feat(gateway): inject timestamps into agent handler messages

Messages arriving through the gateway agent method (TUI, web, spawned
subagents, sessions_send, heartbeats) now get a timestamp prefix
automatically. This gives all agent contexts date/time awareness
without modifying the system prompt (which is cached for stability).

Channel messages (Discord, Telegram, etc.) already have timestamps
via envelope formatting in a separate code path and never reach
the agent handler, so there is no double-stamping risk.

Cron jobs also inject their own 'Current time:' prefix and are
detected and skipped.

Extracted as a pure function (injectTimestamp) with 12 unit tests
covering: timezone handling, 12/24h format, midnight boundaries,
envelope detection, cron detection, and empty messages.

Integration test verifies the agent handler wires it in correctly.

Closes #3658
Refs: #1897, #1928, #2108

src/gateway/server-methods/agent.ts

@@ -2,6 +2,7 @@ import { randomUUID } from "node:crypto";
 import { agentCommand } from "../../commands/agent.js";
 import { listAgentIds } from "../../agents/agent-scope.js";
 import { loadConfig } from "../../config/config.js";
+import { injectTimestamp, timestampOptsFromConfig } from "./agent-timestamp.js";
 import {
   resolveAgentIdFromSessionKey,
   resolveExplicitAgentSessionKey,
@@ -136,6 +137,13 @@ export const agentHandlers: GatewayRequestHandlers = {
         return;
       }
     }
+
+    // Inject timestamp into messages that don't already have one.
+    // Channel messages (Discord, Telegram, etc.) get timestamps via envelope
+    // formatting in a separate code path — they never reach this handler.
+    // See: https://github.com/moltbot/moltbot/issues/3658
+    message = injectTimestamp(message, timestampOptsFromConfig(cfg));
+
     const isKnownGatewayChannel = (value: string): boolean => isGatewayMessageChannel(value);
     const channelHints = [request.channel, request.replyChannel]
       .filter((value): value is string => typeof value === "string")