Render math in chat messages with KaTeX

Add a marked extension that tokenizes math delimiters and renders them
with KaTeX: `$$ ... $$` / `\[ ... \]` as block, `$ ... $` / `\( ... \)`
and mid-paragraph `$$ ... $$` as inline.

Inline `$ ... $` requires a non-space after the opening `$`, no newline
inside, and a non-digit after the closing `$`, so prose like "$100 から
$200" is left alone; `\$` still escapes a literal dollar sign. Parse
errors fall back to KaTeX's red inline message instead of throwing, and
math whose closing delimiter has not streamed in yet stays plain text.

DOMPurify strips `semantics` and `annotation` by default, which would
leak the raw TeX out of KaTeX's MathML output, so both are allowed
explicitly.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

frontend/package.json

@@ -11,6 +11,7 @@
   },
   "dependencies": {
     "dompurify": "^3.1.0",
+    "katex": "^0.16.47",
     "marked": "^12.0.0",
     "pinia": "^2.1.7",
     "vue": "^3.4.0",