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>

src/claude-usage-backoff.test.ts

@@ -102,6 +102,32 @@ describe('Claude usage 429 Retry-After backoff', () => {
     expect(second[0].usageRateLimited).toBe(true);
   });
 
+  it('falls back to the default cooldown when Retry-After is 0 (does not disable backoff)', async () => {
+    // Production regression: the usage endpoint was returning 429 with
+    // `retry-after: 0`, which made `retryAfterMs ?? DEFAULT` evaluate to 0 — a
+    // ~5s cooldown. The poller then re-hit the endpoint every 60s and re-tripped
+    // the 429 indefinitely. A non-positive Retry-After must use the default.
+    vi.useFakeTimers();
+    vi.setSystemTime(new Date('2026-06-20T00:00:00Z'));
+
+    const fetchMock = vi.fn(async () =>
+      makeResponse(429, { 'retry-after': '0' }),
+    );
+    vi.stubGlobal('fetch', fetchMock);
+
+    const { fetchAllClaudeUsage } = await import('./claude-usage.js');
+
+    await fetchAllClaudeUsage();
+    expect(fetchMock).toHaveBeenCalledTimes(1);
+
+    // +70s: past the 60s throttle. With the bug (0 cooldown) this would refetch;
+    // with the default 5-min cooldown it must still be held.
+    vi.setSystemTime(Date.now() + 70_000);
+    const held = await fetchAllClaudeUsage();
+    expect(fetchMock).toHaveBeenCalledTimes(1);
+    expect(held[0].usageRateLimited).toBe(true);
+  });
+
   it('keeps backing off after the 60s throttle elapses but before the cooldown closes', async () => {
     // This is the core regression: a 93s Retry-After must outlast the 60s
     // MIN_FETCH_INTERVAL throttle. Without the cooldown the poller would refetch