feat: guest access, pricing modal, UI polish, LaTeX prompt fix

- Remove login gate, allow guest browsing with Sign in link
- Add favicon (book logo)
- Add pricing modal (Free/Standard/Exam) with hover animations
- Dynamic course list from DB instead of hardcoded
- Enforce LaTeX in AI trio generation prompt
- UI improvements: homepage animations, analytics donut chart, error book cards
- Fix error book locked state for guests

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

backend/app/services/paper_processor.py

@@ -144,6 +144,7 @@ Solution requirements:
 - Mark common mistakes with <div class="common-error">...</div>
 
 KaTeX formula rules:
+- CRITICAL: ALL math expressions MUST use LaTeX inside $ or $$. NEVER use Unicode symbols like ⁿ, ≥, ≠, ², ×, ∑, ∈. Use $n$, $\geq$, $\neq$, $^2$, $\times$, $\sum$, $\in$ instead.
 - Block formula: $$ on its own line, with blank lines before and after
 - Inline formula: $x^2$ no line break
 - Matrix: \\begin{{bmatrix}} ... \\end{{bmatrix}}
@@ -174,6 +175,7 @@ Rules:
 - Keep each item matched to the same question_number
 - All text must be in English
 - HTML only, KaTeX compatible
+- CRITICAL LaTeX requirement: ALL mathematical expressions MUST use LaTeX notation wrapped in $ (inline) or $$ (display block). NEVER use Unicode math symbols like ⁿ, ≥, ≠, ², ×, ∑, ∈, ⊆, etc. Instead use $n$, $\geq$, $\neq$, $^2$, $\times$, $\sum$, $\in$, $\subseteq$, etc. Every variable, number in a formula, operator, and equation must be inside $ delimiters.
 - For MC questions, explain why the correct option is right and why the others are wrong
 - For long questions, show a complete derivation or reasoning chain
 - Use <ol> or numbered steps in solution when appropriate