[FEATURE] Auto mode: fire PushNotification automatically when classifier denies an action

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet. Closest neighbors (#38500, #37153, #60433) cover hooks / mobile approval for manual permission prompts; none target automatic out-of-band notification on auto-mode classifier denial.
• [x] This is a single feature request.

Problem Statement

In practice I leave an auto-mode session running, walk away, and come back hours later to find that the classifier blocked the third action of the run. Everything since has been Claude trying alternatives or sitting paused after the 3-consecutive / 20-total fallback. The whole productivity win of unattended runs is lost because I didn't know I needed to act.

This bites hardest exactly where auto mode is most valuable: long-running tasks, overnight runs, and operators using Remote Control / mobile.

Proposed Solution

When auto mode's classifier denies an action, Claude Code should automatically invoke the existing PushNotification tool so the operator gets pinged on whatever surface they've configured (desktop, and via Remote Control, mobile).

Priority

High - Significant impact on productivity

Feature Category

Interactive mode (TUI)

Use Case Example

1. I kick off claude --permission-mode auto on a refactor that will run ~45 minutes and walk to lunch.
2. Mid-run, Claude tries to install a transitive dep not in the lockfile. Classifier denies it (correctly).
3. Claude tries two alternative installs; classifier denies both. Fallback trips, auto mode pauses.
4. Today: no signal reaches me until I'm back at my desk. The run has been idle for 30 minutes.
5. With this feature: I get a PushNotification on my Claude app to grant permission. I approve from my phone and the run resumes immediately.