[FEATURE] Cloud sessions: keep the container warm for a grace window after activity (30-60s resume triggered by even a 2-minute pause)

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

I use Claude Code on the web heavily (cloud sessions, accessed through the Windows desktop app and claude.ai/code, also from iPad/iPhone). The cloud container suspends after a very short period of inactivity — short enough that even a ~2-minute pause while I compose my next message triggers it. My next message then waits 30-60 seconds on "resuming session" before Claude begins responding.

The result is that in an ordinary interactive conversation, nearly every message pays the resume penalty: a 2-minute human pause costs a ~1-minute machine delay. Over a working session this adds up to a large share of total wait time. We measured the session's actual model and tool round-trips at ~3 seconds each, so the resume thaw dominates perceived latency for conversational use.

I checked the docs (claude-code-on-the-web, cloud-environments): suspension after inactivity is documented, but the timeout length is not, and no setting, environment option, or flag controls it today.

### Proposed Solution

Keep the session's container warm for a grace window after the last activity — even 5 minutes would eliminate the penalty for normal conversational pacing (a human reading a reply and typing the next message). Suspension after genuinely long idle periods can stay exactly as it is; the request is only that a brief between-turns pause not trigger it.

### Alternative Solutions

Alternatively (or additionally): expose the idle timeout as a per-environment or per-session setting, so users can choose the latency/cost tradeoff themselves.

Workarounds tried: none exist today. The docs document suspension but not the timeout length, and no setting, environment option, or flag controls it. Issue #32050 indicates even long-running background tasks don't prevent the idle timeout, so there is no way to hold the container open from inside a session either.

### Priority

High - Significant impact on productivity

### Feature Category

Performance and speed

### Use Case Example

Example scenario:
1. I ask Claude to make a change; it works for a couple of minutes and replies.
2. I read the reply and type my next message — this routinely takes me 2-5 minutes (I proofread longer prompts).
3. By the time I press Enter, the container has already suspended, and my message waits 30-60 seconds on "resuming session" before Claude even starts.
4. This repeats on nearly every turn of the conversation.
5. With a ~5-minute grace window, step 3's delay would disappear for normal conversational pacing, and the measured ~3-second model round-trips would be what I actually experience.

### Additional Context

Measurements from a long-running cloud session (reading its own transcript): model round-trip median ~3.1 s, tool execution median ~0 s — the container itself is fast; the resume thaw is the dominant wait for interactive use.

Related but distinct existing issues: #32050 (closed as stale — idle timeout disconnects during long-running background tasks; confirms the timeout ignores in-session activity) and #66115 (auto-compact on idle timeout to prevent cache-expiry cost — a different mitigation of the same idle boundary).

Cloud Claude Code is excellent — this is the one rough edge that most affects the day-to-day feel of working in it.

No implementation file, test, or entry point is named; start by reading the claude-code-on-the-web and cloud-environments documentation and tracing the cloud session suspension and resume behavior. Done means a brief idle pause no longer causes the resume delay, while longer idle suspension remains available, with any configurable timeout behavior defined and validated.

Do mô hình lập chỉ mục viết ra từ nội dung của issue.