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

backend/regen_comp2711h_trio.py

@@ -0,0 +1,47 @@
+"""
+One-off script: clear AI trio for COMP2711H paper and regenerate with LaTeX-enforced prompt.
+Run inside the backend Docker container or locally with .env loaded.
+"""
+import asyncio
+import json
+import sys
+import os
+
+sys.path.insert(0, os.path.dirname(__file__))
+
+from dotenv import load_dotenv
+load_dotenv(os.path.join(os.path.dirname(__file__), "..", ".env"))
+
+from app.services.supabase_client import get_supabase
+from app.services.paper_processor import _resume_ai_trio
+
+PAPER_ID = "5ee87a62-65bf-4952-be40-fcdf9ba7ca63"  # COMP2711H 2025 fall final
+
+
+async def main():
+    sb = get_supabase()
+
+    # 1. Clear existing AI trio fields so _resume_ai_trio will regenerate them
+    questions = sb.table("paper_questions").select("id, question_number, solution").eq("paper_id", PAPER_ID).execute().data
+    print(f"Found {len(questions)} questions for paper {PAPER_ID[:8]}")
+
+    cleared = 0
+    for q in questions:
+        if q.get("solution"):
+            sb.table("paper_questions").update({
+                "knowledge_reminder": "",
+                "ai_hint": "",
+                "solution": "",
+            }).eq("id", q["id"]).execute()
+            cleared += 1
+            q["solution"] = None  # so _resume_ai_trio picks it up
+
+    print(f"Cleared AI trio for {cleared} questions. Regenerating...")
+
+    # 2. Regenerate
+    await _resume_ai_trio(sb, PAPER_ID, questions)
+    print("Done! AI trio regenerated with LaTeX-enforced prompt.")
+
+
+if __name__ == "__main__":
+    asyncio.run(main())