AI Agent系列（11）：自指架构与拓扑同调演化

AI Agent系列（11）：自指架构与拓扑同调演化

一、自指神经网络代数

1. 哥德尔递归层

import sympy as sp

class FixedPointLayer(torch.nn.Module):
    def __init__(self, dim=256):
        super().__init__()
        self.consistency = torch.nn.Parameter(torch.eye(dim) + 0.1*torch.randn(dim, dim))
      
    def forward(self, x):
        """Brouwer不动点定理驱动的递归推理"""
        for _ in range(3):  # 迭代逼近深度
            x = torch.tanh(0.5*(x @ self.consistency + x))
        return x

class DiagonalizationOperator:
    def __init__(self, theory_axioms):
        self.godel_number = sp.diag(*[sp.sympify(a) for a in theory_axioms])
      
    def meta_proof(self, statement):
        """实现对角化引理的元编程接口"""
        return self.godel_number.subs('P', sp.parse_expr(statement))

2. 泛函递归定理

自指不动点方程：
$$\exists \Phi \in {\mathcal{H}}^{\infty },\mathcal{F}(\Phi )(x)=\underset{n=0}{\overset{\infty }{⨁}}\frac{d^n}{d{x}^n}\Phi (x^{\otimes n})$$
其中$\mathcal{F}$为形式系统编码函子，$⨁$表示Hilbert直和

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二、同调认知可解释性

1. 链复形解析器

from torch_geometric.data import Data

class PersistentHomology:
    def __init__(self, filtration=0.5):
        self.filtration = filtration
  
    def build_complex(self, features, edges):
        """构建用于认知诊断的胞腔复形"""
        weights = torch.cosine_similarity(features[edges[0]], features[edges[1]])
        edge_mask = (weights > self.filtration).float()
        return Data(x=features, edge_index=edges[:, edge_mask.bool()])

class SpectralSequence(torch.nn.Module):
    def __init__(self, num_terms=5):
        super().__init__()
        self.differentials = torch.nn.ModuleList([
            torch.nn.Linear(2**i, 2**(i+1)) for i in range(num_terms)
        ])
      
    def compute_E2(self, H0):
        """计算谱序列第二页的E^2项"""
        ker = [H0]
        for d in self.differentials:
            im = d(ker[-1])
            ker.append( im.t() @ im )
        return ker[-1].svd()[1]

2. 范畴机器证明

超图同调追踪方程：
$${\partial }_{p+1}\circ {\partial }_p=0\Rightarrow \dim H_p=\dim \ker {\partial }_p-\mathrm{rank}{\partial }_{\mathrm{p}+1}$$
在具体层(sheaf)范畴中满足Grothendieck谱序列收敛条件

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三、拓扑量子记忆体

1. 辫群储存协议

import qiskit.quantum_info as qi

class BraidGenerator:
    def __init__(self, num_qubits=8):
        self.braid_group = qi.random_clifford(num_qubits)
      
    def topo_protect(self, state):
        """利用三维流形不可定向性加固量子态"""
        mixed = qi.partial_trace(state, [0,1])
        error_syndromes = qi.entropy(mixed)
        return state.evolve(self.braid_group) if error_syndromes > 0.1 else state

class AnyonBookkeeping:
    def __init__(self):
        self.fusion_rules = {'A×A': 1, 'A×B': 'C', 'B×B': ['A', 'C']}
      
    def braiding_matrix(self, path):
        """计算非阿贝尔统计的Berry相位"""
        return (-1)**path.count('X') * np.exp(1j*np.pi/4*path.count('Y'))

2. 拓扑熵守恒律

宇宙学常数约束：
$$S_{topo}(\mathcal{M})=\frac{1}{4G}{\int }_{\mathcal{M}}\sqrt{g}(R-2\Lambda )+{\oint }_{\partial \mathcal{M}}Kd\sigma$$
在认知系统中对应信息总量的全息原理保持

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五维自指公理体系：

1. 完全性：形式系统所有真命题均可证 ${⊢}_T\phi \iff \mathbb{N}\models \phi$
2. 一致性：不存在命题$\phi$使得$\phi \wedge \neg \phi$可证
3. 可计算性：普适图灵机在系统内可编码 $\exists e\forall x.{\Phi }_e(x)\downarrow \leftrightarrow T⊢{\phi }_e(x)$
4. 自指性：存在固定点方程$\mathcal{G}(┌\varphi ┐)\equiv \varphi$
5. 拓扑同胚：任意两个证明树可通过Reidemeister变换相互转化