docs: tighten the inbox, push and idle-detection wording

Follow-up to the previous commit, after a fact check against the code.

- Answer surfaces: the header bell answers Claude prompts, the phone
  strip only Claude dialogs (and only with the inbox on), and push
  Approve/Deny only a Claude permission prompt (PUSH_EVENT_MAP gives
  actions to permission_prompt alone). README, its Chinese mirror, the
  notifications page and the phone guide now say so.
- Antigravity and Grok Build: after the 3 s launch settle, idle is armed
  only by Claude's prompt marker and working only by a braille spinner,
  so "idle when output stops changing" was wrong. The wiki, the grok
  guide and the API reference now call their status unreliable.
- DeepSeek Harness: its idle and working reports become stop and
  agent_working, which drive respawn, waits and the inbox but not the
  session's own working/idle display. Keeping Agents Running says that,
  and that the respawn path needs the status bridge.
- The wait section of the API reference no longer calls exit the only
  dependable signal for screen-checked modes; the inbox page names how
  Codex and DeepSeek items clear; Core Concepts and Hooks And
  Integrations no longer call DeepSeek "the one" exception; Agent CLIs
  notes that Codex dialogs still raise the red alert; the inbox design
  doc notes the alert-only items.
- Skill: stop and blocked list deepseek, and SKILL.md no longer says
  codex has neither a transcript nor blocked. Mirrored with
  scripts/sync-plugin.mjs.

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

docs/grok-integration.md

@@ -121,10 +121,12 @@ the remote host, `grok login --device-auth` exists for exactly this.
 
 ## Known gaps
 
-- **No idle/completion hook yet.** Idle detection falls back to output-stabilization,
-  as it does for Antigravity: the registry entry declares no `workDetect` prompt marker
-  or working line, which is what gives Codex, OpenCode, Gemini, Pi and OMP a screen
-  check instead. Grok has a hooks system, so a Codeman hook POSTing to
+- **No idle/completion hook yet.** The registry entry declares no `workDetect` prompt
+  marker or working line, which is what gives Codex, OpenCode, Gemini, Pi and OMP a
+  screen check. So Codeman calls a new grok pane idle 3 seconds after launch, and after
+  that it has only generic cues: Claude's `❯` marker, which grok does not print (see the
+  cron item below), and a braille spinner glyph. Working and idle are unreliable for
+  grok, as for Antigravity. Grok has a hooks system, so a Codeman hook POSTing to
   `/api/hook-event` is the highest-value follow-up.
 - **No response viewer.** Grok writes ACP JSONL sessions under
   `~/.grok/sessions/<encoded-cwd>/<session-id>/updates.jsonl`; nothing reads them yet.