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/discord-gateway/src/modules/ai-moderation/ai-analysis-worker.ts

@@ -316,11 +316,13 @@ async function processBatch(job: {
   // The orchestrator handles text/media split + caching + parallel paths
   // internally, so a 20-message batch = 1 text LLM call (+1 media call
   // when media is present), not N per-message calls.
+  const analysisStart = Date.now();
   const moderationResult = await runModerationAnalysis({
     targets: readyMessages,
     contextBlock,
     attachments,
   });
+  const analysisDurationMs = Date.now() - analysisStart;
 
   const results = moderationResult.results.map((r) =>
     normalizeResult(
@@ -342,6 +344,7 @@ async function processBatch(job: {
       confidence: result.confidence,
       recommendedAction: result.recommendedAction,
       analyzedAt: Date.now(),
+      analysisDurationMs,
       error: result.status === "error" ? result.analysis : null,
     },
   }));

services/discord-gateway/src/modules/message-capture/messagesAnalysis.ts

@@ -26,6 +26,8 @@ export interface AIAnalysisUpdate {
   confidence?: number | null;
   recommendedAction?: MessageRecord["ai_recommended_action"] | null;
   analyzedAt?: number | null;
+  /** Wall-clock time the AI analysis (LLM call) took, in milliseconds. */
+  analysisDurationMs?: number | null;
   error?: string | null;
 }
 
@@ -42,6 +44,7 @@ function buildAIAnalysisSet(result: AIAnalysisUpdate, now?: number) {
     ai_confidence: result.confidence ?? result.score ?? null,
     ai_recommended_action: result.recommendedAction ?? null,
     ai_analyzed_at: result.analyzedAt ?? now ?? Date.now(),
+    ai_analysis_duration_ms: result.analysisDurationMs ?? null,
     ai_error: result.error ?? null,
   };
 }

services/discord-gateway/src/shared/database/schema.ts

@@ -62,6 +62,9 @@ export const pgMessagesTable = pgTable(
       enum: ["none", "monitor", "warn", "review", "delete", "escalate"],
     }),
     ai_analyzed_at: pgBigint("ai_analyzed_at", { mode: "number" }),
+    ai_analysis_duration_ms: pgBigint("ai_analysis_duration_ms", {
+      mode: "number",
+    }),
     ai_error: pgText("ai_error"),
   },
   (table) => ({