Implement exclusive lock protocol for daemon/external tool coexistence

- Add ExclusiveLock struct with JSON marshaling and validation
- Implement IsProcessAlive() with EPERM fail-safe behavior
- Add ShouldSkipDatabase() with stale lock cleanup
- Integrate lock checking into daemon sync cycle
- Return holder name on stale removal for better logging
- Case-insensitive hostname comparison
- Comprehensive unit tests (89.3% coverage)
- Documentation updates (ADVANCED.md, AGENTS.md)
- Add .beads/.exclusive-lock to .gitignore

Closes bd-115, bd-116, bd-117, bd-118, bd-119, bd-120, bd-121, bd-122

Amp-Thread-ID: https://ampcode.com/threads/T-0b835739-0d79-4ef9-aa62-8446a368c42d
Co-authored-by: Amp <amp@ampcode.com>

internal/types/process.go

@@ -0,0 +1,47 @@
+package types
+
+import (
+	"errors"
+	"os"
+	"strings"
+	"syscall"
+)
+
+// IsProcessAlive checks if a process with the given PID is alive on the given hostname.
+// If hostname doesn't match the current host, it returns true (cannot verify remote, assume alive).
+// If hostname matches the current host, it checks if the PID exists.
+// Permission errors are treated as "alive" (fail-safe: better to skip than wrongly remove a lock).
+func IsProcessAlive(pid int, hostname string) bool {
+	currentHost, err := os.Hostname()
+	if err != nil {
+		// Can't determine current hostname, assume process is alive (fail-safe)
+		return true
+	}
+
+	// Case-insensitive hostname comparison to handle FQDN vs short name differences
+	if !strings.EqualFold(hostname, currentHost) {
+		return true
+	}
+
+	// Check if process exists on local host
+	process, err := os.FindProcess(pid)
+	if err != nil {
+		// On Unix, FindProcess always succeeds, so this is unlikely
+		return false
+	}
+
+	// Send signal 0 to check if process exists without actually sending a signal
+	err = process.Signal(syscall.Signal(0))
+	if err == nil {
+		return true
+	}
+
+	// Only mark as dead on ESRCH (no such process)
+	// EPERM (permission denied) and other errors => assume alive (fail-safe)
+	var errno syscall.Errno
+	if errors.As(err, &errno) && errno == syscall.ESRCH {
+		return false
+	}
+
+	return true
+}