fix(mobile): show background monitoring status

[Gigioxx]

Mobile looked idle after a turn completed even when the server still reported background monitoring or work. Both thread lists now show Monitoring or Working, and the open thread reuses the same status resolver for its floating label. Monitoring stays static and disappears when liveness clears. Existing approval, input, error, and plan priorities are preserved.

Fixes #10372.

Verified the regression fails on the original code and passes with the fix: 62 focused tests, mobile typecheck, and scoped lint with the same 23 existing warnings. React Doctor's before/after diagnostics are unchanged. On an iPhone 17 Pro simulator, checked Monitoring, Working, and cleared liveness in the default list and open thread. The legacy list is covered by resolver tests; Android and iPad share the changed components but were not run separately.

Screenshots use a disposable environment with a completed parent turn and controlled server liveness. They verify native presentation, not a live provider watch loop. No server or contract changes ship.

Default list: before	Default list: after

Open thread: before	Open thread: after

Model: GPT-6. Harness: Codex in T3 Code.

[macroscopeapp]

Approvability

Verdict: Would Approve

Macroscope's review found this PR approvable — This is a localized mobile presentation fix that surfaces an existing background-liveness state in thread lists and the open-thread status control. It preserves existing priorities and processing behavior, with focused coverage and no schema, infrastructure, security, or configuration changes.

Not approved because:

• Monthly spending limit reached (workspace setting). Approvability relies on correctness review in order to determine eligibility

^{Review your spending limits in Billing settings. You can add or adjust custom eligibility rules. Learn more.}

[cristip73]

@Gigioxx thanks for this fix. It stopped applying after #13183 retired the legacy thread list and threadPresentation.ts, so it is now conflicting on 4 files.

I ported it onto current main in #13803, with you as co-author on the commit. The logic is yours. It follows the triage on #10372 and also closes #4962, the "subagents still running, phone says idle" case. If you would rather rebase this PR yourself, say so and I will close mine. The goal is just to get one of them mergeable.