fix(textbooks): убраны лишние слэши в LaTeX-формулах (over-escaping)

Формулы в JS-литералах имели \\\\dfrac / \\\\\\\\dfrac (4/8 слэшей) вместо
\\dfrac (2). После JS-анескейпа KaTeX получал \\dfrac, трактовал \\ как
перенос строки и печатал dfrac/cdot/sqrt/pi как текст (карточка пирамиды и
конуса в geometry_11_ch2, и др.).

Схлопнуты прогоны слэшей кратные 4 перед LaTeX-командой -> 2. Прогоны из
3 слэшей (\\ перенос строки + \cmd в \begin{cases}) и перед x/цифрой не
тронуты. 150 правок в 7 файлах (algebra_11_ch1/ch2/ch3, geometry_11_ch1..ch4).

БД чиста: questions (1398) text/explanation/correct_text + options (5187) -
0 багов. Скрипт: backend/scripts/fix_overescaped_latex.js (идемпотентный,
dry-run по умолчанию, --apply, с KaTeX-валидацией).

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

backend/scripts/fix_overescaped_latex.js

@@ -0,0 +1,108 @@
+/*
+ * Fix OVER-ESCAPED LaTeX backslashes in textbook HTML.
+ *
+ * BUG: some formulas in JS string literals have too many backslashes, e.g.
+ *      "$V=\\\\dfrac{1}{3}S_{осн}\\\\cdot h$"   (4 backslashes)
+ * After JS unescaping KaTeX receives  \\dfrac  -> it renders "\\" as a LINE
+ * BREAK and prints "dfrac"/"cdot" as plain text (exactly the screenshot).
+ * The correct literal is 2 backslashes ("\\dfrac" -> value \dfrac).
+ *
+ * PARITY RULE (critical — protects legitimate row separators):
+ *   literal-run length  value backslashes  meaning
+ *      2                  1   \cmd                         OK   keep
+ *      4                  2   \\  + "cmd"(text)            BUG  -> 2
+ *      6                  3   \\ + \cmd (rowbreak+cmd)     OK   keep
+ *      8                  4   \\\\ + "cmd"(text)           BUG  -> 2
+ *   => collapse ONLY runs whose length is a multiple of 4, AND only when the
+ *      run is immediately followed by a known LaTeX command. Runs before "x",
+ *      digits, etc. (real \\ row separators inside cases/array) are untouched.
+ *
+ * Usage:  node backend/scripts/fix_overescaped_latex.js          (dry run)
+ *         node backend/scripts/fix_overescaped_latex.js --apply  (write)
+ */
+'use strict';
+const fs = require('fs');
+const path = require('path');
+
+const APPLY = process.argv.includes('--apply');
+
+// Known LaTeX commands observed at 4/8 backslashes (exact-match whitelist).
+const CMDSET = new Set([
+  'dfrac','tfrac','frac','sqrt','cdot','pi','log','ln','lg','alpha','beta','gamma',
+  'delta','Delta','theta','lambda','mu','sigma','phi','varphi','omega','infty',
+  'iff','in','notin','ne','neq','ge','geq','le','leq','mathbb','mathrm',
+  'leftrightarrow','rightarrow','leftarrow','times','div','vec','overline',
+  'perp','parallel','cos','sin','tan','cot','ldots','cdots','pm','mp','angle','triangle',
+]);
+
+let katex = null;
+try { katex = require('katex'); } catch { /* validation optional */ }
+function mathRegions(t) {
+  const out = []; let i = 0;
+  while (i < t.length) {
+    const a = t.indexOf('$', i); if (a < 0) break;
+    const dbl = t[a + 1] === '$'; const s = a + (dbl ? 2 : 1);
+    let b = dbl ? t.indexOf('$$', s) : t.indexOf('$', s);
+    if (b < 0 && dbl) b = t.indexOf('$', s);
+    if (b < 0) break;
+    out.push(t.slice(s, b));
+    i = b + (dbl && t.slice(b, b + 2) === '$$' ? 2 : 1);
+  }
+  return out;
+}
+// These math strings live in JS literals; KaTeX sees them AFTER one level of JS
+// unescaping. Emulate that so validation reflects what the browser renders.
+function jsUnescape(s) {
+  return s.replace(/\\\\/g, '\\');
+}
+function katexErrors(t) {
+  if (!katex) return null;
+  let bad = 0;
+  for (const inner of mathRegions(t)) {
+    const expr = jsUnescape(inner);
+    try { katex.renderToString(expr, { throwOnError: true }); }
+    catch { bad++; }
+  }
+  return bad;
+}
+
+const dir = path.join(__dirname, '..', '..', 'frontend', 'textbooks');
+const files = ['algebra_11_ch1.html','algebra_11_ch3.html','geometry_11_ch3.html',
+  'geometry_11_ch2.html','geometry_11_ch1.html','algebra_11_ch2.html','algebra_8.html',
+  'algebra_7_ch4.html','geometry_11_ch4.html'];
+
+const report = [];
+report.push('MODE: ' + (APPLY ? 'APPLY' : 'DRY-RUN'));
+let grandFixes = 0;
+
+for (const f of files) {
+  const p = path.join(dir, f);
+  const t = fs.readFileSync(p, 'utf8');
+  const before = katexErrors(t);
+
+  const perCmd = {};
+  let fixes = 0;
+  const next = t.replace(/(\\{4,})([A-Za-z]+)/g, (whole, bs, word) => {
+    if (bs.length % 4 !== 0) return whole;        // 6,10,... rowbreak+command -> keep
+    if (!CMDSET.has(word)) return whole;          // x / begin / unknown -> keep
+    fixes++;
+    perCmd[word] = (perCmd[word] || 0) + 1;
+    return '\\\\' + word;                         // collapse to two backslashes
+  });
+
+  // validate by emulating browser render of the FIXED text
+  const after = katexErrors(next);
+  grandFixes += fixes;
+  report.push('');
+  report.push(f + ': fixes=' + fixes + '  katexErrors before=' + before + ' after=' + after +
+    (fixes ? '  cmds=' + JSON.stringify(perCmd) : ''));
+  if (after !== null && before !== null && after > before)
+    report.push('  !! WARNING: katex errors INCREASED — not writing this file');
+
+  if (APPLY && fixes > 0 && !(after > before)) fs.writeFileSync(p, next, 'utf8');
+}
+
+report.push('');
+report.push('TOTAL fixes: ' + grandFixes);
+fs.writeFileSync(path.join(__dirname, 'fix_overescaped_latex.report.txt'), report.join('\n'), 'utf8');
+console.log(report.join('\n'));