fix(dependencies): double-checked locking pattern is a data race under free-threaded Python 3.13+ (PEP 703) #296

Summary

The singleton initialization pattern in backend/app/api/dependencies.py uses double-checked locking: check _cache is None, acquire lock, check again, create. This is safe under CPython's GIL but becomes a data race under Python 3.13+ free-threaded mode (PEP 703, nogil). Two threads can both pass the outer None check and construct duplicate SQLAlchemy engines, leaking connection pools.

Fix

backend/app/api/dependencies.py: move the singleton initialization to module level (eager init at import time) or initialize inside the lock without the outer None check. This removes the TOCTOU window entirely.

Impact

Under free-threaded Python 3.13+, multiple competing SQLAlchemy engine instances are created, each with their own connection pool, causing pool exhaustion and connection leaks under concurrent startup.