diplomail (Stage B): admin/owner sends + lifecycle hooks

Item 7 of the spec wants game-state and membership-state changes to
land as durable inbox entries the affected players can re-read after
the fact — push alone times out of the 5-minute ring buffer. Stage B
adds the admin-kind send matrix (owner-driven via /user, site-admin
driven via /admin) plus the lobby lifecycle hooks: paused / cancelled
emit a broadcast system mail to active members, kick / ban emit a
single-recipient system mail to the affected user (which they keep
read access to even after the membership row is revoked, per item 8).

Migration relaxes diplomail_messages_kind_sender_chk so an owner
sending kind=admin keeps sender_kind=player; the new
LifecyclePublisher dep on lobby.Service is wired through a thin
adapter in cmd/backend/main, mirroring how lobby's notification
publisher is plumbed today.

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

backend/internal/lobby/games.go

@@ -10,6 +10,7 @@ import (
 	"galaxy/cronutil"
 
 	"github.com/google/uuid"
+	"go.uber.org/zap"
 )
 
 // CreateGameInput is the parameter struct for Service.CreateGame.
@@ -441,9 +442,43 @@ func (s *Service) transition(ctx context.Context, callerUserID *uuid.UUID, calle
 			return updated, fmt.Errorf("post-commit %s: %w", rule.Reason, err)
 		}
 	}
+	s.emitGameLifecycleMail(ctx, updated, callerIsAdmin, rule)
 	return updated, nil
 }
 
+// emitGameLifecycleMail asks the diplomail publisher to drop a
+// system-mail entry whenever a state change is user-visible. Only
+// the `paused` and `cancelled` transitions emit mail today (the spec
+// names them explicitly); `running`/`finished`/etc. are signalled by
+// other channels and do not need a durable inbox entry.
+func (s *Service) emitGameLifecycleMail(ctx context.Context, game GameRecord, callerIsAdmin bool, rule transitionRule) {
+	var kind string
+	switch rule.To {
+	case GameStatusPaused:
+		kind = LifecycleKindGamePaused
+	case GameStatusCancelled:
+		kind = LifecycleKindGameCancelled
+	default:
+		return
+	}
+	actor := "the game owner"
+	if callerIsAdmin {
+		actor = "an administrator"
+	}
+	ev := LifecycleEvent{
+		GameID: game.GameID,
+		Kind:   kind,
+		Actor:  actor,
+		Reason: rule.Reason,
+	}
+	if err := s.deps.Diplomail.PublishLifecycle(ctx, ev); err != nil {
+		s.deps.Logger.Warn("publish lifecycle mail failed",
+			zap.String("game_id", game.GameID.String()),
+			zap.String("kind", kind),
+			zap.Error(err))
+	}
+}
+
 // checkOwner enforces ownership semantics:
 //
 // - callerIsAdmin == true → always allowed (admin force-start, etc.).