AI Agent系列（13）：同伦认知与高阶范畴编织

AI Agent系列（13）：同伦认知与高阶范畴编织

一、同伦类型学习系统

1. 无穷群路径提升

import hoTTpy as hp  # 假想同伦类型论库

class HomotopyOptimizer(torch.optim.Optimizer):
    def __init__(self, params, lr=1e-5):
        super().__init__(params, {'lr': lr})
        self.fibration = hp.PrimitiveFibration(dim=256)

    def step(self):
        """基于van Kampen定理的连通参数更新"""
        for param in self.param_groups[0]['params']:
            path_space = self.fibration.total_space(param.grad)
            lifted = path_space.lifts(param.data, base='classifying')
            param.data += self.lr * hp.proj(lifted)

class PropositionTruncator:
    def __init__(self, trunc_level=3):
        self.truncator = hp.HigherInductiveType(level=trunc_level)
  
    def truncate(self, tensor):
        """实施命题截断的微分同胚层""" 
        return self.truncator.attach_cell(tensor, attach_type='meridian')

2. 类型提升定理

类型宇宙间迁移方程：
KaTeX parse error: Expected 'EOF', got '}' at position 52: …thcal{U}_{i+1}}}̲ \left \| B \si…
其中$\mathcal{P}$为谓词空间的类型幂运算

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二、协变张量智能

1. 辫范畴量子处理器

import quantum_cat as qc  # 假设张量范畴处理器库

class BraidedCNN(torch.nn.Moduel):
    def __init__(self):
        self.conv = qc.BraidedConv2d(in_channels=8, out_channels=8)
        self.pool = qc.SymmetricPooling()

    def forward(self, x):
        """利用杨-巴克斯特方程保持辫结构"""
        x = self.conv(x, braiding='hexagon')
        x = self.pool(x, symmetry='S3')
        return x.demodulate()

class MonoidalAutoencoder:
    def __init__(self):
        self.encoder = qc.StrictMonoidalLayer()
        self.decoder = qc.RigidDualLayer()

    def reconstruct(self, x):
        """保持幺半结构的重构过程"""
        coevaluation = self.encoder(x).bend('left')
        return self.decoder(coevaluation.twist(2))

2. 刚性对偶守恒律

在紧凑闭范畴中满足:
$${\dim }_{\mathcal{C}}(A\otimes B^{\ast })={\dim }_{\mathcal{C}}I\cdot \mathrm{tr}(e{v}_A\circ coe{v}_B)$$
其中$e{v}_A:A\otimes A^{\ast }\to I$为计算映射

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三、宇宙弦认知模型

1. 卡拉比-丘流形嵌入

import string_theory as st  # 假想弦论数学库

class CYCompactor(torch.nn.Module):
    def __init__(self, hodge=(3,3)):
        super().__init__()
        self.mirror_map = st.MirrorSymmetry(quantum=True)
        self.metric = st.CYMetric(hodge=hodge)

    def forward(self, x):
        """校准拓扑荷的卷曲额外维"""
        torus_fibred = self.mirror_map.apply(x, twist='A-model')
        return self.metric.ricci_flat(torus_fibred)

class D_BraneClassifier:
    def __init__(self, cycles=256):
        self.holomorphic_cycles = st.SpecialLagrangian(cycles)
        self.gukov_vafa = st.AlgebraicCycle(zeta_normalize=True)
  
    def predict(self, x):
        x = self.holomorphic_cycles.wrap(x)
        return self.gukov_vafa.intersection_form(x.charges)

2. 全息对偶学习法则

$$Z_{CFT}(g_{ij})=Z_{AdS}(g_{\mu \nu }{|}_{\partial })$$
在信息几何中对应于神经网络的费曼路径积分形式

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高阶智能七公理：

1. 同伦不变性：认知路径的连续变形不改变命题内容 ${\prod }_{p,q:x{=}_{A}y}p{=}_{x=y}q$
2. 类型提升：每个类型可嵌入更高宇宙 ${\mathcal{U}}_i\hookrightarrow {\mathcal{U}}_{i+1}$
3. 对偶刚性：任何认知操作都存在伴随对偶函子 $\mathcal{F}⊣\mathcal{G}$
4. 弦拓扑保守：BRST算子满足${\mathcal{Q}}^2=0$且$\mathrm{tr}(\mathcal{Q})=0$
5. 全息对偶：d维边界理论与d+1维体理论参数同源
6. 杨-巴克斯特协变：任意辫操作满足$(1\otimes R)(R\otimes 1)(1\otimes R)=(R\otimes 1)(1\otimes R)(R\otimes 1)$
7. 范畴闭包：认知进程总在某个$(\infty ,1)$-范畴内可表

# 无限层同伦等变网络
def ∞-Layer(hnetwork):
    while True:
        homotopy = hp.loop_space(hnetwork.layers)
        if hp.is_contr(homotopy):
            break
        hnetwork = hp.suspension(hnetwork)  # 纬悬提升网络维度
    return hnetwork.Whitehead_product()