The learning rule

In the book: Chapter 1, §"Implementing a Learning Algorithm" and §"Why the Perceptron Works". Builds on: The formal model.

Planned. Formalizes the mistake-driven update $\mathbf{w}\leftarrow \mathbf{w}+y\mathbf{x},b\leftarrow b+y$: proves the score-improvement identity the book computes (the functional margin $y({\mathbf{w}}^{\top }\mathbf{x}+b)$ increases by exactly ${\mathbf{x}}^{\top }\mathbf{x}+1$ on the triggering example), relates the book's code form $e=y-\hat{y}$ to the analyzed form, and shows why constant scaling of updates cannot change the boundary found.