Postgres connection pool, clean 503 under overload, time indexes, 4 workers

Postgres here allows 60 connections, but each of the 96 gunicorn threads held its
own persistent one: at 100 concurrent users (local Postgres with the same limit)
21% of requests were 500 "remaining connection slots are reserved". Now each worker
has a psycopg pool (8), so the same load gives 0 errors and 714 req/s (was 460).

- A full pool answers 503 + Retry-After (DRF handler and middleware), not 500.
- The AI tutor and the OTP SMS call return their connection before the network wait.
- statement_timeout 30 s, idle-in-transaction 60 s; bad pool settings and the old
  DB_CONN_MAX_AGE stop startup instead of failing every request.
- /api/health dbPool: per-worker pool stats and workers x pool + 6 vs max_connections.
- Indexes on the columns global counters filter by time (weekly leaderboard,
  global SMS cap, global AI cap); the leaderboard recomputes once per expiry.
- Worker count moves into gunicorn.conf.py (4 on a shared 8-core box).
- security doc: Redis item closed (server uses our own Redis since 2026-10-08).

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

ai_tutor/models.py

@@ -30,7 +30,9 @@ class Message(models.Model):
     # به API نرسید). سهمیه‌ی رایگانِ امروز = تعدادِ «free»های امروز.
     CHARGE_CHOICES = [('free', 'رایگانِ روزانه'), ('credit', 'اعتبارِ بسته'), ('none', 'بی‌هزینه')]
     charge = models.CharField(max_length=6, choices=CHARGE_CHOICES, blank=True, default='')
-    created_at = models.DateTimeField(auto_now_add=True)
+    # ایندکس: سقفِ روزانه‌ی کلِ سامانه (_check_global_daily_cap) در هر سوال پیام‌های امروزِ همه را
+    # می‌شمارد؛ این سریع‌ترین جدولِ در حالِ رشد است.
+    created_at = models.DateTimeField(auto_now_add=True, db_index=True)
 
     # --- حاشیه‌نویسیِ تولید (فقط برای پیام‌های role='assistant') --------------
     # سینیور خواست لایک/دیسلایک بعداً قابلِ تحلیل باشد: «کدام جواب بهتر بوده».