[Bug]: Claude provider shows auth timeout warning on Windows despite being authenticated

[Heinz-G]

Bug description

On Windows, the Claude provider shows a yellow "Needs attention – Could not verify Claude authentication status. Timed out while running command." warning even when the CLI is correctly installed and the user is fully authenticated.

Root cause

claude auth status is run with DEFAULT_TIMEOUT_MS = 4_000 (4s). On Windows, Node.js CLI tools have higher cold-start overhead, and auth status involves disk/account lookups that can exceed this limit on first run. The timeout fires before the command completes, producing the false-positive warning.

The --version probe (which is fast) shares the same 4s timeout — that one is fine. Only auth status needs more headroom.

Steps to reproduce

1. Install T3 Code on Windows
2. Install Claude CLI (npm i -g @anthropic-ai/claude-code) and authenticate (claude)
3. Open T3 Code settings → Providers
4. Observe Claude provider shows "Needs attention – Could not verify Claude authentication status. Timed out while running command."

Expected behaviour

Provider shows green "Ready" / authenticated status.

Environment

• OS: Windows 11
• Claude CLI: v2.0.20
• T3 Code: latest

Fix

PR #2272 adds AUTH_PROBE_TIMEOUT_MS = 10_000 and uses it specifically for the auth status check, leaving the --version probe unchanged.