Archive/delete of Claude Code sessions is per-client — desktop's archive flag is invisible to the VS Code extension

Using the Claude Desktop Code tab and the Claude Code VS Code extension against the same project is a constant cleanup tax: I archive a conversation in the desktop app, it still shows up in the extension; clicking it there silently un-archives it back in the desktop app; deleting it in the extension leaves it sitting in the desktop list. Every tidy-up has to be done twice, and the two clients actively undo each other.

The cause looks structural: both clients read the same ~/.claude/projects/<project>/*.jsonl files, but the archived state is stored only in the desktop app's Electron localStorage (%APPDATA%\Claude\Local Storage\leveldb contains "archived":false records). Nothing in the shared ~/.claude/ tree records it, so the extension has no way to read or write that flag.

The CLI is a third case with no cleanup at all: /resume enumerates every .jsonl regardless, and the only removal tool is claude project purge, which wipes an entire project's transcripts, tasks, and file history.

Request: move archive/delete state into shared state under ~/.claude/ (a per-project sidecar, or a field in the session index) that every client reads and writes — then the extension can hide archived sessions and offer Archive alongside Delete, and /resume can honor it too.

Env: Windows 11, Claude Code 2.1.237 (desktop Code tab) / 2.1.234 (CLI), VS Code extension.

Refs #62980, #65674, #66229, #49775 (all closed and locked; filing new per the bot's instruction).