[BUG] DEC private mode escape sequences leak as visible text in Ghostty (`[?630088]`, `[?630306]`)

Preflight Checklist

• [x] I have searched existing issues and this hasn't been reported yet
• [x] This is a single bug report (please file separate reports for different bugs)
• [x] I am using the latest version of Claude Code

What's Wrong?

DEC private mode escape sequences emitted by Claude Code are rendered as visible text artifacts in Ghostty terminal. Multiple high-numbered private mode sequences leak into the visible output:

• [?630088] — appears in the terminal output area
• [?630306] — appears in the status line / prompt area

These are CSI sequences (\e[?<number>h or \e[?<number>l) that Claude Code sends for terminal capability detection. Ghostty does not recognize these non-standard private mode numbers and renders them as literal text instead of silently discarding them.

This is similar to the previously reported (and closed) OSC color query leak (#12910), but involves DEC private mode set/reset sequences rather than OSC color queries.

Expected Behavior

These escape sequences should either:

1. Not be emitted to terminals that don't support them (e.g., detect Ghostty and skip), or
2. Be sent as proper queries (DECRQM) so the terminal can respond with "not recognized" rather than printing raw codes

How to Reproduce

1. Open Ghostty terminal (macOS, latest stable)
2. Run claude to start Claude Code
3. Interact normally — the [?630088] and [?630306] sequences appear as visible text artifacts in the output

Environment

• Claude Code version: latest (installed via npm)
• Terminal: Ghostty (stable, macOS)
• OS: macOS (Darwin 25.2.0)
• Shell: zsh with Starship prompt

Ghostty Config (relevant)

shell-integration = zsh
shell-integration-features = cursor,sudo,title

Screenshots

The artifacts appear inline with terminal output, e.g.:

~ main [?630306]  Opus 4.6  22% ctx  12:50