Fix owner context loss after finalize; green the test suite

Owner continuity:
- Seed a freshly created owner task with the previous task's latest owner
  final on a cold start after the previous task already closed, so a user
  reply to a finalized TASK_DONE no longer produces a "no context" answer.
  Activated for this deployment via PAIRED_CARRY_FORWARD_LATEST_OWNER_FINAL
  (.env); carried text is injected as clearly-marked background only.
- Skip intermediate STEP_DONE outputs when picking the carry-forward anchor.

Single-mode routing:
- enforceRoomModeOnLease strips a stale reviewer/arbiter lease from a room
  switched back to single, preventing single-mode messages from stalling in
  the paired path on a stuck execution lease.

Session auth / credentials:
- Pre-sync Claude credentials into each session dir before the agent spawns.
- Honor CLAUDE_CREDENTIALS_PATH in setup/login.ts (per-service isolation).
- Add a relogin-required gate so a permanently logged-out claude-code room
  asks the user to re-login instead of spawning a doomed agent.

Other:
- Arbiter verdicts written in the user's language (verdict keyword stays EN).
- status-dashboard chatName field; runtime-inventory credential path resolver.

Tests (make suite fully green: 1595 pass / 3 skip):
- service-routing: default owner is now the claude service and reviewer is
  codex-review; update the 7 failover/default expectations accordingly.
- migrate-room-registrations: owner inferred as claude-code (configured
  OWNER_AGENT_TYPE) for a dual legacy room; reviewer becomes codex.
- register: mock paired-workspace provisioning + reload signal (registration
  now provisions a workspace and hot-reloads); assert RELOADED status.
- paired-execution-context: force a claude-code reviewer to exercise the
  Claude read-only branch regardless of the deployment default.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

prompts/arbiter-paired-room.md

@@ -43,3 +43,8 @@ You may receive reference opinions from external models appended to your prompt.
 - If both sides are saying the same thing but not acting on it, call it out and direct the owner to act
 - If the conversation shows the owner asking the user a question (not the reviewer), always ESCALATE — the arbiter cannot answer on behalf of the user
 - If you see a prior arbiter verdict of PROCEED in the history but the same issue persists, do NOT repeat PROCEED — use ESCALATE instead
+
+## Language
+
+- Write your verdict in the user's language (Korean for this deployment) unless the user wrote in another language. The user must be able to read your verdict.
+- Keep ONLY the leading verdict keyword in English — `PROCEED` / `REVISE` / `RESET` / `ESCALATE` — exactly as specified above, because the system parses that first token. Write everything after it (reasoning, evidence, the required action for the owner) in Korean.