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T16-5 φ-时空拓扑形式化规范

1. 基础数学对象​

1.1 φ-拓扑空间​

class PhiTopologicalSpace:
    def __init__(self, manifold: 'PhiManifold'):
        self.manifold = manifold
        self.phi = (1 + np.sqrt(5)) / 2
        self.dimension = manifold.dimension
        
    def euler_characteristic(self) -> 'PhiNumber':
        """计算φ-欧拉特征数 χ^φ"""
        
    def genus(self) -> 'PhiNumber':
        """计算φ-亏格 g^φ = (2-χ^φ)/2"""
        
    def verify_no_11_constraint(self) -> bool:
        """验证拓扑不变量满足no-11约束"""
        
    def is_allowed_topology(self) -> bool:
        """检查是否为允许的拓扑类型"""

1.2 φ-拓扑不变量​

class PhiTopologicalInvariants:
    def __init__(self, space: 'PhiTopologicalSpace'):
        self.space = space
        self.phi = (1 + np.sqrt(5)) / 2
        
    def betti_numbers(self, k: int) -> 'PhiNumber':
        """计算第k个φ-Betti数 b_k^φ"""
        
    def fundamental_group(self) -> 'PhiFundamentalGroup':
        """计算φ-基本群 π_1^φ"""
        
    def homotopy_groups(self, n: int) -> 'PhiHomotopyGroup':
        """计算第n个φ-同伦群 π_n^φ"""
        
    def characteristic_classes(self) -> Dict[str, 'PhiCohomologyClass']:
        """计算示性类（陈类、Pontryagin类等）"""

1.3 φ-基本群​

class PhiFundamentalGroup:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.generators = []  # φ-生成元
        self.relations = []   # φ-关系
        
    def add_generator(self, gen: 'PhiGroupElement') -> bool:
        """添加生成元（检查no-11约束）"""
        
    def add_relation(self, rel: 'PhiGroupRelation') -> bool:
        """添加关系（检查no-11约束）"""
        
    def presentation(self) -> str:
        """返回群的表示 <a1,...,an | R>"""
        
    def is_abelian(self) -> bool:
        """检查是否为交换群"""
        
    def order(self) -> 'PhiNumber':
        """计算群的阶（可能无限）"""

2. 拓扑分类系统​

2.1 φ-拓扑分类器​

class PhiTopologyClassifier:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.allowed_topologies = {}
        self._compute_allowed_types()
        
    def _compute_allowed_types(self):
        """计算所有满足no-11约束的拓扑类型"""
        
    def classify(self, space: 'PhiTopologicalSpace') -> str:
        """分类给定空间的拓扑类型"""
        
    def is_homeomorphic(self, space1: 'PhiTopologicalSpace', 
                       space2: 'PhiTopologicalSpace') -> bool:
        """判断两个空间是否同胚"""
        
    def compute_moduli_space(self, topology_type: str) -> 'PhiModuliSpace':
        """计算给定拓扑类型的模空间"""

2.2 φ-同调群​

class PhiHomologyGroup:
    def __init__(self, space: 'PhiTopologicalSpace', degree: int):
        self.space = space
        self.degree = degree
        self.phi = (1 + np.sqrt(5)) / 2
        
    def rank(self) -> 'PhiNumber':
        """计算同调群的秩（Betti数）"""
        
    def torsion(self) -> List['PhiNumber']:
        """计算挠部分"""
        
    def generators(self) -> List['PhiChain']:
        """返回生成元"""
        
    def poincare_dual(self) -> 'PhiCohomologyGroup':
        """Poincaré对偶"""

2.3 φ-上同调群​

class PhiCohomologyGroup:
    def __init__(self, space: 'PhiTopologicalSpace', degree: int):
        self.space = space
        self.degree = degree
        self.phi = (1 + np.sqrt(5)) / 2
        
    def cup_product(self, alpha: 'PhiCocycle', 
                    beta: 'PhiCocycle') -> 'PhiCocycle':
        """杯积运算"""
        
    def cohomology_ring(self) -> 'PhiRing':
        """上同调环结构"""
        
    def characteristic_class(self, bundle: 'PhiVectorBundle') -> 'PhiCocycle':
        """计算向量丛的示性类"""

3. 拓扑相变机制​

3.1 φ-拓扑相变​

class PhiTopologicalTransition:
    def __init__(self, initial: 'PhiTopologicalSpace', 
                 final: 'PhiTopologicalSpace'):
        self.initial = initial
        self.final = final
        self.phi = (1 + np.sqrt(5)) / 2
        
    def transition_allowed(self) -> bool:
        """检查相变是否满足no-11约束"""
        
    def euler_change(self) -> 'PhiNumber':
        """计算欧拉特征数变化 Δχ^φ"""
        
    def recursive_depth_jump(self) -> 'PhiNumber':
        """计算递归深度跃迁"""
        
    def critical_point(self) -> 'PhiParameter':
        """找到相变临界点"""
        
    def order_parameter(self) -> 'PhiNumber':
        """拓扑序参量"""

3.2 φ-拓扑缺陷​

class PhiTopologicalDefect:
    def __init__(self, defect_type: str, space: 'PhiTopologicalSpace'):
        self.type = defect_type  # 'string', 'wall', 'texture', etc.
        self.space = space
        self.phi = (1 + np.sqrt(5)) / 2
        
    def homotopy_class(self) -> 'PhiHomotopyClass':
        """缺陷的同伦分类"""
        
    def energy_density(self) -> 'PhiNumber':
        """缺陷能量密度"""
        
    def stability(self) -> bool:
        """拓扑稳定性"""
        
    def interaction(self, other: 'PhiTopologicalDefect') -> 'PhiProcess':
        """缺陷相互作用"""

4. 具体拓扑结构​

4.1 φ-球面​

class PhiSphere:
    def __init__(self, dimension: int):
        self.n = dimension
        self.phi = (1 + np.sqrt(5)) / 2
        self._verify_dimension_allowed()
        
    def _verify_dimension_allowed(self):
        """验证维度满足no-11约束"""
        
    def euler_characteristic(self) -> 'PhiNumber':
        """S^n的欧拉特征数"""
        
    def homotopy_groups(self) -> Dict[int, 'PhiHomotopyGroup']:
        """计算所有同伦群"""
        
    def hopf_fibration(self) -> 'PhiFibration':
        """Hopf纤维化（如果存在）"""

4.2 φ-环面​

class PhiTorus:
    def __init__(self, dimension: int):
        self.n = dimension
        self.phi = (1 + np.sqrt(5)) / 2
        
    def modular_group(self) -> 'PhiModularGroup':
        """模群 SL(n,Z_φ)"""
        
    def flat_metrics(self) -> 'PhiModuliSpace':
        """平坦度量的模空间"""
        
    def theta_functions(self) -> List['PhiThetaFunction']:
        """θ函数"""

4.3 φ-亏格曲面​

class PhiRiemannSurface:
    def __init__(self, genus: 'PhiNumber'):
        self.g = genus
        self.phi = (1 + np.sqrt(5)) / 2
        self._verify_genus_allowed()
        
    def _verify_genus_allowed(self):
        """验证亏格满足no-11约束"""
        
    def teichmuller_space(self) -> 'PhiTeichmullerSpace':
        """Teichmüller空间"""
        
    def mapping_class_group(self) -> 'PhiMappingClassGroup':
        """映射类群"""
        
    def period_matrix(self) -> 'PhiMatrix':
        """周期矩阵"""

5. 拓扑场论​

5.1 φ-TQFT​

class PhiTopologicalQFT:
    def __init__(self, dimension: int):
        self.dim = dimension
        self.phi = (1 + np.sqrt(5)) / 2
        
    def partition_function(self, manifold: 'PhiManifold') -> 'PhiNumber':
        """配分函数 Z(M)"""
        
    def correlation_functions(self, operators: List['PhiOperator']) -> 'PhiNumber':
        """关联函数"""
        
    def state_space(self, boundary: 'PhiManifold') -> 'PhiHilbertSpace':
        """边界的态空间"""
        
    def gluing_axiom(self, M1: 'PhiManifold', 
                     M2: 'PhiManifold') -> bool:
        """验证粘合公理"""

5.2 φ-Chern-Simons理论​

class PhiChernSimons:
    def __init__(self, gauge_group: 'PhiGaugeGroup', level: 'PhiNumber'):
        self.G = gauge_group
        self.k = level  # 必须满足no-11约束
        self.phi = (1 + np.sqrt(5)) / 2
        
    def action(self, connection: 'PhiConnection') -> 'PhiNumber':
        """Chern-Simons作用量"""
        
    def wilson_loop(self, knot: 'PhiKnot') -> 'PhiNumber':
        """Wilson圈期望值"""
        
    def knot_invariant(self, knot: 'PhiKnot') -> 'PhiPolynomial':
        """结不变量（Jones多项式等）"""

6. 物理应用​

6.1 φ-拓扑物态​

class PhiTopologicalPhase:
    def __init__(self, hamiltonian: 'PhiHamiltonian'):
        self.H = hamiltonian
        self.phi = (1 + np.sqrt(5)) / 2
        
    def topological_invariant(self) -> 'PhiNumber':
        """拓扑不变量（陈数等）"""
        
    def edge_states(self) -> List['PhiEdgeState']:
        """边缘态"""
        
    def bulk_boundary_correspondence(self) -> bool:
        """体边对应"""
        
    def phase_diagram(self) -> 'PhiPhaseDiagram':
        """相图"""

6.2 φ-量子霍尔效应​

class PhiQuantumHallEffect:
    def __init__(self, magnetic_field: 'PhiNumber'):
        self.B = magnetic_field
        self.phi = (1 + np.sqrt(5)) / 2
        
    def hall_conductance(self) -> 'PhiNumber':
        """霍尔电导 σ_xy^φ"""
        
    def filling_factors(self) -> List['PhiNumber']:
        """允许的填充因子"""
        
    def composite_fermions(self) -> 'PhiQuasiparticle':
        """复合费米子"""

7. 验证函数​

7.1 理论一致性检查​

def verify_no_11_topology(space: 'PhiTopologicalSpace') -> bool:
    """验证拓扑空间满足no-11约束"""
    
def verify_poincare_duality(manifold: 'PhiManifold') -> bool:
    """验证Poincaré对偶"""
    
def verify_index_theorem(operator: 'PhiDifferentialOperator', 
                        manifold: 'PhiManifold') -> bool:
    """验证指标定理"""

7.2 数值计算检查​

def check_euler_formula(complex: 'PhiSimplicialComplex') -> bool:
    """检查欧拉公式 V-E+F=χ"""
    
def check_gauss_bonnet(manifold: 'PhiRiemannianManifold') -> float:
    """检查Gauss-Bonnet定理"""

8. 关键常数​

# 基础常数
PHI = (1 + np.sqrt(5)) / 2

# 特殊拓扑不变量
SPHERE_EULER = {
    0: PhiNumber(2),   # S^0
    1: PhiNumber(0),   # S^1
    2: PhiNumber(2),   # S^2
    3: PhiNumber(0),   # S^3
}

# 禁止的拓扑特征
FORBIDDEN_EULER = [3, 6, 7, 11, 12, 13, 14, 15, ...]  # 包含连续11的数

# 量子霍尔平台
QH_PLATEAUS = [
    PhiNumber(1),           # ν = 1
    PhiNumber(1/2),         # ν = 1/2
    PhiNumber(1/3),         # ν = 1/3
    PhiNumber(2/5),         # ν = 2/5 (Fibonacci)
    PhiNumber(3/8),         # ν = 3/8 (Fibonacci)
]

9. 错误处理​

class PhiTopologyError(Exception):
    """拓扑计算错误基类"""
    
class ForbiddenTopologyError(PhiTopologyError):
    """禁止的拓扑类型"""
    
class No11ViolationError(PhiTopologyError):
    """违反no-11约束"""
    
class TopologicalTransitionError(PhiTopologyError):
    """非法的拓扑相变"""