Install-method detection

Nested lingtai-update reference. Detection is deliberately conservative.

1. Read ~/.lingtai-tui/install.json (or the configured global directory) and

validate the lingtai.tui.install/v1 metadata. Matching the running executable, including the managed lingtai-tui alias, makes this a source/user-local install.

1. Otherwise, recognize Homebrew from the executable path and Homebrew

prefixes/environment, including /opt/homebrew, /usr/local, and /home/linuxbrew/.linuxbrew patterns.

1. Otherwise classify it as unknown/other. Do not run brew merely because it

is available, and do not infer source ownership from a path alone when source metadata says otherwise.

Symlinks matter: a manually linked or development binary may not be the Homebrew Cellar copy. Check lingtai-tui doctor output and the resolved executable before choosing a manual command. This metadata is diagnostic and routing information, not a credential store.

A Homebrew-detected install is not upgraded through brew: /update-tui and lingtai-tui self-update migrate it to the native installer instead (see reference/update-tui/SKILL.md). Migrating installs and verifies a native binary, but does not by itself guarantee the shell will run it: on a standard Apple-Silicon Homebrew host, /opt/homebrew/bin is earlier on PATH than install.sh's default native bin dir, so the resolved lingtai-tui can still be the old Homebrew one even after a "successful" install. Detection accounts for this directly — when valid native (install.json) metadata exists but does not match the currently resolved executable, and that executable still resolves as Homebrew, detection reports DuplicateNativeInstall instead of silently treating it as a plain fresh Homebrew install: the method stays Homebrew (the resolved binary really is still Homebrew's), and every update entry point surfaces "native installed, migration not complete, remove Homebrew yourself" instead of re-prompting or re-running the installer. Only once the resolved executable actually matches the native metadata does detection report the install as source/user-local from then on — until then, the Homebrew Cellar copy stays on disk. Removing it always requires a concrete interactive "y" to the "Remove the old Homebrew installation now?" prompt at startup (see reference/update-tui/SKILL.md); self-update and doctor never uninstall it themselves, only report the manual command.