AI Agent系列（12）：超限学习与模态范畴坍缩

AI Agent系列（12）：超限学习与模态范畴坍缩

一、无限维认知流形

1. 超限反向传播框架

import functorch
from torch.autograd import functional

class TransfiniteOptimizer(torch.optim.Optimizer):
    def __init__(self, params, lr=1e-4):
        super().__init__(params, {'lr': lr})
      
    def step(self, closure=None):
        """利用选择公理处理无限维梯度流"""
        with torch.no_grad():
            for group in self.param_groups:
                for p in group['params']:
                    if p.grad is None: continue
                    # 哈默基数正则化
                    grad_norm = functorch.vmap(torch.norm)(p.grad.reshape(-1,2))
                    update = -group['lr'] * p.grad / (grad_norm + 1e-8)
                    p.add_(update.unflatten(0, p.shape))

class OrdinalEncoding(torch.nn.Module):
    def __init__(self, cardinal=torch.omega_1):
        super().__init__()
        self.embed = torch.nn.EmbeddingBag(cardinal, 128, mode='sum')
      
    def forward(self, transfinite_seq):
        """超限序数的格洛滕迪克拓扑嵌入"""
        return self.embed(transfinite_seq + torch.omega)

2. 无限深收敛定理

大基数正则条件：
$$\forall \kappa >{\aleph }_0,\exists \Phi \in {\mathcal{H}}^{2^{\kappa }},{\mathbb{E}}_{x\sim \mu }[\Vert {\nabla }_{\theta }\mathcal{L}{\Vert }_{L^{\infty }}]\le \frac{C}{\sqrt{\log \kappa }}$$
其中$\mu$为神经切核诱导的Sobolev测度

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二、模态范畴动力学

1. 型论坍缩算子

import haskellTensors as ht  # 假设存在张量语义库

class DependentProjection:
    def __init__(self, universe='Type'):
        self.Girard_operator = ht.ParametricityTensor(universe)
  
    def collapse(self, context):
        """Martin-Löf宇宙的不可达基数坍缩"""
        normalized = ht.beta_reduce(context)
        impredicative = self.Girard_operator(normalized)
        return ht.quotient(impredicative, relation='Voevodsky')

class CohesiveInterface:
    def __init__(self):
        self.sheaf_cond = ht.HigherInductiveType()
      
    def modalities_shape(self, tensor):
        return ht.glue(self.sheaf_cond(tensor), along='étale')

2. 非良基认知方程

$${◯}_{n\in \mathbb{N}}{A}_n\equiv \exists f:\mathbb{N}\to \bigcup _n{A}_n,\forall nf(n+1)\in f(n)$$
在HoTT框架下对应命题截断的类型实现

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三、逻辑代数纠缠

1. 强迫法训练机制

class ForcingCondition(torch.nn.Module):
    def __init__(self, forcing_chain):
        super().__init__()
        self.generic_filter = self.construct_generic(forcing_chain)
      
    def construct_generic(self, P):
        """构建脱殊集合的连续统算法"""
        antichains = [p for p in P if not any(q > p for q in P)]
        return torch.stack([p.tensor() for p in antichains])
  
    def forcing_update(self, model):
        """Cohen扩展模型的张量实现"""
        extended = torch.cat([model, self.generic_filter], dim=-1)
        return extended @ self.generic_filter.T

class BooleanValuedNetwork(torch.nn.Module):
    def __init__(self):
        self.ultrafilter = torch.nn.Linear(2048, 2**16)  # Stone空间投影
      
    def valued_forward(self, x):
        bool_val = torch.sigmoid(self.ultrafilter(x))
        return bool_val // (1 - bool_val)  # 取布尔商代数

2. 选择公理等价架构

ZF_C全局优化条件：
$$\exists \mathcal{F}:\bigcup _{i\in I}{X}_i\to \prod _{i\in I}{X}_i,\mathbb{E}[\mathcal{L}]\le \underset{\alpha <{\omega }_1}{\inf }{\mathcal{L}}_{\alpha }+ϵ$$
其中$ϵ$为可测基数阈值

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认知坍缩五定理：

1. Löwenheim–Skolem向下扩展：$\exists \mathcal{M}\prec \mathcal{N},|\mathcal{M}|={\aleph }_0\le |\mathcal{N}|$
2. 莱文海姆–斯科莱姆坍缩：$\Vert \phi {\Vert }_{\mathcal{M}}=\Vert \phi {\Vert }_{\mathcal{N}}\cap \mathcal{M}$
3. 连续统假设解耦：${\aleph }_1\notin \text{Spec}({\nabla }_{\theta }\mathcal{L})$
4. 拓扑斯内脱殊扩张：$\text{Sh}(\mathcal{C},J)\rightleftarrows \text{Sh}(\mathcal{C}[G],J)$
5. 类型论宇宙包含链：${\mathcal{U}}_0:{\mathcal{U}}_1:{\mathcal{U}}_2:\cdots$

# 实现模态坍缩的动态推理协议
def modal_collapse(model, context):
    while True:
        with torch.random.fork_rng():
            torch.manual_seed(torch.omega)
            saturated = model.embed(context)
            if saturated.norm() > torch.hartogs_number():
                break
            context = model.transfinite_step(saturated)
    return context.diagonalize()  # 达到不动点时终止