Add API cost tracking to admin dashboard

Track estimated Anthropic API costs per request across all Claude API
call sites (recommender, analyze, artist-dive, generate-playlist, crate,
rabbit-hole, playlist-fix, timeline, compatibility). Log token usage and
estimated cost to the app logger. Aggregate costs in admin stats endpoint
and display total/today costs and token usage in the admin dashboard.

backend/app/api/endpoints/compatibility.py

@@ -1,6 +1,9 @@
 import json
+import logging
 
 import anthropic
+
+api_logger = logging.getLogger("app")
 from fastapi import APIRouter, Depends, HTTPException
 from pydantic import BaseModel
 from sqlalchemy import select
@@ -144,6 +147,12 @@ Return ONLY the JSON. Include exactly 5 recommendations."""
         messages=[{"role": "user", "content": prompt}],
     )
 
+    # Track API cost (Haiku: $0.80/M input, $4/M output)
+    input_tokens = message.usage.input_tokens
+    output_tokens = message.usage.output_tokens
+    cost = (input_tokens * 0.80 / 1_000_000) + (output_tokens * 4 / 1_000_000)
+    api_logger.info(f"API_COST|model=claude-haiku|input={input_tokens}|output={output_tokens}|cost=${cost:.4f}|user=system|endpoint=compatibility")
+
     try:
         text = message.content[0].text.strip()
         if text.startswith("```"):