## Summary

Fixes the `mise install` regression where a non-root user gets
`Permission denied` while rebuilding runtime symlinks against a
read-only system installs dir (typical Docker pattern: root populates
`/usr/local/share/mise/installs/` at build time, non-root user runs
`mise install` at runtime). Reported in
[#8596](#8596) and addressed by
[#9408](#9408) — this PR is an
alternative.

## Root cause

`rebuild_symlinks_in_dir` always called `remove_all` +
`make_symlink_or_file` on every existing runtime symlink, because the
"stale runtime dir" branch fired for symlinks too:

```rust
if from.exists() {
    if is_runtime_symlink(&from) && file::resolve_symlink(&from)?.unwrap_or_default() != to {
        // Branch 1 — only fires when the symlink target is wrong
        ...
    } else if from.file_name() != to.file_name() && !concrete_installs.contains(&from_name) {
        // Branch 2 — fires for ANY symlink because file_name("latest") != file_name("22.5.0")
        file::remove_all(&from)?;
    }
    ...
}
```

For a perfectly healthy `latest -> ./22.5.0`:

- Branch 1 fails (`target == to`).
- Branch 2 enters because `from.file_name()` (`"latest"`) differs from
`to.file_name()` (`"22.5.0"`) and `concrete_installs` doesn't include
`"latest"`.
- `remove_all` then runs on a healthy symlink — wasted I/O on every
install in the common case, hard failure when the dir is read-only.

## Fix

- Branch 1 now handles all runtime-symlink cases: rewrite if target
differs, otherwise `continue`.
- Branch 2 only fires for non-symlink stale dirs (its actual purpose —
cleanup for the 2026.4 regression that created real `latest/`
directories).
- Drop the position-based permission tolerance added in #8722. With
healthy symlinks taking the no-op path, read-only system dirs aren't
touched at all. If a write IS genuinely required and can't happen, fail
loudly so the user knows to fix permissions rather than silently ending
up with a stale `latest` after `mise install --system`.

## Why not [#9408](#9408)

That PR replaces position-based tolerance with category-based tolerance
(any non-`Local` dir gets permission errors swallowed) and silently
skips system/shared installs from user shims. Both:

- Mask the real bug (the false-positive `remove_all` above) instead of
fixing it. Even with category-based tolerance the install does I/O it
shouldn't, just hidden under a `debug!`.
- Trade one form of silent behavior for another. Reviewers (greptile,
gemini) flagged the shims change as a regression for the "root installs
once, all users get shims" pattern.
- Don't match the maintainer's stated policy: if `mise install` doesn't
touch system tools, it should work fine; if it does and the dir isn't
writable, it should fail loudly. This PR achieves both via a delta-based
no-op rather than tolerance.

## Tests

- `e2e/cli/test_install_system_readonly` — sets up `tiny@1.0.0` in user
dir, `tiny@2.1.0` in a fake system dir, `chmod -R a-w` the system dir,
then runs `mise install --force tiny@1.0.0` and `mise install
dummy@1.0.0`. Asserts both succeed AND that the `skipping symlink
update` / `Permission denied` / `failed rm` warnings do NOT appear in
output. Verified to fail before this fix (with `WARN skipping symlink
update for ...: failed rm -rf ... Permission denied`) and pass after.

## Test plan

- [x] `cargo test --bin mise` — 774 pass
- [x] `mise run lint-fix` — clean
- [x] `mise run test:e2e test_install_system_readonly` — pass
- [x] `mise run test:e2e test_install_system test_shared_install_dirs
test_install_before_ignores_stale_latest_dirs test_upgrade_latest_stale
test_reshim_with_shims_on_path test_uninstall test_install_before
test_install_short_ignores_full_backend_config test_install_dry_run
test_install_concurrent_via_exec` — all pass

🤖 Generated with [Claude Code](https://claude.com/claude-code)

<!-- CURSOR_SUMMARY -->
---

> [!NOTE]
> **Medium Risk**
> Changes runtime symlink rebuild behavior across both user and
shared/system install dirs, which could affect installs/reshims if edge
cases exist. It also removes permission-error swallowing for shared
dirs, so genuine mismatches in read-only system installs will now fail
loudly.
> 
> **Overview**
> Prevents `mise install` from rewriting existing runtime symlinks when
they already point at the desired target, avoiding unnecessary
deletes/recreates that previously caused `Permission denied` failures on
read-only system install directories.
> 
> Refactors shared/system install-dir enumeration into
`install_dirs_for` and removes the prior "skip on permission error"
behavior; shared/system dirs are now treated like user dirs, but the
rebuild path should be a no-op unless a change is actually needed. Adds
an E2E regression test (`test_install_system_readonly`) that simulates a
read-only system installs dir and asserts installs succeed without
system-dir warnings.
> 
> <sup>Reviewed by [Cursor Bugbot](https://cursor.com/bugbot) for commit
acf5798. Bugbot is set up for automated
code reviews on this repo. Configure
[here](https://www.cursor.com/dashboard/bugbot).</sup>
<!-- /CURSOR_SUMMARY -->

---------

Co-authored-by: Claude Opus 4.7 (1M context) <noreply@anthropic.com>