Pet overlay forces macOS to auto-switch back to Codex's Space when navigating to another Space

What platform is your computer?

macOS (Darwin 25.5.0)

What issue are you seeing?

After waking the Codex pet, navigating to any other macOS Space causes the system to immediately auto-switch back to the Space where Codex's main window lives. Every attempt to focus another Space yanks me back to Codex within ~1–2 seconds, making multi-Space workflow impractical while the pet is awake.

Observations from a screen recording of the repro (attached as a comment):

• The pet sprite is visible on every Space, consistent with canJoinAllSpaces-style window behavior.
• Switching to a Space that contains only a Finder window still triggers the auto-switch back.
• The trip back happens fast enough that it interrupts the Mission Control transition animation.

Hypothesis: the pet overlay is not actually a fully non-activating panel, so when it animates / gains focus / receives input on the new Space, it activates the Codex app process. macOS then applies the "When switching to an application, switch to a Space with open windows for the application" rule (System Settings → Desktop & Dock → Mission Control) and slides back to Codex's home Space.

This may share a root cause with #20706, which logs NSWindow does not support nonactivating panel styleMask 0x80 — suggesting the intended non-activating mask is not being applied on macOS. See also #20831 for an unrelated but pet-overlay-adjacent issue.

What steps can reproduce the bug?

1. On a multi-Space macOS setup (e.g. 6 Spaces), assign Codex's main window to one Space.
2. Wake the pet with /pet.
3. Switch to a different Space that does not contain Codex's main window — via Mission Control, swipe, or Ctrl+arrows. A Space with only a Finder window is enough.
4. Within ~1–2 seconds, macOS automatically slides back to Codex's home Space.

What is the expected behavior?

Switching to another Space should stay on that Space. The pet overlay should never implicitly reactivate the Codex app or trigger Spaces switching when the user navigates away.

Additional information

• macOS Spaces: 6 Spaces, Codex pinned to one of them; pet overlay floats on all Spaces.
• Likely-related setting: System Settings → Desktop & Dock → Mission Control → "When switching to an application, switch to a Space with open windows for the application". Disabling this may be a workaround, but the overlay should not be triggering app activation in the first place.
• Screen recording reproducing the bug will be attached as a comment.