feat: surface AI analysis duration across gateway, backend, and FE

Adds per-message AI moderation analysis time (ai_analysis_duration_ms)
so operators can see how long the LLM took to moderate each message.

Gateway:
- messagesTable: new ai_analysis_duration_ms (bigint) column.
- AIAnalysisUpdate + buildAIAnalysisSet: carry analysisDurationMs through
  both single and bulk update paths.
- ai-analysis-worker: measure wall-clock time around runModerationAnalysis
  and attach it to every result in the batch.

Backend:
- Mirror schema column; messageMapper maps ai_analysis_duration_ms;
  moderation-types + MappedMessage expose it.

Frontend:
- message.ts type gains ai_analysis_duration_ms.
- AiBadge (messages view) shows 'status · 1.2s' when duration is present;
  analysis view badge mirrors the same formatting.

DB:
- scripts/add-ai-analysis-duration.sql (idempotent ADD COLUMN IF NOT EXISTS).

No behavior change for moderation logic; null until new gateway build
records values.

services/frontend/src/app/(dashboard)/analysis/view.tsx

@@ -18,6 +18,12 @@ function aiTone(
   return "neutral";
 }
 
+/** Human-readable analysis duration, e.g. 850ms / 1.2s / 3.4s. */
+function formatAnalysisDuration(ms: number): string {
+  if (ms < 1000) return `${Math.round(ms)}ms`;
+  return `${(ms / 1000).toFixed(1)}s`;
+}
+
 export function AnalysisView() {
   const [query, setQuery] = useState("");
   const search = useMessageSearch(query, query.trim().length >= 2);
@@ -99,7 +105,10 @@ export function AnalysisView() {
                       </span>
                       {m.ai_status && (
                         <Badge tone={aiTone(m.ai_status)} className="ml-auto">
-                          {m.ai_status}
+                          {m.ai_analysis_duration_ms &&
+                          m.ai_analysis_duration_ms > 0
+                            ? `${m.ai_status} · ${formatAnalysisDuration(m.ai_analysis_duration_ms)}`
+                            : m.ai_status}
                         </Badge>
                       )}
                     </div>

services/frontend/src/app/(dashboard)/messages/view.tsx

@@ -174,7 +174,10 @@ export function MessagesView({
                       )}
                     </div>
                   </div>
-                  <AiBadge status={m.ai_status} />
+                  <AiBadge
+                    status={m.ai_status}
+                    durationMs={m.ai_analysis_duration_ms}
+                  />
                 </button>
               ))}
             </div>
@@ -207,7 +210,13 @@ export function MessagesView({
   );
 }
 
-function AiBadge({ status }: { status?: AiStatus | null }) {
+function AiBadge({
+  status,
+  durationMs,
+}: {
+  status?: AiStatus | null;
+  durationMs?: number | null;
+}) {
   if (!status) return null;
   const tone = aiTone(status);
   const icon =
@@ -222,14 +231,25 @@ function AiBadge({ status }: { status?: AiStatus | null }) {
     ) : (
       <AlertTriangle className="size-3" />
     );
+  const label =
+    durationMs && durationMs > 0
+      ? `${status} · ${formatDuration(durationMs)}`
+      : status;
+
   return (
     <Badge tone={tone} dot={status === "processing" || status === "pending"}>
       {icon}
-      {status}
+      {label}
     </Badge>
   );
 }
 
+/** Human-readable analysis duration, e.g. 850ms / 1.2s / 3.4s. */
+function formatDuration(ms: number): string {
+  if (ms < 1000) return `${Math.round(ms)}ms`;
+  return `${(ms / 1000).toFixed(1)}s`;
+}
+
 function MessageDetail({
   m,
   attachments,
@@ -250,7 +270,10 @@ function MessageDetail({
           </div>
         </div>
         <div className="ml-auto">
-          <AiBadge status={m.ai_status} />
+          <AiBadge
+            status={m.ai_status}
+            durationMs={m.ai_analysis_duration_ms}
+          />
         </div>
       </div>
 

services/frontend/src/lib/types/message.ts

@@ -130,6 +130,7 @@ export interface MessageRecord {
   ai_recommended_action?: AiRecommendedAction | null;
   ai_error?: string | null;
   ai_analyzed_at?: number | null;
+  ai_analysis_duration_ms?: number | null;
   /** Detail-only: number of past edits (message_edits snapshots) */
   edit_count?: number;
   /** Detail-only: previous content snapshots, newest first */