Add Markdown+TeX validator (validateMarkdown library + vitest gate)

web/.prettierignore

@@ -3,4 +3,8 @@ node_modules
 out
 dist
 coverage
-public
+public
+
+# Validator fixtures contain intentionally malformed markdown; Prettier would
+# auto-correct unclosed bold, angle-bracket dangerous URLs, etc.
+scripts/__fixtures__/validate-md

web/package.json

@@ -61,14 +61,18 @@
     "rehype-katex": "^7.0.1",
     "rehype-raw": "^7.0.0",
     "rehype-sanitize": "^6.0.0",
+    "rehype-stringify": "^10.0.1",
     "remark-breaks": "^4.0.0",
     "remark-directive": "^4.0.0",
     "remark-gfm": "^4.0.1",
     "remark-math": "^6.0.0",
+    "remark-parse": "^11.0.0",
+    "remark-rehype": "^11.1.2",
     "resend": "^6.9.1",
     "sonner": "^2.0.7",
     "svix": "^1.84.1",
     "tailwind-merge": "^3.4.0",
+    "unified": "^11.0.5",
     "unist-util-visit": "^5.1.0",
     "use-long-press": "^3.3.0",
     "zod": "^4.3.6",

web/scripts/__fixtures__/validate-md/invalid-katex-bad-environment.md

@@ -0,0 +1,3 @@
+Unknown environment name — KaTeX does not know `notathing`.
+
+$$\begin{notathing} x = 1 \end{notathing}$$

web/scripts/__fixtures__/validate-md/invalid-katex-empty-subscript.md

@@ -0,0 +1,3 @@
+Subscript with no argument — `_` followed immediately by the closing delimiter.
+
+$x_$

web/scripts/__fixtures__/validate-md/invalid-katex-in-list.md

@@ -0,0 +1,6 @@
+A KaTeX error inside a bullet-list item must surface (not be discarded with the surrounding markdown).
+
+- First valid item.
+- Second valid item with formula $a + b$.
+- Third item has a broken formula: $\frac{a}$.
+- Fourth item.

web/scripts/__fixtures__/validate-md/invalid-katex-in-spoiler.md

@@ -0,0 +1,5 @@
+A KaTeX error nested inside a spoiler must propagate, not get swallowed by the directive transformer.
+
+:::spoiler[Hint]
+Try the substitution $\foo{u}$.
+:::

web/scripts/__fixtures__/validate-md/invalid-katex-in-table.md

@@ -0,0 +1,6 @@
+A KaTeX error inside a GFM table cell must surface — confirms cell rendering does not silently drop math errors.
+
+| Operation | Formula       |
+| --------- | ------------- |
+| valid     | $a + b$       |
+| broken    | $\unknowncmd$ |

web/scripts/__fixtures__/validate-md/invalid-katex-mismatched-env.md

@@ -0,0 +1,3 @@
+Mismatched begin/end — `aligned` opened, `cases` closed.
+
+$$\begin{aligned} x &= 1 \end{cases}$$

web/scripts/__fixtures__/validate-md/invalid-katex-missing-arg.md

@@ -0,0 +1,3 @@
+`\frac` is a two-argument macro; supplying only one is a parse error.
+
+$\frac{a}$

web/scripts/__fixtures__/validate-md/invalid-katex-nested.md

@@ -0,0 +1,3 @@
+Outer command (`\sqrt`) is valid, inner command (`\foo`) is not — confirms errors surface from nested expressions.
+
+$\sqrt{\foo{x}}$

web/scripts/__fixtures__/validate-md/invalid-katex-stray-amp.md

@@ -0,0 +1,3 @@
+Stray `&` outside an alignment environment — only legal inside `aligned`, `array`, etc.
+
+$x & y$

web/scripts/__fixtures__/validate-md/invalid-katex-unbalanced-braces.md

@@ -0,0 +1,3 @@
+Unbalanced braces — opening `{` after `a` is never closed.
+
+$\frac{a{b}$

web/scripts/__fixtures__/validate-md/invalid-katex-unclosed-brace.md

@@ -0,0 +1,3 @@
+Unclosed brace inside inline math — KaTeX parser hits end-of-input expecting `}`.
+
+$x{$

web/scripts/__fixtures__/validate-md/invalid-katex-unclosed-env.md

@@ -0,0 +1,3 @@
+Unclosed environment — `aligned` is opened but never closed.
+
+$$\begin{aligned} x &= 1$$

web/scripts/__fixtures__/validate-md/invalid-katex-unknown-cmd.md

@@ -0,0 +1,3 @@
+Unknown control sequence — KaTeX should reject `\foo`.
+
+$\foo{bar}$

web/scripts/__fixtures__/validate-md/invalid-katex-unmatched-left.md

@@ -0,0 +1,3 @@
+`\left(` opens a dynamic delimiter group with no matching `\right`.
+
+$\left( x + y$

web/scripts/__fixtures__/validate-md/valid-blockquote.md

@@ -0,0 +1,5 @@
+A blockquote with embedded inline math, multi-line continuation:
+
+> The Cauchy-Schwarz inequality states that for any real numbers $a_i$ and $b_i$,
+> $\left(\sum a_i b_i\right)^2 \le \left(\sum a_i^2\right) \left(\sum b_i^2\right)$,
+> with equality if and only if the sequences are proportional.

web/scripts/__fixtures__/validate-md/valid-code.md

@@ -0,0 +1,8 @@
+Inline `code` span and a fenced code block — supported by the schema even if rare in problem statements.
+
+```python
+def gcd(a, b):
+    while b:
+        a, b = b, a % b
+    return a
+```

web/scripts/__fixtures__/validate-md/valid-display-math.md

@@ -0,0 +1,7 @@
+Single-line display math:
+
+$$\int_0^1 x \, dx = \frac{1}{2}$$
+
+The canonical archive example combining several means:
+
+$$a=\frac{x+y}{2},\ g=\sqrt{xy},\ h=\frac{2xy}{x+y},\ k=\sqrt{\frac{x^2+y^2}{2}}.$$

web/scripts/__fixtures__/validate-md/valid-image.md

@@ -0,0 +1,11 @@
+Plain relative image reference (the bulk-import authoring shape):
+
+![Triangle ABC](images/fig1.png)
+
+Same image with a scale URL parameter (already supported in the renderer):
+
+![Scaled diagram](images/fig2.png?scale=75)
+
+Media-protocol URL pointing at an R2-hosted asset:
+
+![](media:abc123?scale=50)

web/scripts/__fixtures__/validate-md/valid-inline-math.md

@@ -0,0 +1,13 @@
+Battery of inline math drawn from the SKMO archive.
+
+Plain inequality: $x \ne y$.
+
+Fraction: $\frac{a}{b}$.
+
+Degrees and radicals: $60^\circ$, $\sqrt{xy}$.
+
+Interval notation: $\langle a, b\rangle$ and $\langle b, \infty)$.
+
+Subscripted point names: $k_1(S_1, 3\,{\text{cm}})$.
+
+Powers of trig functions: $\tg^2\gamma \cdot \tg\frac{\gamma}{2}$.

web/scripts/__fixtures__/validate-md/valid-link.md

@@ -0,0 +1,11 @@
+Link variants the renderer's url-allow-list accepts:
+
+External HTTPS: [Wikipedia](https://en.wikipedia.org/wiki/Mathematical_olympiad).
+
+Mailto: [Send a question](mailto:olympiada@example.com).
+
+Anchor in the same page: [Jump to the proof](#proof).
+
+Media protocol (R2-hosted asset): [PDF version](media:abc123).
+
+Domain-only link without protocol: [example.com](example.com).

web/scripts/__fixtures__/validate-md/valid-list-bullet.md

@@ -0,0 +1,6 @@
+Bullet list with mixed plain and math items:
+
+- First item in the list.
+- Second item with an embedded formula $a^2 + b^2 = c^2$.
+- Third item with several math symbols: $\alpha$, $\beta$, $\gamma$.
+- Final item.

web/scripts/__fixtures__/validate-md/valid-list-nested.md

@@ -0,0 +1,6 @@
+Nested list — outer bullet, inner numbered. Mirrors case-analysis structure in real solutions.
+
+- We split into two cases by the sign of $a$.
+  1. If $a > 0$, then $\sqrt{a^2} = a$.
+  2. If $a < 0$, then $\sqrt{a^2} = -a$.
+- In both cases $\sqrt{a^2} = |a|$.

web/scripts/__fixtures__/validate-md/valid-list-numbered.md

@@ -0,0 +1,6 @@
+Numbered list, common in problem solutions:
+
+1. First we show that $a > 0$.
+2. Therefore $\sqrt{a}$ is well-defined.
+3. We apply the AM-GM inequality.
+4. Conclusion: equality holds if and only if $a = b$.

web/scripts/__fixtures__/validate-md/valid-math-align.md

@@ -0,0 +1,9 @@
+Aligned multi-line equation common in solutions:
+
+$$
+\begin{aligned}
+(a + b)^2 &= a^2 + 2ab + b^2 \\
+(a - b)^2 &= a^2 - 2ab + b^2 \\
+(a + b)(a - b) &= a^2 - b^2
+\end{aligned}
+$$

web/scripts/__fixtures__/validate-md/valid-math-mixed-flow.md

@@ -0,0 +1,5 @@
+For given positive numbers $x \ne y$ consider the means
+
+$$a=\frac{x+y}{2},\ g=\sqrt{xy},\ h=\frac{2xy}{x+y},\ k=\sqrt{\frac{x^2+y^2}{2}}.$$
+
+(These are the arithmetic, geometric, harmonic, and quadratic means of $x$ and $y$.)

web/scripts/__fixtures__/validate-md/valid-prose.md

@@ -0,0 +1,3 @@
+A simple paragraph with **bold**, _italic_, and ~~strikethrough~~ text.
+
+A second paragraph follows with **_bold italic_** and a `code` span.

web/scripts/__fixtures__/validate-md/valid-realistic-problem.md

@@ -0,0 +1,19 @@
+For given positive numbers $x \ne y$ consider the means
+
+$$a=\frac{x+y}{2},\ g=\sqrt{xy},\ h=\frac{2xy}{x+y},\ k=\sqrt{\frac{x^2+y^2}{2}}.$$
+
+(These are the arithmetic, geometric, harmonic, and quadratic means of $x$ and $y$.)
+
+Prove that the following chain of inequalities holds:
+
+1. $h < g$,
+2. $g < a$,
+3. $a < k$.
+
+For intuition, the four means can be visualised on the figure below:
+
+![Four means of two numbers](images/means.png?scale=80)
+
+:::spoiler[Hint]
+Start with $h < g$: it suffices to show $\frac{2xy}{x+y} < \sqrt{xy}$, which after clearing denominators reduces to $(\sqrt{x} - \sqrt{y})^2 > 0$.
+:::

web/scripts/__fixtures__/validate-md/valid-spoiler.md

@@ -0,0 +1,11 @@
+Problem statement asking the reader to find a closed form.
+
+:::spoiler[Hint]
+Try the substitution $u = x^2$. After substituting you get
+
+$$\int u \, du = \frac{u^2}{2} + C.$$
+
+Convert back to the original variable to obtain the result.
+:::
+
+After solving, compare your work with the model solution.

web/scripts/__fixtures__/validate-md/valid-table.md

@@ -0,0 +1,7 @@
+GFM table — needed for solutions per §3 format-coverage check.
+
+| Operation      | Formula     | Example         |
+| -------------- | ----------- | --------------- |
+| addition       | $a + b$     | $2 + 3 = 5$     |
+| multiplication | $a \cdot b$ | $2 \cdot 3 = 6$ |
+| exponentiation | $a^b$       | $2^3 = 8$       |