by a coordinator. This is DeepMind’s primary weapon for large-scale conquest of Erdős problems.

3. Core Algorithm: LLM + Lean Formalized Proof Loop

3.1 Workflow Details

AI generates proof draft → Lean compiler verification → Failure provides error feedback → AI fixes → Verify again → Loop until success

The core of this loop is compiler feedback’s anchoring effect on LLM reasoning. Compared to traditional methods, the Lean compiler provides strict formal verification, ensuring AI-generated proofs are absolutely correct with no “hallucination” space.

3.2 Python Implementation: Complete Proof Loop

import asyncio
from typing import Optional, Tuple
import anthropic

class FormalProofLoop:
    """
    Formal proof loop
    Core: LLM generates → Lean verifies → Feedback fixes → Loop
    """
    
    def __init__(
        self,
        lean_path: str = "/usr/local/bin/lean",
        model: str = "claude-sonnet-4-20250514"
    ):
        self.lean_path = lean_path
        self.client = anthropic.Anthropic()
        self.model = model
    
    async def prove_theorem(
        self,
        theorem_name: str,
        theorem_statement: str,
        max_iterations: int = 3000
    ) -> Tuple[bool, str, int]:
        """
        Prove theorem
        
        Returns:
            (success, lean_code, iterations)
        """
        lean_code = f"""theorem {theorem_name}
{theorem_statement}
:= 
begin
  -- Proof begins
  
end
"""
        
        iteration = 0
        error_history = []
        
        while iteration < max_iterations:
            is_valid, error_msg = await self._verify_lean(lean_code)
            
            if is_valid:
                return True, lean_code, iteration
            
            error_history.append({
                "iteration": iteration,
                "error": error_msg,
                "code": lean_code
            })
            
            lean_code = await self._fix_with_error_feedback(
                theorem_name,
                theorem_statement,
                lean_code,
                error_msg,
                error_history
            )
            
            iteration += 1
        
        return False, lean_code, iteration
    
    async def _verify_lean(self, lean_code: str) -> Tuple[bool, Optional[str]]:
        """Verify proof using Lean compiler"""
        import subprocess
        import tempfile
        import os
        
        with tempfile.NamedTemporaryFile(
            mode='w',
            suffix='.lean',
            delete=False
        ) as f:
            f.write(lean_code)
            temp_path = f.name
        
        try:
            result = subprocess.run(
                [self.lean_path, temp_path],
                capture_output=True,
                text=True,
                timeout=60
            )
            
            if result.returncode == 0:
                return True, None
            else:
                return False, self._parse_lean_error(result.stderr)
        finally:
            os.unlink(temp_path)
    
    def _parse_lean_error(self, error: str) -> str:
        """Parse Lean error message"""
        lines = error.split('\n')
        for line in lines:
            if 'error:' in line.lower():
                return line
        return error[:500]
    
    async def _fix_with_error_feedback(
        self,
        theorem_name: str,
        theorem_statement: str,
        current_code: str,
        error: str,
        error_history: list
    ) -> str:
        """Fix proof using error feedback"""
        
        error_summary = "\n".join([
            f"Iteration {e['iteration']}: {e['error'][:200]}"
            for e in error_history[-3:]
        ])
        
        prompt = f"""You are a Lean 4 proof assistant. Please fix errors in the following Lean proof code.

Theorem name: {theorem_name}
Theorem statement: {theorem_statement}

Current Lean code:
```lean
{current_code}

Lean compiler error: {error}

Recent error history: {error_summary}

Please generate the corrected Lean proof code. Ensure:

Fix all syntax errors
Resolve logical issues
Use appropriate proof tactics (rw, simp, apply, cases, induction, etc.)
Do not use sorry

Only return Lean code, no explanations."""

    response = self.client.messages.create(
        model=self.model,
        max_tokens=4096,
        messages=[{"role": "user", "content": prompt}]
    )
    
    code = response.content[0].text
    
    if "```lean" in code:
        start = code.index("```lean") + 7
        end = code.index("```", start)
        code = code[start:end]
    elif "```" in code:
        start = code.index("```") + 3
        end = code.rindex("```")
        code = code[start:end]
    
    return code.strip()

async def main(): prover = FormalProofLoop()

success, code, iterations = await prover.prove_theorem(
    theorem_name="simple_example",
    theorem_statement="(n : ℕ) → n + 0 = n",
    max_iterations=100
)

if success:
    print(f"Proof succeeded! Used {iterations} iterations")
    print(code)
else:
    print(f"Proof failed after {iterations} iterations")

if name == “main”: asyncio.run(main())


### 3.3 Key Finding: Simple Agents Can Solve Complex Problems

DeepMind discovered a surprising conclusion: **Even the simplest Agent A can solve all 9 Erdős problems!**

This means:
- Agent A and Agent B perform nearly identically on most problems
- Agent D's advantage mainly shows on the hardest problems, with 2-5x cost efficiency
- LLM capability improvement is the key factor
- **Compiler feedback plays a powerful role in anchoring LLM reasoning**

## 4. Deep Dive: Technical Principles and Innovations

### 4.1 Why Is the Lean Compiler So Important?

Lean is a proof assistant and functional programming language developed by Microsoft Research. Its key features:

1. **Formal verification**: Every step must strictly follow mathematical logic
2. **Type safety**: Ensures completeness and consistency of proofs
3. **Checkability**: Anyone can verify proof correctness

Traditional AI proof problems: ┌─────────────────────────────────────────┐ │ AI generates “seemingly correct” proof │ │ ↓ │ │ Human expert verification → May have logical flaws │ │ ↓ │ │ Difficult to detect “hallucination” errors │ └─────────────────────────────────────────┘

AlphaProof Nexus approach: ┌─────────────────────────────────────────┐ │ AI generates proof draft │ │ ↓ │ │ Lean compiler strict verification → Detects all errors │ │ ↓ │ │ Feedback to AI for correction → Gradually approach correct proof │ └─────────────────────────────────────────┘


### 4.2 Role of Evolutionary Algorithms in Proof Search

Evolutionary algorithms enhance proof quality through:

1. **Diversity preservation**: Maintaining diverse proof strategies in population
2. **Elite preservation**: Retaining best individuals to avoid degradation
3. **Mutation and crossover**: Exploring new proof paths
4. **ELO rating**: Evaluating proof quality based on adversarial comparison

### 4.3 Unique Value of AlphaProof Reinforcement Learning

AlphaProof is specifically designed for mathematical proofs:

- **Tree search**: Efficient search in vast proof space
- **Value evaluation**: Assessing distance from complete proof
- **Strategy learning**: Learning to select most effective proof tactics

## 5. Experimental Results and Case Analysis

### 5.1 Erdős #12: Classic Problem for 56 Years

**Problem**: Does there exist an infinite set A satisfying "for any three distinct elements a<b<c, a+b≠c"?

**AI's proof**:
- Brilliantly combines Chinese Remainder Theorem and three-term arithmetic progression-free sets
- Constructs carefully designed "blocks" satisfying density conditions
- Complete proof spans over 200 lines of Lean code

### 5.2 Erdős #125: Lower Density Problem

**Problem**: In specific number systems, is the lower density of the sumset of sets positive?

**AI's answer**: No, lower density is zero

**Core proof strategy**:
- Inductive sparsification argument
- Utilizes Diophantine approximation properties of 3^m and 4^k
- Key property: log₄/log₃ is irrational

### 5.3 Erdős #846: Miracle of Geometric Construction

**Problem**: Collinearity properties in planar point sets

**AI's construction is breathtaking**:
- Maps each edge of complete graph K∞ to a point in the plane
- Encodes coordinates using quadratic polynomials
- Completes proof using infinite Ramsey theorem

## 6. Code Implementation: Building Your Own Mathematical Proof Agent

### 6.1 Complete Python Implementation

```python
"""
AlphaProof Nexus Simplified Implementation
For educational and research purposes
"""

import asyncio
import re
import subprocess
from dataclasses import dataclass, field
from typing import List, Optional, Dict, Tuple
from enum import Enum
import anthropic

# ============ Configuration ============
class Config:
    ANTHROPIC_MODEL = "claude-sonnet-4-20250514"
    MAX_ITERATIONS = 3000
    LEAN_PATH = "/usr/local/bin/lean"
    TEMPERATURE = 0.7

# ============ Data Structures ============
class ProofStatus(Enum):
    UNKNOWN = "unknown"
    PROVING = "proving"
    PROVED = "proved"
    FAILED = "failed"

@dataclass
class ProofState:
    theorem_name: str
    theorem_statement: str
    lean_code: str
    status: ProofStatus = ProofStatus.UNKNOWN
    error_message: Optional[str] = None
    iteration: int = 0
    proof_steps: List[str] = field(default_factory=list)

@dataclass
class LeanVerificationResult:
    is_valid: bool
    error_message: Optional[str] = None
    error_line: Optional[int] = None

# ============ Lean Verifier ============
class LeanVerifier:
    """Lean proof verifier"""
    
    def __init__(self, lean_path: str = Config.LEAN_PATH):
        self.lean_path = lean_path
    
    def verify(self, lean_code: str) -> LeanVerificationResult:
        """
        Verify Lean proof
        
        Args:
            lean_code: Lean 4 proof code
            
        Returns:
            LeanVerificationResult: Verification result
        """
        import tempfile
        import os
        
        with tempfile.NamedTemporaryFile(
            mode='w',
            suffix='.lean',
            delete=False
        ) as f:
            f.write(lean_code)
            temp_path = f.name
        
        try:
            result = subprocess.run(
                [self.lean_path, temp_path],
                capture_output=True,
                text=True,
                timeout=60
            )
            
            if result.returncode == 0:
                return LeanVerificationResult(is_valid=True)
            else:
                error_info = self._parse_error(result.stderr)
                return LeanVerificationResult(
                    is_valid=False,
                    error_message=error_info['message'],
                    error_line=error_info.get('line')
                )
        except subprocess.TimeoutExpired:
            return LeanVerificationResult(
                is_valid=False,
                error_message="Verification timeout"
            )
        finally:
            try:
                os.unlink(temp_path)
            except:
                pass
    
    def _parse_error(self, stderr: str) -> Dict:
        """Parse Lean error message"""
        # Format: file.lean:line:col: error: message
        pattern = r'([^:]+):(\d+):(\d+):\s*error:\s*(.+)'
        match = re.search(pattern, stderr)
        
        if match:
            return {
                'line': int(match.group(2)),
                'col': match.group(3),
                'message': match.group(4).strip()
            }
        
        return {'message': stderr[:500].strip()}

# ============ Proof Generator ============
class ProofGenerator:
    """Generate Lean proofs using LLM"""
    
    def __init__(self, model: str = Config.ANTHROPIC_MODEL):
        self.client = anthropic.Anthropic()
        self.model = model
    
    async def generate(
        self,
        theorem_name: str,
        theorem_statement: str,
        current_code: str,
        error_message: Optional[str],
        proof_hints: Optional[List[str]] = None
    ) -> str:
        """
        Generate or fix Lean proof
        
        Args:
            theorem_name: Theorem name
            theorem_statement: Theorem statement
            current_code: Current Lean code
            error_message: Verification error
            proof_hints: Additional proof hints
            
        Returns:
            Fixed Lean code
        """
        system_prompt = """You are a professional Lean 4 proof assistant. Your task is to fix proof code based on Lean compiler feedback.

Common Lean 4 proof tactics:
- `intro` / `intros`: Introduce variables and hypotheses
- `rw` / `rewrite`: Rewrite equations
- `simp`: Use simplification rules
- `apply`: Apply lemmas or theorems
- `exact`: Specify exact term
- `use`: Use existence proof
- `cases`: Case analysis
- `induction`: Mathematical induction
- `refl`: Prove equality
- `omega`: Solve linear arithmetic

Important principles:
1. Every reasoning step must have explicit justification
2. Do not use `sorry`
3. Ensure all referenced theorems or lemmas exist
4. Keep code structure clear"""

        user_prompt = f"""Theorem: {theorem_name}
Statement: {theorem_statement}

Current Lean code:

{current_code}


        if error_message:
            user_prompt += f"""

Lean compiler error:

{error_message}


Please fix errors in the above code."""

        if proof_hints:
            user_prompt += f"""

Proof hints:
"""
            for i, hint in enumerate(proof_hints, 1):
                user_prompt += f"{i}. {hint}\n"

        user_prompt += """

Please only return the fixed Lean code, wrapped in ```lean code block```."""

        response = self.client.messages.create(
            model=self.model,
            max_tokens=8192,
            temperature=Config.TEMPERATURE,
            system=system_prompt,
            messages=[{"role": "user", "content": user_prompt}]
        )
        
        code = response.content[0].text
        return self._extract_lean_code(code)
    
    def _extract_lean_code(self, text: str) -> str:
        """Extract Lean code from LLM output"""
        patterns = [
            r'```lean\s*(.*?)\s*```',
            r'```\s*(theorem.*?)```',
            r'(theorem.*)',
        ]
        
        for pattern in patterns:
            match = re.search(pattern, text, re.DOTALL)
            if match:
                return match.group(1).strip()
        
        return text.strip()

# ============ Main Proof Loop ============
class AlphaProofNexusLite:
    """
    AlphaProof Nexus Simplified Implementation
    
    This class implements the core proof loop described in the paper:
    1. LLM generates proof
    2. Lean verification
    3. Error feedback
    4. Iterative correction
    """
    
    def __init__(
        self,
        lean_path: str = Config.LEAN_PATH,
        model: str = Config.ANTHROPIC_MODEL
    ):
        self.verifier = LeanVerifier(lean_path)
        self.generator = ProofGenerator(model)
    
    async def prove(
        self,
        theorem_name: str,
        theorem_statement: str,
        initial_template: Optional[str] = None,
        max_iterations: int = Config.MAX_ITERATIONS
    ) -> ProofState:
        """
        Prove theorem
        
        Args:
            theorem_name: Theorem name
            theorem_statement: Theorem statement
            initial_template: Initial Lean template (optional)
            max_iterations: Maximum iterations
            
        Returns:
            ProofState: Final proof state
        """
        if initial_template:
            lean_code = initial_template
        else:
            lean_code = f"""theorem {theorem_name}
{theorem_statement}
:= 
begin
  sorry
end
"""
        
        state = ProofState(
            theorem_name=theorem_name,
            theorem_statement=theorem_statement,
            lean_code=lean_code,
            status=ProofStatus.PROVING
        )
        
        error_history = []
        
        for iteration in range(max_iterations):
            result = self.verifier.verify(state.lean_code)
            
            if result.is_valid:
                state.status = ProofStatus.PROVED
                state.iteration = iteration
                return state
            
            error_history.append(result.error_message)
            
            lean_code = await self.generator.generate(
                theorem_name,
                theorem_statement,
                state.lean_code,
                result.error_message,
                proof_hints=self._get_proof_hints(error_history)
            )
            
            state.lean_code = lean_code
            state.error_message = result.error_message
            state.iteration = iteration
            
            if iteration > 0 and lean_code == state.lean_code:
                break
        
        state.status = ProofStatus.FAILED
        return state
    
    def _get_proof_hints(self, error_history: List[str]) -> List[str]:
        """Generate proof hints based on error history"""
        hints = []
        
        if any("unknown identifier" in e.lower() for e in error_history):
            hints.append("Check if all referenced theorem and variable names are correct")
        
        if any("type mismatch" in e.lower() for e in error_history):
            hints.append("Check type matching, use `congr` for isomorphic types")
        
        if any("tactic failed" in e.lower() for e in error_history):
            hints.append("Proof tactic failed, try other tactics or decompose the problem")
        
        if len(error_history) > 3:
            hints.append("Consider redesigning the proof strategy from scratch")
        
        return hints

# ============ Usage Example ============
async def example_proof():
    """Example: Prove simple addition property"""
    
    prover = AlphaProofNexusLite()
    
    state = await prover.prove(
        theorem_name="add_zero_right",
        theorem_statement="∀ (n : ℕ), n + 0 = n"
    )
    
    print(f"Status: {state.status.value}")
    print(f"Iterations: {state.iteration}")
    
    if state.status == ProofStatus.PROVED:
        print("Proof succeeded!")
        print(state.lean_code)
    else:
        print("Proof failed")
        print(f"Last error: {state.error_message}")

if __name__ == "__main__":
    asyncio.run(example_proof())

6.2 Go Version Lean Integration

package main

import (
	"bufio"
	"context"
	"fmt"
	"os"
	"os/exec"
	"strings"
	"time"
)

// Lean4Proof represents a Lean proof
type Lean4Proof struct {
	TheoremName    string
	TheoremStmt   string
	ProofCode     string
	ProofSteps    []string
	IsComplete    bool
	LastError     string
	Iteration     int
}

// Lean4Engine interacts with Lean 4 compiler
type Lean4Engine struct {
	leanPath string
	timeout  time.Duration
}

// NewLean4Engine creates a new Lean 4 engine
func NewLean4Engine(leanPath string) *Lean4Engine {
	if leanPath == "" {
		leanPath = "lean"
	}
	return &Lean4Engine{
		leanPath: leanPath,
		timeout:  60 * time.Second,
	}
}

// VerificationResult represents verification result
type VerificationResult struct {
	IsValid bool
	Error   string
	Line    int
}

// Verify verifies Lean proof
func (e *Lean4Engine) Verify(proof *Lean4Proof) VerificationResult {
	tmpFile, err := os.CreateTemp("", "proof_*.lean")
	if err != nil {
		return VerificationResult{IsValid: false, Error: err.Error()}
	}
	defer os.Remove(tmpFile.Name())
	defer tmpFile.Close()

	_, err = tmpFile.WriteString(proof.ProofCode)
	if err != nil {
		return VerificationResult{IsValid: false, Error: err.Error()}
	}
	tmpFile.Close()

	ctx, cancel := context.WithTimeout(context.Background(), e.timeout)
	defer cancel()

	cmd := exec.CommandContext(ctx, e.leanPath, tmpFile.Name())
	output, err := cmd.CombinedOutput()

	if err == nil {
		return VerificationResult{IsValid: true}
	}

	return e.parseLeanError(string(output))
}

// parseLeanError parses Lean error message
func (e *Lean4Engine) parseLeanError(output string) VerificationResult {
	scanner := bufio.NewScanner(strings.NewReader(output))
	
	for scanner.Scan() {
		line := scanner.Text()
		
		if strings.Contains(line, "error:") {
			parts := strings.Split(line, ":")
			if len(parts) >= 2 {
				var lineNum int
				fmt.Sscanf(parts[1], "%d", &lineNum)
				
				idx := strings.Index(line, "error:")
				message := strings.TrimSpace(line[idx+6:])
				
				return VerificationResult{
					IsValid: false,
					Error:   message,
					Line:    lineNum,
				}
			}
		}
	}
	
	return VerificationResult{
		IsValid: false,
		Error:   output,
	}
}

// GenerateProof generates or fixes proof
func (e *Lean4Engine) GenerateProof(
	theoremName string,
	theoremStmt string,
	currentProof string,
	errorMsg string,
) string {
	if errorMsg != "" {
		if strings.Contains(errorMsg, "unknown identifier") {
			return fmt.Sprintf(`theorem %s
%s
:= 
begin
  -- Check if identifier is correctly defined
  sorry
end
`, theoremName, theoremStmt)
		}
	}
	
	return fmt.Sprintf(`theorem %s
%s
:= 
begin
  -- Please add your proof
  sorry
end
`, theoremName, theoremStmt)
}

// ProofLoop complete proof loop
func (e *Lean4Engine) ProofLoop(
	ctx context.Context,
	theoremName string,
	theoremStmt string,
	maxIterations int,
) *Lean4Proof {
	proof := &Lean4Proof{
		TheoremName:  theoremName,
		TheoremStmt:  theoremStmt,
		ProofCode:    fmt.Sprintf("theorem %s\n%s\n:= \nbegin\n  sorry\nend\n", theoremName, theoremStmt),
		IsComplete:   false,
		Iteration:    0,
	}

	for proof.Iteration < maxIterations {
		select {
		case <-ctx.Done():
			proof.LastError = "timeout"
			return proof
		default:
		}

		result := e.Verify(proof)
		
		if result.IsValid {
			proof.IsComplete = true
			return proof
		}

		proof.ProofCode = e.GenerateProof(
			theoremName,
			theoremStmt,
			proof.ProofCode,
			result.Error,
		)
		
		proof.LastError = result.Error
		proof.Iteration++
	}

	return proof
}

func main() {
	engine := NewLean4Engine("")

	ctx := context.Background()
	proof := engine.ProofLoop(
		ctx,
		"add_zero_right",
		"∀ (n : ℕ), n + 0 = n",
		100,
	)

	fmt.Printf("Theorem: %s\n", proof.TheoremName)
	fmt.Printf("Iterations: %d\n", proof.Iteration)
	fmt.Printf("Complete: %v\n", proof.IsComplete)
	
	if proof.IsComplete {
		fmt.Println("Proof succeeded!")
		fmt.Println(proof.ProofCode)
	} else {
		fmt.Printf("Last error: %s\n", proof.LastError)
	}
}

7. Industry Impact and Future Outlook

7.1 Revolutionary Impact on Mathematical Research

AlphaProof Nexus’s success foreshadows AI becoming a standard research tool for mathematicians:

Accelerate conjecture verification: Quickly verify or refute mathematical conjectures
Discover new theorems: AI may discover mathematical structures unnoticed by humans
Fill proof gaps: Complete proofs unfinished by human mathematicians
Educational assistance: Help students learn formal proof methods

7.2 Insights for AI Agent Technology

This paper reveals important principles for AI Agent design:

Formal verification > Probabilistic guessing: Compiler feedback is more reliable than human feedback
Simple architecture + Strong model = Good results: Agent A’s success shows base model capability is key
Iterative correction > One-shot generation: Multi-round feedback loops are more effective than single generation
Evolutionary algorithms complement reinforcement learning: Both methods have advantages on different difficulty problems

7.3 Open Source and Openness

DeepMind has open-sourced all Lean proof code for all 9 problems on GitHub:

Repository: google-deepmind/alphaproof-nexus-results
License: Apache 2.0 (code) + CC-BY 4.0 (documentation)

This provides valuable resources for the research community.

7.4 Future Development Directions

Stronger LLMs: Improved base models will directly enhance proof capability
Larger-scale testing: Cover more Erdős problems and other mathematical domains
Human-AI collaboration: AI assists human mathematicians in creative research
Cross-domain applications: Apply formal proof methods to computer science, physics, and other fields

8. Conclusion

AlphaProof Nexus is a milestone in AI mathematical research. It demonstrates that through the combination of LLM + Formal verification + Iterative feedback, AI can:

✅ Solve problems that have troubled mathematicians for decades
✅ Provide absolutely correct, hallucination-free proofs
✅ Complete proofs at extremely low cost (a few hundred dollars per problem)
✅ Maintain proof completeness and verifiability

More importantly, this technology reveals core principles of AI Agent design: Simple architecture combined with powerful base models, plus strict formal verification, can produce astonishing results.

Fields Medal laureate’s comment—“If this paper were submitted to the Annals of Mathematics by a human, I would recommend its acceptance without hesitation”—perhaps marks AI’s formal establishment as an indispensable tool in mathematical research.

References:

arXiv:2605.22763v1 - “Advancing Mathematics Research with AI-Driven Formal Proof Search”
GitHub: google-deepmind/alphaproof-nexus-results
Original report: “Google DeepMind AlphaProof Nexus Solves 9 Erdős Centenary Problems”

Related Resources:

Lean 4 Official: https://leanprover.github.io/