Public page SEO and production analytics (#125)

## Summary
- Closes the public-page gaps from #124: each route gets its own title, description, canonical, heading, links, and JSON-LD, and unknown URLs have a real 404 document.
- Adds HeyCatch and the Google tag (`G-V12DLRLPWR`) only on the production site and the iOS and Android apps. Tianji still loads on beta.
- Needs the server-infra change that stops serving the app shell for missing paths.

## Test plan
- [ ] Build production and confirm `/` and `/pricing/` HTML include a unique title, description, canonical, one h1, social tags, related links, and Organization JSON-LD.
- [ ] Confirm a nonsense path is served as `404.html` once the nginx change is deployed.
- [ ] Confirm beta and local web do not load HeyCatch or gtag, and that a production or native build does.
- [ ] Open the live site after deploy and click around, then mark HeyCatch installed in its dashboard.

Reviewed-on: #125

llm-fe/scripts/sitemap.js

@@ -0,0 +1,153 @@
+/**
+ * Sitemap lastmod comes from the page source, not from the build clock.
+ * Changelog uses the newest entry date. Terms use the "Last Updated" line.
+ * Other public pages use the last git commit that touched that file.
+ */
+const { execFileSync } = require('child_process');
+const fs = require('fs');
+const path = require('path');
+
+const ORIGIN = 'https://hesychia.ai';
+const INDEXNOW_KEY = '8d1667dd71c6ef137171989c26eeb632';
+const INDEXNOW_HOST = 'hesychia.ai';
+
+const llmFe = path.resolve(__dirname, '..');
+const repoRoot = path.resolve(llmFe, '..');
+
+/** Public pages only. Same URLs the site already lists for crawlers. */
+const SITEMAP_PAGES = [
+  {
+    path: '/',
+    file: 'llm-fe/src/llm-fe/pages/Landing/LandingPage.tsx',
+    lastmod: 'git',
+  },
+  {
+    path: '/pricing/',
+    file: 'llm-fe/src/llm-fe/pages/Pricing/PricingPage.tsx',
+    lastmod: 'git',
+  },
+  {
+    path: '/privacy/',
+    file: 'llm-fe/src/llm-fe/pages/Privacy/PrivacyPage.tsx',
+    lastmod: 'git',
+  },
+  {
+    path: '/changelog/',
+    file: 'llm-fe/src/llm-fe/pages/Changelog/changelogEntries.ts',
+    lastmod: 'changelog',
+  },
+  {
+    path: '/terms_of_service/',
+    file: 'llm-fe/src/llm-fe/components/legal/TermsOfServiceDocument.tsx',
+    lastmod: 'terms',
+  },
+  {
+    path: '/account-deletion/',
+    file: 'llm-fe/src/llm-fe/pages/AccountDeletion/AccountDeletionPage.tsx',
+    lastmod: 'git',
+  },
+];
+
+const MONTHS = {
+  January: '01',
+  February: '02',
+  March: '03',
+  April: '04',
+  May: '05',
+  June: '06',
+  July: '07',
+  August: '08',
+  September: '09',
+  October: '10',
+  November: '11',
+  December: '12',
+};
+
+function newestChangelogDate(source) {
+  const dates = [...source.matchAll(/date:\s*'(\d{4}-\d{2}-\d{2})'/g)].map((match) => match[1]);
+  if (dates.length === 0) {
+    throw new Error('changelog entries have no dates');
+  }
+  return dates.sort().at(-1);
+}
+
+function termsUpdated(source) {
+  const match = source.match(/Last Updated:\s+([A-Z][a-z]+)\s+(\d{1,2}),\s+(\d{4})/);
+  if (!match) {
+    throw new Error('terms have no Last Updated date');
+  }
+  const month = MONTHS[match[1]];
+  if (!month) {
+    throw new Error(`unknown month in terms date: ${match[1]}`);
+  }
+  return `${match[3]}-${month}-${match[2].padStart(2, '0')}`;
+}
+
+function gitLastmod(file) {
+  const out = execFileSync('git', ['log', '-1', '--format=%cs', '--', file], {
+    cwd: repoRoot,
+    encoding: 'utf8',
+  }).trim();
+  if (!/^\d{4}-\d{2}-\d{2}$/.test(out)) {
+    throw new Error(`no git commit date for ${file}`);
+  }
+  return out;
+}
+
+function lastmodFor(page) {
+  const abs = path.join(repoRoot, page.file);
+  const source = fs.readFileSync(abs, 'utf8');
+  if (page.lastmod === 'changelog') return newestChangelogDate(source);
+  if (page.lastmod === 'terms') return termsUpdated(source);
+  return gitLastmod(page.file);
+}
+
+function sitemapEntries() {
+  return SITEMAP_PAGES.map((page) => ({
+    loc: `${ORIGIN}${page.path}`,
+    lastmod: lastmodFor(page),
+  }));
+}
+
+function renderSitemap(entries) {
+  const urls = entries
+    .map(
+      (entry) => `  <url>
+    <loc>${entry.loc}</loc>
+    <lastmod>${entry.lastmod}</lastmod>
+  </url>`,
+    )
+    .join('\n');
+  return `<?xml version="1.0" encoding="UTF-8"?>
+<urlset xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
+${urls}
+</urlset>
+`;
+}
+
+function writeSitemap() {
+  const xml = renderSitemap(sitemapEntries());
+  fs.writeFileSync(path.join(llmFe, 'public', 'sitemap.xml'), xml);
+  return xml;
+}
+
+function indexNowBody(key, urls) {
+  return {
+    host: INDEXNOW_HOST,
+    key,
+    keyLocation: `https://${INDEXNOW_HOST}/${key}.txt`,
+    urlList: urls,
+  };
+}
+
+module.exports = {
+  INDEXNOW_KEY,
+  SITEMAP_PAGES,
+  newestChangelogDate,
+  termsUpdated,
+  gitLastmod,
+  sitemapEntries,
+  renderSitemap,
+  writeSitemap,
+  indexNowBody,
+};