OpenAI破解80年数学猜想：AI首次完成原创性科学突破

"""
Erdős单位距离问题建模
Erdős Unit Distance Problem Modeling

作者：AI研究员
日期：2026-05-23
"""

import numpy as np
from typing import List, Tuple, Set
from collections import defaultdict
import matplotlib.pyplot as plt
from dataclasses import dataclass
import math


@dataclass
class Point:
    """二维平面上的点"""
    x: float
    y: float
    
    def distance_to(self, other: 'Point') -> float:
        """计算到另一个点的欧氏距离"""
        return math.sqrt((self.x - other.x) ** 2 + (self.y - other.y) ** 2)
    
    def is_unit_distance(self, other: 'Point', tolerance: float = 1e-6) -> bool:
        """判断是否为单位距离（带容差）"""
        return abs(self.distance_to(other) - 1.0) < tolerance
    
    def __hash__(self):
        return hash((round(self.x, 6), round(self.y, 6)))
    
    def __eq__(self, other):
        if not isinstance(other, Point):
            return False
        return (abs(self.x - other.x) < 1e-6 and 
                abs(self.y - other.y) < 1e-6)


class UnitDistanceProblem:
    """单位距离问题求解器"""
    
    def __init__(self, points: List[Point]):
        self.points = points
        self.n = len(points)
        self._unit_pairs: Set[Tuple[int, int]] = set()
    
    def count_unit_distances(self) -> int:
        """统计单位距离点对数量"""
        self._unit_pairs.clear()
        
        for i in range(self.n):
            for j in range(i + 1, self.n):
                if self.points[i].is_unit_distance(self.points[j]):
                    self._unit_pairs.add((i, j))
        
        return len(self._unit_pairs)
    
    def get_unit_distance_pairs(self) -> List[Tuple[Point, Point]]:
        """获取所有单位距离点对"""
        return [(self.points[i], self.points[j]) 
                for i, j in self._unit_pairs]
    
    def theoretical_upper_bound(self) -> float:
        """理论上界: O(n^(4/3))"""
        return self.n ** (4/3)
    
    def ai_proposed_lower_bound(self, delta: float = 0.014) -> float:
        """AI提出的新下界: n^(1+δ), δ ≈ 0.014"""
        return self.n ** (1 + delta)


def generate_square_grid(n: int) -> List[Point]:
    """
    生成正方形网格点（传统最优构造）
    
    复杂度: u(n) ≈ n^(1+o(1))
    """
    side = int(math.ceil(math.sqrt(n)))
    points = []
    
    for i in range(side):
        for j in range(side):
            if len(points) < n:
                points.append(Point(i, j))
    
    return points


def generate_triangular_grid(n: int) -> List[Point]:
    """
    生成三角网格点（可能更优的结构）
    
    三角网格比正方形网格有更多的邻近点
    """
    points = []
    row = 0
    
    while len(points) < n:
        # 第row行有row+1个点
        for col in range(row + 1):
            if len(points) >= n:
                break
            # 使用等边三角形的顶点坐标
            x = col + 0.5 * row
            y = row * math.sqrt(3) / 2
            points.append(Point(x, y))
        row += 1
    
    return points


def visualize_points(points: List[Point], title: str = "点分布"):
    """可视化点分布"""
    fig, ax = plt.subplots(figsize=(10, 10))
    
    xs = [p.x for p in points]
    ys = [p.y for p in points]
    
    ax.scatter(xs, ys, s=50, c='blue', alpha=0.6)
    
    # 绘制单位距离连线
    udp = UnitDistanceProblem(points)
    pairs = udp.get_unit_distance_pairs()
    
    for p1, p2 in pairs[:100]:  # 限制绘制数量
        ax.plot([p1.x, p2.x], [p1.y, p2.y], 'r-', alpha=0.3, linewidth=0.5)
    
    ax.set_title(f"{title}\n单位距离点对数: {len(pairs)}")
    ax.set_xlabel('X')
    ax.set_ylabel('Y')
    ax.set_aspect('equal')
    ax.grid(True, alpha=0.3)
    
    return fig


# ==================== 测试代码 ====================

if __name__ == "__main__":
    # 测试不同规模的问题
    sizes = [100, 500, 1000, 2000]
    
    print("=" * 70)
    print("Erdős单位距离问题 实验分析")
    print("=" * 70)
    
    for n in sizes:
        # 正方形网格
        square_points = generate_square_grid(n)
        udp_square = UnitDistanceProblem(square_points)
        count_square = udp_square.count_unit_distances()
        
        # 三角网格
        triangle_points = generate_triangular_grid(n)
        udp_triangle = UnitDistanceProblem(triangle_points)
        count_triangle = udp_triangle.count_unit_distances()
        
        # 理论值
        upper_bound = udp_square.theoretical_upper_bound()
        ai_bound = udp_square.ai_proposed_lower_bound()
        
        print(f"\nn = {n}")
        print("-" * 50)
        print(f"正方形网格单位距离对数: {count_square}")
        print(f"三角网格单位距离对数:   {count_triangle}")
        print(f"理论上界 O(n^(4/3)):     {upper_bound:.2f}")
        print(f"AI新下界 n^(1+0.014):   {ai_bound:.2f}")
        
        # 增长率分析
        if n > sizes[0]:
            prev_n = sizes[sizes.index(n) - 1]
            prev_square = UnitDistanceProblem(generate_square_grid(prev_n))
            prev_square.count_unit_distances()
            growth = count_square / len(prev_square._unit_pairs) if prev_square._unit_pairs else 0
            print(f"相比 n={prev_n} 的增长率: {growth:.2f}x")

"""
代数数域扩张构造器
Algebraic Number Field Extension Constructor

这是OpenAI证明Erdős猜想的核心创新部分
使用代数数论工具构造更高对称性的数域结构
"""

import numpy as np
from typing import List, Tuple, Optional
from dataclasses import dataclass
from fractions import Fraction
import cmath
import matplotlib.pyplot as plt


@dataclass
class AlgebraicInteger:
    """
    代数整数表示（基于最小多项式）
    形如 a_0 + a_1*θ + a_2*θ² + ... + a_{n-1}*θ^{n-1}
    """
    coefficients: List[complex]  # [a_0, a_1, ..., a_{n-1}]
    theta: complex  # 生成元θ
    min_poly_degree: int  # 最小多项式次数
    
    def __add__(self, other: 'AlgebraicInteger') -> 'AlgebraicInteger':
        """加法运算"""
        max_len = max(len(self.coefficients), len(other.coefficients))
        coeffs = [0] * max_len
        
        for i, c in enumerate(self.coefficients):
            if i < max_len:
                coeffs[i] += c
        
        for i, c in enumerate(other.coefficients):
            if i < max_len:
                coeffs[i] += c
        
        return AlgebraicInteger(coeffs, self.theta, max(self.min_poly_degree, other.min_poly_degree))
    
    def __mul__(self, other: 'AlgebraicInteger') -> 'AlgebraicInteger':
        """乘法运算"""
        deg = self.min_poly_degree + other.min_poly_degree - 1
        coeffs = [0] * deg
        
        for i, a in enumerate(self.coefficients):
            for j, b in enumerate(other.coefficients):
                if i + j < deg:
                    coeffs[i + j] += a * b
        
        return AlgebraicInteger(coeffs, self.theta, deg)
    
    def norm(self) -> complex:
        """
        计算代数整数的范数
        范数 = 所有共轭的乘积
        这是代数数论中的核心概念
        """
        conjugates = self._get_conjugates()
        norm_value = 1
        for conj in conjugates:
            norm_value *= conj._eval()
        return norm_value
    
    def _get_conjugates(self) -> List['AlgebraicInteger']:
        """获取所有共轭"""
        conjugates = []
        for i in range(self.min_poly_degree):
            theta_conj = cmath.rect(1, 2 * np.pi * i / self.min_poly_degree + cmath.phase(self.theta))
            conj = AlgebraicInteger(self.coefficients, theta_conj, self.min_poly_degree)
            conjugates.append(conj)
        return conjugates
    
    def _eval(self) -> complex:
        """求值"""
        result = 0
        theta_pow = 1
        for coeff in self.coefficients:
            result += coeff * theta_pow
            theta_pow *= self.theta
        return result


class GaussianInteger(AlgebraicInteger):
    """
    高斯整数：形如 a + bi 的复数（a, b为整数）
    这是埃尔德什原始构造使用的数域
    """
    
    def __init__(self, a: int, b: int):
        # 高斯整数使用虚数单位 i 作为生成元
        super().__init__(
            coefficients=[a, b],  # a + b*i
            theta=1j,  # i
            min_poly_degree=2
        )
        self.a = a
        self.b = b
    
    @classmethod
    def from_algebraic(cls, alg: AlgebraicInteger) -> Optional['GaussianInteger']:
        """尝试从代数整数转换回高斯整数"""
        if len(alg.coefficients) == 2:
            a = int(round(alg.coefficients[0].real))
            b = int(round(alg.coefficients[1].real))
            return cls(a, b)
        return None
    
    def to_point(self, scale: float = 1.0) -> Tuple[float, float]:
        """转换为平面坐标"""
        return (self.a * scale, self.b * scale)


class ClassFieldTower:
    """
    类域塔构造器
    
    这是OpenAI证明的核心工具之一
    通过构建无穷类域塔来获得更丰富的数域结构
    """
    
    def __init__(self, base_field: str = "Q"):
        self.base_field = base_field
        self.field_extensions: List[str] = [base_field]
        self.class_numbers: List[int] = []
        self.galois_groups: List[int] = []  # 伽罗瓦群的阶
    
    def extend_field(self, extension_degree: int, is_cyclic: bool = True) -> 'ClassFieldTower':
        """
        扩展数域
        
        参数:
            extension_degree: 扩张次数
            is_cyclic: 是否为循环扩张
        
        返回:
            新的类域塔
        """
        new_tower = ClassFieldTower(self.base_field)
        new_tower.field_extensions = self.field_extensions.copy()
        new_tower.class_numbers = self.class_numbers.copy()
        new_tower.galois_groups = self.galois_groups.copy()
        
        # 生成新的域扩张
        new_field = f"K{len(new_tower.field_extensions)}"
        new_tower.field_extensions.append(new_field)
        
        # 模拟计算类数（实际计算非常复杂）
        if self.class_numbers:
            # Golod-Shafarevich条件判断是否有无穷类域塔
            h = self.class_numbers[-1]  # 前一层类数
            d = extension_degree  # 扩张次数
            
            # 如果 h > 4 + 2√(d-1)，则可能有无穷类域塔
            threshold = 4 + 2 * np.sqrt(d - 1)
            if h > threshold:
                new_class_num = h * 2  # 类数增长
            else:
                new_class_num = min(h + 1, 100)  # 类数受限
        else:
            new_class_num = 1  # 初始类数为1
        
        new_tower.class_numbers.append(new_class_num)
        new_tower.galois_groups.append(extension_degree)
        
        return new_tower
    
    def has_infinite_tower(self) -> bool:
        """
        判断是否存在无穷类域塔
        
        使用Golod-Shafarevich定理
        """
        if len(self.class_numbers) < 3:
            return False
        
        # 检查类数是否持续增长
        growth_count = 0
        for i in range(1, len(self.class_numbers)):
            if self.class_numbers[i] > self.class_numbers[i-1]:
                growth_count += 1
        
        # 如果连续3次以上增长，认为存在无穷类域塔
        return growth_count >= 3
    
    def generate_points_for_unit_distance(self, scale: float = 1.0) -> List[Tuple[float, float]]:
        """
        使用类域塔生成点集
        
        这些点具有特殊的代数结构，可能产生更多的单位距离对
        """
        points = []
        
        # 使用高斯整数作为基础
        range_val = int(np.sqrt(100))  # 生成100个点
        
        for a in range(-range_val, range_val + 1):
            for b in range(-range_val, range_val + 1):
                gi = GaussianInteger(a, b)
                
                # 计算这个高斯整数的范数
                norm = gi.norm()
                
                # 如果范数接近某个特定值，则包含在点集中
                # 这是通过数域结构来选择点的方法
                if norm is not None and norm.real > 0:
                    # 根据代数数论结构选择点
                    # 实际选择依赖于复杂的数论计算
                    if abs(norm.real % 5) < 2:  # 示例选择条件
                        points.append((a * scale, b * scale))
        
        return points


class GolodShafarevichBuilder:
    """
    Golod-Shafarevich理论实现
    
    这是证明数域存在性的关键理论工具
    """
    
    def __init__(self):
        self.dimensions: List[int] = []
        self.relators: List[int] = []
    
    def add_generator(self, dim: int = 1):
        """添加生成元"""
        self.dimensions.append(dim)
    
    def add_relator(self, count: int):
        """添加关系子"""
        self.relators.append(count)
    
    def golod_shafarevich_bound(self) -> Optional[int]:
        """
        计算Golod-Shafarevich上界
        
        返回:
            如果存在无穷类域塔，返回无穷（用-1表示）
            否则返回类数的上界估计
        """
        if not self.dimensions or not self.relators:
            return None
        
        d = sum(self.dimensions)  # 生成元数量
        r = len(self.relators)  # 关系子数量
        
        # Golod-Shafarevich不等式
        # h >= d - r + 1 + (r-1)√(d-1)
        # 如果h可以任意大，则存在无穷类域塔
        
        # 计算初始类数下界
        h0 = d - r + 1
        
        # 估算可能的最大类数
        if r <= 2:
            return h0 + 10  # 有界
        else:
            # 存在无穷类域塔的可能性
            sqrt_term = np.sqrt(d - 1) if d > 1 else 0
            growth_factor = (r - 1) * sqrt_term
            
            if growth_factor > 5:  # 超过阈值
                return -1  # 无穷类域塔
            else:
                return int(h0 + growth_factor * 10)
    
    def is_infinite_class_field_tower(self) -> bool:
        """判断是否存在无穷类域塔"""
        bound = self.golod_shafarevich_bound()
        return bound == -1


def construct_counterexample(delta: float = 0.014) -> List[Tuple[float, float]]:
    """
    构造反例点集
    
    这模拟了OpenAI证明中使用的构造方法
    证明u(n)可以达到n^(1+δ)
    
    参数:
        delta: 增长指数，OpenAI证明中δ ≈ 0.014
    
    返回:
        点集坐标列表
    """
    print("=" * 70)
    print(f"构造Erdős猜想反例 (δ = {delta})")
    print("=" * 70)
    
    # 1. 构建类域塔
    tower = ClassFieldTower("Q")
    
    # 通过多次扩张构建塔
    for i in range(5):
        # 选择扩张次数（根据Golod-Shafarevich条件）
        degree = 3 + i % 3  # 3, 4, 5交替
        tower = tower.extend_field(degree)
        
        print(f"第{i+1}层扩张: 度数={degree}, "
              f"类数={tower.class_numbers[-1] if tower.class_numbers else 'N/A'}")
    
    print(f"\n无穷类域塔: {'是' if tower.has_infinite_tower() else '否'}")
    
    # 2. 使用类域塔生成点
    points = tower.generate_points_for_unit_distance(scale=1.0)
    
    # 3. 分析点集的单位距离性质
    print(f"\n生成的点数量: {len(points)}")
    
    # 使用简化的单位距离计算
    ud_count = 0
    n = len(points)
    
    for i in range(min(n, 100)):  # 采样计算
        for j in range(i + 1, min(n, 100)):
            x1, y1 = points[i]
            x2, y2 = points[j]
            dist = np.sqrt((x1 - x2)**2 + (y1 - y2)**2)
            if abs(dist - 1.0) < 0.01:
                ud_count += 1
    
    # 外推到整个点集
    estimated_total = ud_count * (n / 100) * (n / 99) / 2 if n > 100 else ud_count
    
    print(f"估计的单位距离对数: ~{int(estimated_total)}")
    print(f"n^(1+δ) = {n}^{1+delta} = {n**(1+delta):.2f}")
    
    return points


# ==================== 测试代码 ====================

if __name__ == "__main__":
    print("代数数域构造器测试")
    print("-" * 50)
    
    # 测试高斯整数
    gi1 = GaussianInteger(1, 0)
    gi2 = GaussianInteger(0, 1)
    gi3 = gi1 + gi2  # 1 + i
    
    print(f"gi3 = {gi3.a} + {gi3.b}i")
    print(f"gi3的范数 = {gi3.norm()}")
    
    # 测试类域塔
    tower = ClassFieldTower()
    for i in range(3):
        tower = tower.extend_field(4)
    
    print(f"\n类域塔层数: {len(tower.field_extensions)}")
    print(f"各层类数: {tower.class_numbers}")
    print(f"是否为无穷类域塔: {tower.has_infinite_tower()}")
    
    # 测试Golod-Shafarevich
    gsh = GolodShafarevichBuilder()
    gsh.add_generator(3)
    gsh.add_relator(2)
    print(f"\nGolod-Shafarevich上界: {gsh.golod_shafarevich_bound()}")
    print(f"存在无穷类域塔: {gsh.is_infinite_class_field_tower()}")
    
    # 构造反例
    print("\n" + "=" * 70)
    print("构造Erdős猜想反例")
    print("=" * 70)
    
    # 生成较大规模的点集
    tower = ClassFieldTower("Q")
    for i in range(8):  # 8层扩张
        tower = tower.extend_field(4)
    
    # 生成1000个点
    all_points = tower.generate_points_for_unit_distance(scale=0.1)
    print(f"生成的点数量: {len(all_points)}")
    
    # 简化验证
    n = min(len(all_points), 500)
    sample_points = all_points[:n]
    
    # 计算实际单位距离数
    count = 0
    for i in range(n):
        for j in range(i + 1, n):
            x1, y1 = sample_points[i]
            x2, y2 = sample_points[j]
            if abs(np.sqrt((x1-x2)**2 + (y1-y2)**2) - 1.0) < 0.01:
                count += 1
    
    print(f"\n样本分析 (n={n}):")
    print(f"  实际单位距离对数: {count}")
    print(f"  n^(1+0.014): {n**(1.014):.2f}")
    print(f"  传统上界 n^(4/3): {n**(4/3):.2f}")

"""
通用推理引擎核心实现
General Purpose Reasoning Engine - Core Implementation

支持长链推理、符号操作和证明搜索
"""

from typing import List, Dict, Any, Optional, Tuple
from dataclasses import dataclass, field
from enum import Enum
import heapq
import random


class ReasoningStepType(Enum):
    """推理步骤类型"""
    SYMBOLIC = "symbolic"
    SEARCH = "search"
    RETRIEVAL = "retrieval"
    VERIFICATION = "verification"
    SYNTHESIS = "synthesis"


@dataclass
class ReasoningStep:
    """推理步骤"""
    step_type: ReasoningStepType
    content: str
    confidence: float
    parent_steps: List[int] = field(default_factory=list)
    cost: float = 1.0
    
    def __lt__(self, other):
        # 优先级队列比较（成本越低越优先）
        return self.cost < other.cost


@dataclass
class ProofState:
    """证明状态"""
    hypothesis: str
    current_steps: List[ReasoningStep]
    unexplored_branches: List[str]
    verified: bool = False
    score: float = 0.0


class GeneralPurposeReasoningEngine:
    """
    通用推理引擎
    
    这是一个概念性实现，展示AI数学推理系统的核心机制
    """
    
    def __init__(self, knowledge_base: Dict[str, Any]):
        self.kb = knowledge_base
        self.reasoning_history: List[List[ReasoningStep]] = []
        self.current_depth = 0
        self.max_depth = 50
        
        # 推理策略权重
        self.strategy_weights = {
            ReasoningStepType.SYMBOLIC: 0.3,
            ReasoningStepType.SEARCH: 0.25,
            ReasoningStepType.RETRIEVAL: 0.2,
            ReasoningStepType.VERIFICATION: 0.15,
            ReasoningStepType.SYNTHESIS: 0.1,
        }
    
    def solve(self, problem: str, context: Optional[Dict] = None) -> Optional[ProofState]:
        """
        求解数学问题
        
        参数:
            problem: 问题描述
            context: 额外上下文信息
        
        返回:
            证明状态（如果找到证明）
        """
        print(f"开始求解问题: {problem[:100]}...")
        
        # 初始化证明状态
        initial_state = ProofState(
            hypothesis=problem,
            current_steps=[],
            unexplored_branches=[problem],
        )
        
        # 优先队列存储候选推理路径
        priority_queue: List[Tuple[float, int, ProofState]] = []
        heapq.heappush(priority_queue, (0, 0, initial_state))
        
        visited_states = set()
        iteration = 0
        
        while priority_queue and iteration < 1000:
            # 取出最优候选
            cost, _, state = heapq.heappop(priority_queue)
            state_hash = self._hash_state(state)
            
            if state_hash in visited_states:
                continue
            
            visited_states.add(state_hash)
            iteration += 1
            
            # 检查是否达到最大深度
            if len(state.current_steps) >= self.max_depth:
                continue
            
            # 生成下一步推理
            next_steps = self._generate_next_steps(state)
            
            for step in next_steps:
                new_state = self._apply_step(state, step)
                
                # 评估状态
                new_state.score = self._evaluate_state(new_state)
                
                # 检查是否找到有效证明
                if self._is_proof_complete(new_state):
                    print(f"找到证明！共 {len(new_state.current_steps)} 步推理")
                    return new_state
                
                # 加入队列
                priority = new_state.score + step.cost
                heapq.heappush(priority_queue, (priority, iteration, new_state))
        
        print("未找到完整证明")
        return None
    
    def _generate_next_steps(self, state: ProofState) -> List[ReasoningStep]:
        """生成下一步推理选项"""
        steps = []
        
        # 符号推理
        if state.unexplored_branches:
            step = ReasoningStep(
                step_type=ReasoningStepType.SYMBOLIC,
                content=f"符号推理: {state.unexplored_branches[0][:50]}...",
                confidence=0.8,
                cost=1.0
            )
            steps.append(step)
        
        # 搜索策略
        step = ReasoningStep(
            step_type=ReasoningStepType.SEARCH,
            content="搜索可能的证明路径",
            confidence=0.7,
            cost=1.5
        )
        steps.append(step)
        
        # 知识检索
        step = ReasoningStep(
            step_type=ReasoningStepType.RETRIEVAL,
            content="从知识库检索相关定理",
            confidence=0.9,
            cost=0.5
        )
        steps.append(step)
        
        # 验证
        step = ReasoningStep(
            step_type=ReasoningStepType.VERIFICATION,
            content="验证当前推理的正确性",
            confidence=0.95,
            cost=0.8
        )
        steps.append(step)
        
        return steps
    
    def _apply_step(self, state: ProofState, step: ReasoningStep) -> ProofState:
        """应用推理步骤"""
        new_steps = state.current_steps + [step]
        new_branches = state.unexplored_branches[1:] if state.unexplored_branches else []
        
        return ProofState(
            hypothesis=state.hypothesis,
            current_steps=new_steps,
            unexplored_branches=new_branches,
            verified=step.step_type == ReasoningStepType.VERIFICATION,
            score=0.0
        )
    
    def _evaluate_state(self, state: ProofState) -> float:
        """评估证明状态"""
        if not state.current_steps:
            return 0.0
        
        # 基于步骤类型加权
        type_scores = {
            ReasoningStepType.SYMBOLIC: 0.3,
            ReasoningStepType.SEARCH: 0.2,
            ReasoningStepType.RETRIEVAL: 0.15,
            ReasoningStepType.VERIFICATION: 0.25,
            ReasoningStepType.SYNTHESIS: 0.1,
        }
        
        score = 0.0
        for step in state.current_steps:
            score += type_scores.get(step.step_type, 0) * step.confidence
        
        # 深度惩罚（避免过长的证明）
        depth_penalty = len(state.current_steps) * 0.01
        score -= depth_penalty
        
        return max(0.0, score)
    
    def _is_proof_complete(self, state: ProofState) -> bool:
        """检查证明是否完成"""
        # 检查是否有足够的验证步骤
        verified_count = sum(
            1 for step in state.current_steps 
            if step.step_type == ReasoningStepType.VERIFICATION
        )
        
        # 检查推理链长度
        min_steps = 10
        
        return (verified_count >= 3 and 
                len(state.current_steps) >= min_steps and
                state.verified)
    
    def _hash_state(self, state: ProofState) -> str:
        """状态哈希"""
        step_ids = [s.step_type.value for s in state.current_steps[-5:]]
        return str(step_ids)
    
    def get_reasoning_chain(self, state: ProofState) -> List[Dict[str, Any]]:
        """获取推理链"""
        chain = []
        for i, step in enumerate(state.current_steps):
            chain.append({
                "step": i + 1,
                "type": step.step_type.value,
                "content": step.content,
                "confidence": step.confidence
            })
        return chain


def main():
    """主函数 - 测试推理引擎"""
    
    # 初始化知识库
    knowledge_base = {
        "erdos_unit_distance": {
            "description": "单位距离问题",
            "known_results": ["u(n) = O(n^(4/3))", "u(n) >= n^(1+o(1))"],
            "related_theorems": ["Gallai's theorem", "Moser's theorem"]
        },
        "algebraic_number_theory": {
            "class_field_tower": {
                "description": "类域塔理论",
                "golod_shafarevich": "判定无穷类域塔的条件"
            },
            "number_fields": {
                "gaussian_integers": "形如a+bi的数域",
                "cyclotomic_fields": "分圆域"
            }
        }
    }
    
    # 创建推理引擎
    engine = GeneralPurposeReasoningEngine(knowledge_base)
    
    # 求解问题
    problem = """
    在平面上放置n个点，最多能有多少对点之间的距离恰好为1？
    证明或推翻: u(n) = n^(1+o(1))
    """
    
    result = engine.solve(problem)
    
    if result:
        print("\n" + "=" * 70)
        print("推理链详情")
        print("=" * 70)
        
        chain = engine.get_reasoning_chain(result)
        for item in chain:
            print(f"步骤 {item['step']}: [{item['type']}]")
            print(f"  内容: {item['content']}")
            print(f"  置信度: {item['confidence']:.2f}")
            print()


if __name__ == "__main__":
    main()

/*
AI数学推理系统 - Go语言实现
Distributed Mathematical Reasoning System in Go

展示如何构建一个支持分布式推理和验证的系统
*/

package main

import (
	"context"
	"encoding/json"
	"fmt"
	"math"
	"math/rand"
	"sort"
	"sync"
	"time"
)

// ==================== 核心数据结构 ====================

// Point represents a point in 2D plane
type Point struct {
	X, Y float64
}

// UnitDistancePair represents a pair of points with unit distance
type UnitDistancePair struct {
	P1, P2 Point
}

// ProofStep represents a step in the reasoning process
type ProofStep struct {
	ID          int             `json:"id"`
	Type        string          `json:"type"` // symbolic, search, retrieval, verification, synthesis
	Content     string          `json:"content"`
	Confidence  float64         `json:"confidence"`
	Timestamp   time.Time       `json:"timestamp"`
	Metadata    map[string]any  `json:"metadata,omitempty"`
}

// ProofState represents the current state of a proof
type ProofState struct {
	mu            sync.RWMutex
	ID            string                 `json:"id"`
	Problem       string                 `json:"problem"`
	Steps         []ProofStep            `json:"steps"`
	CurrentScore  float64                `json:"score"`
	IsComplete    bool                   `json:"is_complete"`
	Verified      bool                   `json:"verified"`
	CreatedAt     time.Time              `json:"created_at"`
	UpdatedAt     time.Time              `json:"updated_at"`
	Context       map[string]any         `json:"context,omitempty"`
}

// ReasoningEngine is the main reasoning system
type ReasoningEngine struct {
	mu           sync.RWMutex
	config      EngineConfig
	knowledgeKB  KnowledgeBase
	history      []*ProofState
}

// EngineConfig contains engine configuration
type EngineConfig struct {
	MaxDepth          int       `json:"max_depth"`
	MaxIterations     int       `json:"max_iterations"`
	Timeout           duration  `json:"timeout"`
	WorkerCount       int       `json:"worker_count"`
	StrategyWeights   map[string]float64 `json:"strategy_weights"`
}

// KnowledgeBase stores mathematical knowledge
type KnowledgeBase struct {
	mu      sync.RWMutex
	data    map[string]KnowledgeEntry
}

// KnowledgeEntry represents a single knowledge entry
type KnowledgeEntry struct {
	ID          string         `json:"id"`
	Type        string         `json:"type"`
	Content     string         `json:"content"`
	Related     []string       `json:"related,omitempty"`
	Confidence  float64       `json:"confidence"`
}

// AlgebraicNumberField represents an algebraic number field extension
type AlgebraicNumberField struct {
	Degree         int                 `json:"degree"`
	Discriminant  int                 `json:"discriminant"`
	ClassNumber   int                 `json:"class_number"`
	GaloisGroup   []int               `json:"galois_group,omitempty"`
}

// ClassFieldTower represents a class field tower
type ClassFieldTower struct {
	BaseField     string                  `json:"base_field"`
	Layers        []AlgebraicNumberField  `json:"layers"`
	HasInfinite   bool                    `json:"has_infinite_tower"`
}

// ==================== 单位距离问题求解 ====================

// UnitDistanceSolver solves the unit distance problem
type UnitDistanceSolver struct {
	engine *ReasoningEngine
}

// NewUnitDistanceSolver creates a new unit distance solver
func NewUnitDistanceSolver(engine *ReasoningEngine) *UnitDistanceSolver {
	return &UnitDistanceSolver{engine: engine}
}

// CountUnitDistances counts the number of unit distance pairs
func (s *UnitDistanceSolver) CountUnitDistances(points []Point) int {
	count := 0
	n := len(points)
	
	for i := 0; i < n; i++ {
		for j := i + 1; j < n; j++ {
			if IsUnitDistance(points[i], points[j]) {
				count++
			}
		}
	}
	
	return count
}

// IsUnitDistance checks if two points have unit distance
func IsUnitDistance(p1, p2 Point) bool {
	dx := p1.X - p2.X
	dy := p1.Y - p2.Y
	dist := math.Sqrt(dx*dx + dy*dy)
	return math.Abs(dist-1.0) < 1e-6
}

// GenerateSquareGrid generates points in a square grid
func GenerateSquareGrid(n int) []Point {
	points := make([]Point, 0, n)
	side := int(math.Ceil(math.Sqrt(float64(n))))
	
	for i := 0; i < side && len(points) < n; i++ {
		for j := 0; j < side && len(points) < n; j++ {
			points = append(points, Point{X: float64(i), Y: float64(j)})
		}
	}
	
	return points
}

// GenerateTriangularGrid generates points in a triangular grid
func GenerateTriangularGrid(n int) []Point {
	points := make([]Point, 0, n)
	row := 0
	
	for len(points) < n {
		for col := 0; col <= row && len(points) < n; col++ {
			x := float64(col) + 0.5*float64(row)
			y := float64(row) * math.Sqrt(3) / 2
			points = append(points, Point{X: x, Y: y})
		}
		row++
	}
	
	return points
}

// ==================== 代数数论工具 ====================

// GaussianInteger represents a Gaussian integer a + bi
type GaussianInteger struct {
	A, B int
}

// Norm returns the norm of a Gaussian integer
func (g GaussianInteger) Norm() int {
	return g.A*g.A + g.B*g.B
}

// ToPoint converts to a 2D point
func (g GaussianInteger) ToPoint(scale float64) Point {
	return Point{X: float64(g.A) * scale, Y: float64(g.B) * scale}
}

// ClassFieldTowerBuilder builds class field towers
type ClassFieldTowerBuilder struct {
	tower *ClassFieldTower
}

// NewClassFieldTowerBuilder creates a new builder
func NewClassFieldTowerBuilder(baseField string) *ClassFieldTowerBuilder {
	return &ClassFieldTowerBuilder{
		tower: &ClassFieldTower{
			BaseField: baseField,
			Layers:    make([]AlgebraicNumberField, 0),
		},
	}
}

// ExtendField extends the tower with a new field
func (b *ClassFieldTowerBuilder) ExtendField(degree int) *ClassFieldTowerBuilder {
	field := AlgebraicNumberField{
		Degree:        degree,
		Discriminant:  rand.Intn(1000) + 100,
		ClassNumber:   1,
		GaloisGroup:   make([]int, degree),
	}
	
	// Simulate class number growth
	if len(b.tower.Layers) > 0 {
		prevClassNum := b.tower.Layers[len(b.tower.Layers)-1].ClassNumber
		threshold := 4 + 2*math.Sqrt(float64(degree-1))
		
		if float64(prevClassNum) > threshold {
			field.ClassNumber = prevClassNum * 2
		} else {
			field.ClassNumber = prevClassNum + 1
		}
	}
	
	// Fill Galois group
	for i := range field.GaloisGroup {
		field.GaloisGroup[i] = (i + 1) % degree
	}
	
	b.tower.Layers = append(b.tower.Layers, field)
	
	// Update infinite tower status
	b.checkInfiniteTower()
	
	return b
}

// checkInfiniteTower checks if the tower is infinite
func (b *ClassFieldTowerBuilder) checkInfiniteTower() {
	if len(b.tower.Layers) < 3 {
		b.tower.HasInfinite = false
		return
	}
	
	growthCount := 0
	for i := 1; i < len(b.tower.Layers); i++ {
		if b.tower.Layers[i].ClassNumber > b.tower.Layers[i-1].ClassNumber {
			growthCount++
		}
	}
	
	b.tower.HasInfinite = growthCount >= 3
}

// Build returns the constructed tower
func (b *ClassFieldTowerBuilder) Build() *ClassFieldTower {
	return b.tower
}

// GeneratePoints generates points using the class field tower
func (t *ClassFieldTower) GeneratePoints(count int, scale float64) []Point {
	points := make([]Point, 0, count)
	rangeVal := int(math.Sqrt(float64(count)))
	
	for a := -rangeVal; a <= rangeVal && len(points) < count; a++ {
		for b := -rangeVal; b <= rangeVal && len(points) < count; b++ {
			g := GaussianInteger{A: a, B: b}
			norm := g.Norm()
			
			// Select points based on algebraic structure
			if norm > 0 && norm%5 < 2 {
				points = append(points, g.ToPoint(scale))
			}
		}
	}
	
	return points
}

// ==================== 推理引擎 ====================

// NewReasoningEngine creates a new reasoning engine
func NewReasoningEngine(config EngineConfig) *ReasoningEngine {
	if config.StrategyWeights == nil {
		config.StrategyWeights = map[string]float64{
			"symbolic":     0.30,
			"search":       0.25,
			"retrieval":    0.20,
			"verification": 0.15,
			"synthesis":    0.10,
		}
	}
	
	engine := &ReasoningEngine{
		config:     config,
		knowledgeKB: KnowledgeBase{data: make(map[string]KnowledgeEntry)},
		history:    make([]*ProofState, 0),
	}
	
	// Initialize knowledge base
	engine.initKnowledgeBase()
	
	return engine
}

// initKnowledgeBase initializes the mathematical knowledge base
func (e *ReasoningEngine) initKnowledgeBase() {
	e.knowledgeKB.mu.Lock()
	defer e.knowledgeKB.mu.Unlock()
	
	entries := []KnowledgeEntry{
		{
			ID:         "erdos_unit_distance",
			Type:       "problem",
			Content:    "单位距离问题：平面上n个点，最多能有多少对点之间的距离恰好为1",
			Related:    []string{"combinatorial_geometry", "upper_bound"},
			Confidence: 1.0,
		},
		{
			ID:         "erdos_conjecture",
			Type:       "conjecture",
			Content:    "Erdős猜想：u(n) = n^(1+o(1))",
			Related:    []string{"erdos_unit_distance", "square_grid"},
			Confidence: 0.8,
		},
		{
			ID:         "upper_bound_theorem",
			Type:       "theorem",
			Content:    "已知上界：u(n) = O(n^(4/3))",
			Related:    []string{"erdos_unit_distance"},
			Confidence: 1.0,
		},
		{
			ID:         "gaussian_integers",
			Type:       "tool",
			Content:    "高斯整数：形如a+bi的复数，用于构造网格点",
			Related:    []string{"number_fields", "erdos_unit_distance"},
			Confidence: 1.0,
		},
		{
			ID:         "class_field_tower",
			Type:       "tool",
			Content:    "类域塔：代数数域的扩张序列，用于构造更优的点集",
			Related:    []string{"number_fields", "golod_shafarevich"},
			Confidence: 1.0,
		},
		{
			ID:         "golod_shafarevich",
			Type:       "theorem",
			Content:    "Golod-Shafarevich定理：判定是否存在无穷类域塔",
			Related:    []string{"class_field_tower", "number_fields"},
			Confidence: 1.0,
		},
	}
	
	for _, entry := range entries {
		e.knowledgeKB.data[entry.ID] = entry
	}
}

// Solve attempts to solve a mathematical problem
func (e *ReasoningEngine) Solve(ctx context.Context, problem string) (*ProofState, error) {
	state := &ProofState{
		ID:        fmt.Sprintf("proof_%d", time.Now().UnixNano()),
		Problem:   problem,
		Steps:     make([]ProofStep, 0),
		CreatedAt: time.Now(),
		UpdatedAt: time.Now(),
		Context:   make(map[string]any),
	}
	
	fmt.Printf("开始求解问题: %s...\n", problem[:min(50, len(problem))])
	
	// Run reasoning loop
	for iteration := 0; iteration < e.config.MaxIterations; iteration++ {
		select {
		case <-ctx.Done():
			return nil, ctx.Err()
		default:
		}
		
		// Generate and apply reasoning steps
		steps := e.generateNextSteps(state)
		
		for _, step := range steps {
			state.mu.Lock()
			state.Steps = append(state.Steps, step)
			state.UpdatedAt = time.Now()
			state.mu.Unlock()
			
			// Evaluate current state
			score := e.evaluateState(state)
			
			state.mu.Lock()
			state.CurrentScore = score
			state.mu.Unlock()
			
			// Check if proof is complete
			if e.isProofComplete(state) {
				state.mu.Lock()
				state.IsComplete = true
				state.Verified = true
				state.mu.Unlock()
				
				fmt.Printf("找到证明！共 %d 步推理，置信度: %.2f\n", 
					len(state.Steps), state.CurrentScore)
				
				e.mu.Lock()
				e.history = append(e.history, state)
				e.mu.Unlock()
				
				return state, nil
			}
		}
		
		// Early stopping if max depth reached
		if len(state.Steps) >= e.config.MaxDepth {
			break
		}
	}
	
	return state, fmt.Errorf("未能找到完整证明")
}

// generateNextSteps generates candidate reasoning steps
func (e *ReasoningEngine) generateNextSteps(state *ProofState) []ProofStep {
	stepTypes := []string{"symbolic", "search", "retrieval", "verification", "synthesis"}
	steps := make([]ProofStep, 0, len(stepTypes))
	
	for i, stepType := range stepTypes {
		step := ProofStep{
			ID:         len(state.Steps) + i + 1,
			Type:       stepType,
			Content:    fmt.Sprintf("%s reasoning at step %d", stepType, len(state.Steeps)+1),
			Confidence: e.config.StrategyWeights[stepType] * (0.8 + 0.2*rand.Float64()),
			Timestamp:  time.Now(),
			Metadata:   map[string]any{"iteration": len(state.Steps)},
		}
		steps = append(steps, step)
	}
	
	return steps
}

// evaluateState evaluates the current proof state
func (e *ReasoningEngine) evaluateState(state *ProofState) float64 {
	if len(state.Steps) == 0 {
		return 0.0
	}
	
	score := 0.0
	for _, step := range state.Steps {
		weight := e.config.StrategyWeights[step.Type]
		if weight == 0 {
			weight = 0.2
		}
		score += weight * step.Confidence
	}
	
	// Depth penalty
	depthPenalty := float64(len(state.Steps)) * 0.01
	score -= depthPenalty
	
	return math.Max(0.0, score)
}

// isProofComplete checks if the proof is complete
func (e *ReasoningEngine) isProofComplete(state *ProofState) bool {
	verifiedCount := 0
	for _, step := range state.Steps {
		if step.Type == "verification" && step.Confidence > 0.9 {
			verifiedCount++
		}
	}
	
	return verifiedCount >= 3 && len(state.Steps) >= 10
}

// ==================== 主函数 ====================

func main() {
	fmt.Println("=" + strings.Repeat("=", 69))
	fmt.Println("AI数学推理系统 - Erdős单位距离问题求解")
	fmt.Println("=" + strings.Repeat("=", 69))
	
	ctx, cancel := context.WithTimeout(context.Background(), 30*time.Second)
	defer cancel()
	
	// Configure engine
	config := EngineConfig{
		MaxDepth:      50,
		MaxIterations: 100,
		Timeout:       30 * time.Second,
		WorkerCount:   4,
		StrategyWeights: map[string]float64{
			"symbolic":     0.30,
			"search":       0.25,
			"retrieval":    0.20,
			"verification": 0.15,
			"synthesis":    0.10,
		},
	}
	
	// Create engine
	engine := NewReasoningEngine(config)
	
	// Create solver
	solver := NewUnitDistanceSolver(engine)
	
	// Test with different grid sizes
	sizes := []int{100, 500, 1000}
	
	fmt.Println("\n传统网格构造测试:")
	fmt.Println(strings.Repeat("-", 50))
	
	for _, n := range sizes {
		// Square grid
		squarePoints := GenerateSquareGrid(n)
		squareCount := solver.CountUnitDistances(squarePoints)
		
		// Triangular grid
		triPoints := GenerateTriangularGrid(n)
		triCount := solver.CountUnitDistances(triPoints)
		
		// Theoretical bounds
		upperBound := math.Pow(float64(n), 4.0/3.0)
		aiLowerBound := math.Pow(float64(n), 1.014)
		
		fmt.Printf("\nn = %d\n", n)
		fmt.Printf("  正方形网格单位距离对数: %d\n", squareCount)
		fmt.Printf("  三角网格单位距离对数:   %d\n", triCount)
		fmt.Printf("  传统上界 O(n^(4/3)):    %.2f\n", upperBound)
		fmt.Printf("  AI新下界 n^(1+0.014):   %.2f\n", aiLowerBound)
	}
	
	// Test class field tower construction
	fmt.Println("\n\n类域塔构造测试:")
	fmt.Println(strings.Repeat("-", 50))
	
	builder := NewClassFieldTowerBuilder("Q")
	for i := 0; i < 5; i++ {
		builder.ExtendField(3 + i%3)
		tower := builder.Build()
		
		fmt.Printf("\n第 %d 层扩张:\n", i+1)
		fmt.Printf("  类数: %d\n", tower.Layers[i].ClassNumber)
		fmt.Printf("  扩张次数: %d\n", tower.Layers[i].Degree)
	}
	
	fmt.Printf("\n无穷类域塔: %v\n", builder.Build().HasInfinite)
	
	// Generate points using class field tower
	tower := builder.Build()
	points := tower.GeneratePoints(500, 0.1)
	fmt.Printf("类域塔生成点数: %d\n", len(points))
	
	// Run reasoning engine
	fmt.Println("\n\n推理引擎测试:")
	fmt.Println(strings.Repeat("-", 50))
	
	problem := "证明或推翻Erdős单位距离猜想：u(n) = n^(1+o(1))"
	result, err := engine.Solve(ctx, problem)
	
	if err != nil {
		fmt.Printf("推理失败: %v\n", err)
	} else {
		fmt.Printf("\n推理完成:\n")
		fmt.Printf("  证明ID: %s\n", result.ID)
		fmt.Printf("  推理步数: %d\n", len(result.Steps))
		fmt.Printf("  最终评分: %.4f\n", result.CurrentScore)
		fmt.Printf("  证明完成: %v\n", result.IsComplete)
	}
}

// Helper function for Go 1.21+
func min(a, b int) int {
	if a < b {
		return a
	}
	return b
}