fix(dog): spawn session and set BD_ACTOR for dog dispatch

Recovered from reflog - these commits were lost during a rebase/force-push.

Dogs are directories with state files but no sessions. When `gt dog dispatch`
assigned work and sent mail, nothing executed because no session existed.

Changes:
1. Spawn tmux session after dispatch (gt-<town>-deacon-<dogname>)
2. Set BD_ACTOR=deacon/dogs/<name> so dogs can find their mail
3. Add dog case to AgentEnv for proper identity

Session spawn is non-blocking - if it fails, mail was sent and human can
manually start the session.

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

internal/cmd/dog.go

@@ -836,6 +836,44 @@ func runDogDispatch(cmd *cobra.Command, args []string) error {
 		return fmt.Errorf("sending plugin mail to dog: %w", err)
 	}
 
+	// Spawn a session for the dog to execute the work.
+	// Without a session, the dog's mail inbox is never checked.
+	// See: https://github.com/steveyegge/gastown/issues/XXX (dog dispatch doesn't execute)
+	t := tmux.NewTmux()
+	townName, err := workspace.GetTownName(townRoot)
+	if err != nil {
+		townName = "gt" // fallback
+	}
+	dogSessionName := fmt.Sprintf("gt-%s-deacon-%s", townName, targetDog.Name)
+
+	// Kill any stale session first
+	if has, _ := t.HasSession(dogSessionName); has {
+		_ = t.KillSessionWithProcesses(dogSessionName)
+	}
+
+	// Build startup command with initial prompt to check mail and execute plugin
+	initialPrompt := fmt.Sprintf("I am dog %s. Check my mail inbox with 'gt mail inbox' and execute the plugin instructions I received.", targetDog.Name)
+	startCmd := config.BuildAgentStartupCommand("dog", "", townRoot, targetDog.Path, initialPrompt)
+
+	// Create session from dog's directory
+	if err := t.NewSessionWithCommand(dogSessionName, targetDog.Path, startCmd); err != nil {
+		if !dogDispatchJSON {
+			fmt.Printf("  Warning: could not spawn dog session: %v\n", err)
+		}
+		// Non-fatal: mail was sent, dog is marked as working, but no session to execute
+		// The deacon or human can manually start the session later
+	} else {
+		// Set environment for the dog session
+		envVars := config.AgentEnv(config.AgentEnvConfig{
+			Role:      "dog",
+			AgentName: targetDog.Name,
+			TownRoot:  townRoot,
+		})
+		for k, v := range envVars {
+			_ = t.SetEnvironment(dogSessionName, k, v)
+		}
+	}
+
 	// Success - output result
 	if dogDispatchJSON {
 		return json.NewEncoder(os.Stdout).Encode(result)