feat(sessions): keep an idle session's prompt cache warm with a guarded keep-alive turn

Claude Code lets a subscription's one-hour prompt cache expire after an hour idle, and the first turn back re-writes the whole conversation at the cache-write rate; nearly every gap over an hour goes cold, while the saving is concentrated in gaps of one to four hours. Codeman now sends one short keep-alive turn five minutes before the expiry Claude Code reports on its statusline, so the next real prompt still reads the conversation from cache, and stops two hours (configurable) after the last real prompt.

Codeman owns the idle policy for these sessions, as agreed in Discussion #554: a launch with keep-warm on also carries idleCompaction: false in its one --settings JSON, so Claude Code's own idle compaction cannot replace the conversation with a summary a minute before the nudge. It is decided once per launch (keep-warm on, the statusLine exporter injected, Claude Code 2.1.290 or newer, an unknown version fails closed like --name), recorded on the mux record, and picked up by any relaunch; a launch with keep-warm off is byte-identical. A session switched on after launch is kept warm best-effort and the readout says so.

Every guard lives in one pure rulebook: the opt-in (an agent-spawned session does not inherit the app default), a launch that reports cache state, a warm 1h cache with enough tokens at stake and no compaction this gap, a live pane, no usage-limit pause, no respawn, Ralph, task or waiting agent, idle, no open dialog, an empty composer and no recent keystroke. The nudge is a submit and moves lastSubmitAt, which the response viewer's conversation correlation needs; the cap counts from its own persisted lastRealPromptAt instead. Read My Mind drops the exact nudge text, and the turn's stop hook and idle prompt open pre-acknowledged so nothing alerts. The last cache report and the clock survive a restart, so the timer re-arms.

docs/wiki/Settings-Reference.md

@@ -184,7 +184,7 @@ Rebinding for the shortcut registry. See [Keyboard Shortcuts](Keyboard-Shortcuts
 
 ### System
 
-`CLAUDE.md` template for new cases, default working directory, the image watcher, and
+`CLAUDE.md` template for new cases, default working directory, the image watcher, the prompt-cache keep-warm default (**Keep prompt caches warm**, off) and its cap (**Keep warm for at most**, 2 hours after a session's last prompt; see [Keeping Agents Running](Keeping-Agents-Running)), and
 Cloudflare tunnel controls including the tunnel URL. The **Diagnostics** group runs
 `codeman doctor` on the server and lists the agent CLIs, tmux, Node and the optional office
 tools with their versions and install hints (admin only in multi-user mode). In multi-user
@@ -196,7 +196,7 @@ Per session, from the tab.
 
 | Panel            | Contains                                                                                   |
 | ---------------- | ------------------------------------------------------------------------------------------- |
-| **Respawn**      | Auto-resume on usage limit, the respawn cycle configuration, presets, duration. See [Keeping Agents Running](Keeping-Agents-Running). |
+| **Respawn**      | Auto-resume on usage limit, the prompt cache readout and keep-warm switch, the respawn cycle configuration, presets, duration. See [Keeping Agents Running](Keeping-Agents-Running). |
 | **Session**      | Name, working directory, environment overrides, per-tab pop-out override.                    |
 | **Ralph / Todo** | Loop configuration, iteration and todo caps, circuit breaker reset. See [Autonomous Loops](Autonomous-Loops). |
 | **Summary**      | What this session has done: tokens, activity, run summary.                                   |