Open on startup by default and block duplicate workspace names

- LoginItem registers on the first bundled launch, so the app opens on
  startup without the user finding the toggle; a user who turns it off is
  never overridden. A never-registered bundle reports SMAppService .notFound
  rather than .notRegistered, so the guard only treats enabled/approval as
  done. --diag now prints the raw status.
- New Workspace refuses a name that already exists (case-insensitive) and
  says so inline; Return cannot bypass it. The composer's '# Name' still
  switches to the existing workspace, which is the quick-capture gesture.

Sources/Prompt/App.swift

@@ -61,6 +61,7 @@ final class AppDelegate: NSObject, NSApplicationDelegate {
 
         setupStatusItem()
         promptForAccessibilityIfNeeded()
+        LoginItem.autoRegisterOnce()
 
         // Let SwiftUI finish installing its main menu, then augment it.
         DispatchQueue.main.async { self.installMenuItems() }
@@ -140,6 +141,7 @@ final class AppDelegate: NSObject, NSApplicationDelegate {
     @MainActor
     private func writeDiagnostics() {
         var lines = ["Prompt diagnostics", "accessibility: \(AXIsProcessTrusted())"]
+        lines.append("login item status: \(SMAppService.mainApp.status.rawValue)")
         if let window = statusItem?.button?.window {
             lines.append("status item frame: \(window.frame)")
             if let screen = window.screen ?? NSScreen.main {
@@ -266,6 +268,31 @@ enum LoginItem {
         SMAppService.mainApp.status == .enabled
     }
 
+    /// Registers the app as a login item on first launch so Prompt opens on
+    /// startup without the user hunting for the toggle. Runs once (tracked in
+    /// UserDefaults) so a user who disables it is never overridden, and skips
+    /// `swift run`, where there is no bundle to register.
+    static func autoRegisterOnce() {
+        let key = "loginItem.autoRegistered"
+        guard Bundle.main.bundleIdentifier != nil,
+              !UserDefaults.standard.bool(forKey: key) else { return }
+        // A never-registered bundle reports `.notFound` on current macOS
+        // (BTM has no record), not `.notRegistered`, so only skip when there
+        // is nothing left to do.
+        switch SMAppService.mainApp.status {
+        case .enabled, .requiresApproval:
+            UserDefaults.standard.set(true, forKey: key)
+        default:
+            do {
+                try SMAppService.mainApp.register()
+                UserDefaults.standard.set(true, forKey: key)
+                NSLog("Prompt: registered as a login item (%d)", SMAppService.mainApp.status.rawValue)
+            } catch {
+                NSLog("Prompt: login item registration failed: %@", error.localizedDescription)
+            }
+        }
+    }
+
     static func toggle() -> Bool {
         let service = SMAppService.mainApp
         do {