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Neural Network Geometry

Connected and stacked piecewise-smooth manifolds jointly form the geometric structure of the representation space. Node covers act as the local units of these piecewise-smooth manifolds, and their orientations change at every iteration. These piecewise-smooth manifolds are differentiable and integrable.

Neural Network Algebra

The coordinate system evolves with each iteration; counting serves as the most primitive algebraic unit, expressed through the iterated-integral structure of forward propagation; at the same time, activations themselves carry no inherent attributes.

Neural Network Equations

The fixed-point residual is the most fundamental equation of a neural network; the residual itself serves as a boundary condition, and its Lagrangian form further yields the iterative method for computing the fixed point, thereby giving the neural network dynamic, iterative computability.

Neural Network Stochasticity

Stochasticity is expressed through inequalities and group statistics based on summation; this statistical structure enables neural networks to learn the inherently stochastic real world and naturally form stochastic fixed points.

Neural Network Fixed Points

Iterations traverse billions of piecewise-smooth manifolds, giving rise to innumerable fixed points and convergence paths within foundation models. Because the training data themselves contain high-order nonlinearity, curvature (second derivatives), and moderate perturbations, the model can distinguish the correct convergence direction toward a fixed point.

Neural Network Boundary Conditions

Boundary conditions are the sole source of iterative direction and determine the convergence path during training. When a foundation model lacks static fixed points, symmetric, weak, and discrete boundary conditions become necessary to guide the convergence of a high-order nonlinear system.

神经网络几何

连接的、堆叠的分片光滑流形共同构成表示空间的几何结构；节点覆盖作为这些分片光滑流形的局部单元，其取向在每一次迭代中都会发生变化。分片平滑的流形是可微且可积。

神经网络代数

坐标系统随迭代不断变化；计数是最原始的代数单元，通过前向传播中的迭代积分结构体现；同时，激活值本身不携带任何固有属性。

神经网络方程

不动点残差是神经网络最原始的方程形式；残差本身是边界条件，其拉格朗日形式进一步给出了不动点的迭代方法，从而赋予神经网络:动态迭代可计算性。

神经网络随机性

通过不等式和基于求和的群统计方式表现随机性；这种统计结构使神经网络能够学习具有随机本质的真实世界，并自然形成随机不动点。

神经网络不动点

迭代跨越上亿分片光滑流形，从而形成基础模型不可胜数的不动点及收敛路径；由于训练数据本身具有高阶非线性，曲率(二阶导数)与适度扰动，才能分辨不动点收敛方向。 

神经网络边界条件

边界条件是迭代方向的唯一来源，决定训练收敛路径；在基础模型不存在静态不动点的情况下，对称、软弱和离散边界是指导高阶非线性系统收敛所必需的。