feat: persistent per-provider key counters

[boidushya]

Summary

Per-provider counters live in a new counters bucket inside cache.db, updated transactionally with every Set / Delete.
Counter keys are derived from the cache key prefix via a small prefixOf classifier: ttml, kugou, qq, legacy, negative, unknown.
/stats now reads live counts via persistentCache.Counts() in microseconds. New response fields: keys_total, keys_by_provider, last_reconciled_at, last_duration_ms, status (seeding | ready | error), and last_error on failure.
StatsCache was reshaped into a small lifecycle struct. The 24h scan is gone. A weekly background ReconcileCounters corrects drift from rare type-flips (e.g. permanent no-lyrics sentinels stored under provider prefixes).
Clear() wipes the counters bucket in the same transaction as the cache wipe. Restore handler triggers a reconcile so its response reflects the restored DB.

[better-lyrics-harmonizer]

Preview torn down

This PR's ephemeral environment was cleaned up.

Detail	Value
Status	Torn down
Branch	feat/persistent-counter
Last commit	fb73cc3
Closed	2026-06-09 13:35:42 UTC
Workflow	run #27209938516

_{Reopen the PR if you need to spin it back up.}