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.test.ts

@@ -8,6 +8,7 @@ const mocks = vi.hoisted(() => ({
   updateSessionStore: vi.fn(),
   agentCommand: vi.fn(),
   registerAgentRunContext: vi.fn(),
+  loadConfigReturn: {} as Record<string, unknown>,
 }));
 
 vi.mock("../session-utils.js", () => ({
@@ -32,7 +33,7 @@ vi.mock("../../commands/agent.js", () => ({
 }));
 
 vi.mock("../../config/config.js", () => ({
-  loadConfig: () => ({}),
+  loadConfig: () => mocks.loadConfigReturn,
 }));
 
 vi.mock("../../agents/agent-scope.js", () => ({
@@ -115,6 +116,62 @@ describe("gateway agent handler", () => {
     expect(capturedEntry?.claudeCliSessionId).toBe(existingClaudeCliSessionId);
   });
 
+  it("injects a timestamp into the message passed to agentCommand", async () => {
+    vi.useFakeTimers();
+    vi.setSystemTime(new Date("2026-01-29T01:30:00.000Z")); // Wed Jan 28, 8:30 PM EST
+    mocks.agentCommand.mockReset();
+
+    mocks.loadConfigReturn = {
+      agents: {
+        defaults: {
+          userTimezone: "America/New_York",
+          timeFormat: "12",
+        },
+      },
+    };
+
+    mocks.loadSessionEntry.mockReturnValue({
+      cfg: mocks.loadConfigReturn,
+      storePath: "/tmp/sessions.json",
+      entry: {
+        sessionId: "existing-session-id",
+        updatedAt: Date.now(),
+      },
+      canonicalKey: "agent:main:main",
+    });
+    mocks.updateSessionStore.mockResolvedValue(undefined);
+    mocks.agentCommand.mockResolvedValue({
+      payloads: [{ text: "ok" }],
+      meta: { durationMs: 100 },
+    });
+
+    const respond = vi.fn();
+    await agentHandlers.agent({
+      params: {
+        message: "Is it the weekend?",
+        agentId: "main",
+        sessionKey: "agent:main:main",
+        idempotencyKey: "test-timestamp-inject",
+      },
+      respond,
+      context: makeContext(),
+      req: { type: "req", id: "ts-1", method: "agent" },
+      client: null,
+      isWebchatConnect: () => false,
+    });
+
+    // Wait for the async agentCommand call
+    await vi.waitFor(() => expect(mocks.agentCommand).toHaveBeenCalled());
+
+    const callArgs = mocks.agentCommand.mock.calls[0][0];
+    expect(callArgs.message).toMatch(
+      /^\[.*Wednesday.*January 28.*2026.*8:30 PM.*\] Is it the weekend\?$/,
+    );
+
+    mocks.loadConfigReturn = {};
+    vi.useRealTimers();
+  });
+
   it("handles missing cliSessionIds gracefully", async () => {
     mocks.loadSessionEntry.mockReturnValue({
       cfg: {},