feat(email): round-trip test — send via SMTP to the IMAP mailbox and confirm arrival

- emailIntakeService.roundTripTest(): sends a uniquely-tagged email through the
  saved SMTP config to the IMAP mailbox (imap_user), then polls IMAP up to 30s
  for that subject token; deletes the test message on arrival so it never hits
  the accounting inbox. Returns {ok, seconds, recipient} or a typed reason.
- route POST /admin/email/incoming-config/roundtrip (email.send)
- IMAP card: 'Round-trip test' button beside 'Test connection' + Save; toast
  reports recipient + delivery time. Distinct reasons mapped (smtp/imap
  unconfigured, send_failed, not_received→504).
- en/de strings

frontend/src/components/admin/IncomingMailConfigCard.tsx

@@ -11,7 +11,7 @@ import React, { useEffect, useState } from 'react';
 import { useQuery, useMutation, useQueryClient } from '@tanstack/react-query';
 import { useTranslation } from 'react-i18next';
 import { toast } from 'react-toastify';
-import { Save, Server, User, Lock, Eye, EyeOff, FolderSearch, PlugZap } from 'lucide-react';
+import { Save, Server, User, Lock, Eye, EyeOff, FolderSearch, PlugZap, Mailbox } from 'lucide-react';
 import { Button, Card, Input, Loading } from '../common';
 import { emailService, type IncomingMailConfig, type ImapFolder } from '../../services/email.service';
 
@@ -50,6 +50,12 @@ export const IncomingMailConfigCard: React.FC = () => {
     onError: (e: any) => toast.error(e?.response?.data?.error || e.message || t('email.incoming.testFailed', 'Connection failed.')),
   });
 
+  const roundTrip = useMutation({
+    mutationFn: () => emailService.roundTripIncoming(),
+    onSuccess: (r) => toast.success(t('email.incoming.roundTripOk', 'Round-trip OK — delivered to {{recipient}} in {{seconds}}s.', { recipient: r.recipient, seconds: r.seconds })),
+    onError: (e: any) => toast.error(e?.response?.data?.error || e.message || t('email.incoming.roundTripFailed', 'Round-trip test failed.')),
+  });
+
   const detect = useMutation({
     mutationFn: () => emailService.listIncomingFolders(cfg),
     onSuccess: (list) => {
@@ -158,7 +164,7 @@ export const IncomingMailConfigCard: React.FC = () => {
           <p className="mt-1 text-xs text-neutral-500 dark:text-neutral-400">{t('email.incoming.folderHint', 'Enter host, username and password, then Detect to list the mailbox folders.')}</p>
         </div>
 
-        <div className="flex gap-2">
+        <div className="flex flex-wrap gap-2">
           <Button
             variant="outline"
             onClick={() => test.mutate()}
@@ -169,7 +175,18 @@ export const IncomingMailConfigCard: React.FC = () => {
           >
             {t('email.incoming.test', 'Test connection')}
           </Button>
-          <Button variant="primary" onClick={() => save.mutate()} isLoading={save.isPending} leftIcon={<Save className="w-5 h-5" />} className="flex-1">
+          <Button
+            variant="outline"
+            onClick={() => roundTrip.mutate()}
+            isLoading={roundTrip.isPending}
+            disabled={!cfg.imap_host || !cfg.imap_user}
+            leftIcon={<Mailbox className="w-5 h-5" />}
+            className="whitespace-nowrap"
+            title={t('email.incoming.roundTripHint', 'Sends a test email via your SMTP settings to this mailbox and confirms it arrives. Save both first.') as string}
+          >
+            {t('email.incoming.roundTrip', 'Round-trip test')}
+          </Button>
+          <Button variant="primary" onClick={() => save.mutate()} isLoading={save.isPending} leftIcon={<Save className="w-5 h-5" />} className="flex-1 min-w-[12rem]">
             {t('email.incoming.save', 'Save Incoming Mail Settings')}
           </Button>
         </div>

frontend/src/i18n/locales/de.json

@@ -2482,6 +2482,10 @@
       "test": "Verbindung testen",
       "testOk": "Verbunden mit {{folder}} — {{messages}} Nachrichten, {{unseen}} ungelesen.",
       "testFailed": "Verbindung fehlgeschlagen.",
+      "roundTrip": "Rundlauf-Test",
+      "roundTripHint": "Sendet über die SMTP-Einstellungen eine Test-E-Mail an dieses Postfach und prüft, ob sie ankommt. Beide vorher speichern.",
+      "roundTripOk": "Rundlauf OK — an {{recipient}} zugestellt in {{seconds}}s.",
+      "roundTripFailed": "Rundlauf-Test fehlgeschlagen.",
       "port": "Port",
       "security": "Sicherheit",
       "ssl": "SSL/TLS",

frontend/src/i18n/locales/en.json

@@ -2055,6 +2055,10 @@
       "test": "Test connection",
       "testOk": "Connected to {{folder}} — {{messages}} messages, {{unseen}} unread.",
       "testFailed": "Connection failed.",
+      "roundTrip": "Round-trip test",
+      "roundTripHint": "Sends a test email via your SMTP settings to this mailbox and confirms it arrives. Save both first.",
+      "roundTripOk": "Round-trip OK — delivered to {{recipient}} in {{seconds}}s.",
+      "roundTripFailed": "Round-trip test failed.",
       "port": "Port",
       "security": "Security",
       "ssl": "SSL/TLS",

frontend/src/services/email.service.ts

@@ -102,6 +102,12 @@ export interface ImapTestResult {
   unseen: number;
 }
 
+export interface ImapRoundTripResult {
+  ok: boolean;
+  seconds?: number;
+  recipient?: string;
+}
+
 export interface ReceivedEmail {
   id: number;
   message_id: string | null;
@@ -150,6 +156,12 @@ export const emailService = {
     const response = await api.post<ImapTestResult>('/admin/email/incoming-config/test', config || {});
     return response.data;
   },
+  // End-to-end: send via SMTP to the IMAP mailbox and confirm it arrives.
+  // Uses saved config for both sides — no body. May take up to ~30s.
+  async roundTripIncoming(): Promise<ImapRoundTripResult> {
+    const response = await api.post<ImapRoundTripResult>('/admin/email/incoming-config/roundtrip', {});
+    return response.data;
+  },
   async listReceived(params: { page?: number; pageSize?: number } = {}): Promise<ReceivedEmailsResponse> {
     const response = await api.get<ReceivedEmailsResponse>('/admin/email/received', { params });
     return response.data;