---
# Frontmatter for BOOK_PROFILE=academic. See src/content.config.ts for
# the full schema and validation rules.
week: 1
part: foundations
title: "Chapter title — your subject here"
status: scaffolded
# Optional fields:
# description: "Short SEO/meta description, ~140 chars."
# code_path: experiments/jax/week01/some_module.py
# tests_path: experiments/jax/week01/test_some_module.py
# notebook_path: notebooks/week01/companion.ipynb
# roadmap_lines: [10, 42]
---
import NoteBox from '../../components/callouts/NoteBox.astro';
import ExampleBox from '../../components/callouts/ExampleBox.astro';
import InsightBox from '../../components/callouts/InsightBox.astro';
import OpenQuestion from '../../components/callouts/OpenQuestion.astro';
import Theorem from '../../components/Theorem.astro';
import Cite from '../../components/Cite.astro';
import Figure from '../../components/Figure.astro';
import Sidenote from '../../components/Sidenote.astro';

# {frontmatter.title}

## Overview

A half-page motivation: what this chapter covers, why it matters,
and how it fits into the curriculum. Reader should be able to decide
after the Overview whether to read further.<Sidenote>Use Sidenote
for short asides that belong in the right margin on desktop and
reflow inline on mobile.</Sidenote>

<InsightBox>
The single most important takeaway of this chapter, stated as a
short principle. Comes before the Theory so a reader who only
skims gets the heart of the matter.
</InsightBox>

## Theory

The core mathematical content. Definitions, lemmas, propositions,
theorems. Cite primary sources using <Cite key="example-key2024" />.

<Theorem type="definition" id="ch1:def:main">
A *something* is a mathematical object such that … Provide the
precise definition with the key properties explicit.
</Theorem>

<Theorem type="theorem" id="ch1:thm:main" label="Main result">
Under the conditions of Definition <a href="#ch1:def:main">1.1</a>,
the property $P$ holds with $\|x\| \le c \cdot \|y\|$ for some
constant $c > 0$.
</Theorem>

<Theorem type="proof">
Direct calculation. Apply the chain rule, regroup terms, and use
the boundedness assumption from the hypothesis.
</Theorem>

## Examples

Concrete instances that exercise the theory. At least one worked
example per theorem.

<ExampleBox title="Worked example 1">
Take $A = \mathrm{diag}(\lambda_1, \ldots, \lambda_n)$ with all
$|\lambda_i| < 1$. Then the iteration $x_{k+1} = A x_k$ converges
to 0, with rate $\max_i |\lambda_i|$.
</ExampleBox>

<Figure
  src="/figures/example/placeholder.svg"
  caption="Caption explaining what the reader should see in the figure."
  id="ch1-fig-example"
/>

## Reflections

What should the reader walk away knowing? Three to five takeaways.

1. The first key insight, restated in plain language.
2. The second key insight, with a connection to a sibling chapter.
3. The third key insight, often a caveat or limitation.

<OpenQuestion>
A research gap the chapter exposes. What's not yet known? What
would extend this result? Pointers to the wish-list / project-ideas
file for ambitious readers.
</OpenQuestion>

## Forward-map

How this chapter connects to next week. One paragraph; sets up the
reader's mental model for what comes next.

<NoteBox>
Delete this section when no follow-on chapter exists yet, or replace
with a "Synthesis" section in the final chapter of a part.
</NoteBox>