Backfill MongoDB data into PostgreSQL and verify ongoing parity

[coder13]

Goal

Turn PostgreSQL dual writes into a trustworthy migration target by copying historical MongoDB data and continuously checking parity.

Acceptance criteria

• Implement an idempotent, resumable backfill for users/preferences, rooms/participant state, attempts, durable solve results, and sanitized metrics.
• Support dry-run, bounded batches, progress checkpoints, and safe retries.
• Do not copy OAuth access tokens or prohibited metric fields.
• Produce reconciliation output for record counts, missing identifiers, duplicate identifiers, penalty fields, and representative aggregates.
• Verify writes arriving during the backfill are not lost or duplicated.
• Document operational invocation, expected runtime/load, alerts, failure recovery, and how to rerun only failed ranges.
• Add focused tests for deterministic IDs, cursor boundaries, retry behavior, and reconciliation failures.

Dependencies

• Stable production dual writes from PR Add PostgreSQL dual writes and room metrics #171, deployed through Rehearse and execute the modernized production rollout with a tested rollback #176
• The schema and migration history must remain compatible with rollback requirements

Unlocks

• Add a feature-flagged PostgreSQL solve-history read API #174
• Persist room solve history and past sessions #160