fix(focus): probe socket liveness so stale daemon leftovers aren't picked

[rz1989s]

Problem

Launching pi in plain Ghostty printed an extension error every time:

Extension ".../cursor/extensions/cursor.ts"
error: connect ECONNREFUSED /Users/rector/.config/herdr/herdr.sock

…while the same launch inside tmux was silent.

Root cause

detect() in both HerdrFocusProvider and CmuxFocusProvider used fs.access (file-exists) to decide whether the multiplexer was running. A crashed/killed daemon leaves its socket file behind on disk, so exists-based detection mistook the stale leftover for a live server → start() opened a real connection → ECONNREFUSED.

Inside tmux the bug was masked: autoDetect returns at the TMUX_PANE step and never reaches the herdr probe.

Fix

• lib/focus/socket.ts (new) — probeSocket(path): opens a real connection and closes it. connect resolves only when a listener is accept()-ing; stale/missing/non-socket paths reject immediately. Never throws, 300 ms ceiling.
• herdr.detect() / cmux.detect() — use probeSocket instead of access. Path resolution (env/marker/candidates) stays existence-based; only liveness moved to the probe — clean separation.
• herdr.start() — swallows connect failure and degrades to always-focused, matching cmux's existing resilience, so an explicit /cursor provider herdr against a dead socket no longer breaks session_start.

Tests

• tests/focus/socket.test.ts (new): live / missing / stale-file / regular-file / post-close / timeout.
• herdr + cmux detect tests converted from writeFileSync stubs to real listening sockets, plus stale-leftover → false cases.
• index.test.ts: regression guard — stale herdr socket falls through to static.
• herdr: new start() resilience test (explicit dead socket → no throw, no onChange).

pnpm typecheck ✅ · pnpm test:run → 133 pass / 0 fail (was 122 / 1 failing — the pre-existing no env → static was already red on machines with a stale herdr.sock).

Semver

Patch (0.2.2 → 0.2.3): bugfix + resilience hardening, no API change.

After merge

Tag v0.2.3 → CI publishes → bump ~/.pi/agent/settings.json pin @0.2.2 → @0.2.3.