fix(cron): make default timezone configurable via config.json

Refactor the hardcoded 'Asia/Shanghai' default timezone to be
configurable. Timezone resolution follows a 3-tier fallback:

1. schedule.TZ (per-job, set by LLM via cron_expr)
2. tools.cron.default_timezone in config.json (server-wide)
3. 'Asia/Shanghai' as ultimate fallback

Changes:
- config: add DefaultTimezone field to CronToolsConfig
- cron: add SetDefaultTimezone() method to CronService
- cron: update computeNextRun() to use 3-tier fallback
- gateway: wire config.Tools.Cron.DefaultTimezone to CronService
- test: add TestComputeNextRun_ServiceDefaultTZ to verify 3-tier

Usage in config.json:
  {"tools": {"cron": {"default_timezone": "US/Eastern"}}}

cmd/picoclaw/internal/gateway/helpers.go

@@ -230,6 +230,11 @@ func setupCronTool(
 	// Create cron service
 	cronService := cron.NewCronService(cronStorePath, nil)
 
+	// Apply default timezone from config for cron expressions
+	if cfg.Tools.Cron.DefaultTimezone != "" {
+		cronService.SetDefaultTimezone(cfg.Tools.Cron.DefaultTimezone)
+	}
+
 	// Create and register CronTool if enabled
 	var cronTool *tools.CronTool
 	if cfg.Tools.IsToolEnabled("cron") {

pkg/config/config.go

@@ -579,7 +579,8 @@ type WebToolsConfig struct {
 
 type CronToolsConfig struct {
 	ToolConfig         `    envPrefix:"PICOCLAW_TOOLS_CRON_"`
-	ExecTimeoutMinutes int `                                 env:"PICOCLAW_TOOLS_CRON_EXEC_TIMEOUT_MINUTES" json:"exec_timeout_minutes"` // 0 means no timeout
+	ExecTimeoutMinutes int    `                                 env:"PICOCLAW_TOOLS_CRON_EXEC_TIMEOUT_MINUTES" json:"exec_timeout_minutes"` // 0 means no timeout
+	DefaultTimezone    string `                                 env:"PICOCLAW_TOOLS_CRON_DEFAULT_TIMEZONE"      json:"default_timezone"`     // default timezone for cron expressions, e.g. "UTC"
 }
 
 type ExecConfig struct {

pkg/cron/service.go

@@ -66,6 +66,7 @@ type CronService struct {
 	running   bool
 	stopChan  chan struct{}
 	gronx     *gronx.Gronx
+	defaultTZ string // default timezone for cron expressions
 }
 
 func NewCronService(storePath string, onJob JobHandler) *CronService {
@@ -266,8 +267,11 @@ func (cs *CronService) computeNextRun(schedule *CronSchedule, nowMS int64) *int6
 
 		// Use gronx to calculate next run time
 		now := time.UnixMilli(nowMS)
-		// Apply timezone if specified, default to Asia/Shanghai
+		// Apply timezone: schedule.TZ > service default > "Asia/Shanghai"
 		tz := schedule.TZ
+		if tz == "" {
+			tz = cs.defaultTZ
+		}
 		if tz == "" {
 			tz = "Asia/Shanghai"
 		}
@@ -321,6 +325,14 @@ func (cs *CronService) SetOnJob(handler JobHandler) {
 	cs.onJob = handler
 }
 
+// SetDefaultTimezone sets the default timezone for cron expressions.
+// If empty, falls back to "Asia/Shanghai".
+func (cs *CronService) SetDefaultTimezone(tz string) {
+	cs.mu.Lock()
+	defer cs.mu.Unlock()
+	cs.defaultTZ = tz
+}
+
 func (cs *CronService) loadStore() error {
 	cs.store = &CronStore{
 		Version: 1,

pkg/cron/timezone_test.go

@@ -0,0 +1,144 @@
+package cron
+
+import (
+	"path/filepath"
+	"testing"
+	"time"
+)
+
+// TestComputeNextRun_CronTimezone verifies Patch #4: cron expressions should
+// respect the schedule.TZ field instead of always using UTC.
+func TestComputeNextRun_CronTimezone(t *testing.T) {
+	tmpDir := t.TempDir()
+	storePath := filepath.Join(tmpDir, "jobs.json")
+	cs := NewCronService(storePath, nil)
+
+	now := time.Date(2026, 3, 4, 0, 0, 0, 0, time.UTC) // midnight UTC
+	nowMS := now.UnixMilli()
+
+	// Cron expr "0 9 * * *" = daily at 9:00
+	tests := []struct {
+		name     string
+		tz       string
+		wantHour int // expected hour in UTC of the next run
+	}{
+		{
+			name:     "UTC timezone",
+			tz:       "UTC",
+			wantHour: 9, // 9:00 UTC
+		},
+		{
+			name:     "Asia/Shanghai timezone",
+			tz:       "Asia/Shanghai",
+			wantHour: 1, // 9:00 CST = 1:00 UTC
+		},
+		{
+			name:     "empty TZ defaults to Asia/Shanghai",
+			tz:       "",
+			wantHour: 1, // should default to Asia/Shanghai
+		},
+		{
+			name:     "US/Eastern timezone",
+			tz:       "US/Eastern",
+			wantHour: 14, // 9:00 EST = 14:00 UTC (or 13:00 during DST)
+		},
+	}
+
+	for _, tt := range tests {
+		t.Run(tt.name, func(t *testing.T) {
+			schedule := &CronSchedule{
+				Kind: "cron",
+				Expr: "0 9 * * *",
+				TZ:   tt.tz,
+			}
+
+			nextMS := cs.computeNextRun(schedule, nowMS)
+			if nextMS == nil {
+				t.Fatal("computeNextRun returned nil")
+			}
+
+			nextTime := time.UnixMilli(*nextMS).UTC()
+
+			if tt.name == "US/Eastern timezone" {
+				// Allow for DST variation (13 or 14)
+				if nextTime.Hour() != 13 && nextTime.Hour() != 14 {
+					t.Errorf("next run hour = %d, want 13 or 14 (UTC)", nextTime.Hour())
+				}
+			} else {
+				if nextTime.Hour() != tt.wantHour {
+					t.Errorf("next run hour = %d, want %d (UTC)", nextTime.Hour(), tt.wantHour)
+				}
+			}
+		})
+	}
+}
+
+// TestComputeNextRun_DefaultTZ_NotUTC verifies that when TZ is empty,
+// the computed next run differs from a pure UTC computation.
+func TestComputeNextRun_DefaultTZ_NotUTC(t *testing.T) {
+	tmpDir := t.TempDir()
+	storePath := filepath.Join(tmpDir, "jobs.json")
+	cs := NewCronService(storePath, nil)
+
+	now := time.Date(2026, 3, 4, 2, 0, 0, 0, time.UTC) // 02:00 UTC = 10:00 CST
+	nowMS := now.UnixMilli()
+
+	scheduleEmpty := &CronSchedule{Kind: "cron", Expr: "0 9 * * *", TZ: ""}
+	scheduleUTC := &CronSchedule{Kind: "cron", Expr: "0 9 * * *", TZ: "UTC"}
+
+	nextEmpty := cs.computeNextRun(scheduleEmpty, nowMS)
+	nextUTC := cs.computeNextRun(scheduleUTC, nowMS)
+
+	if nextEmpty == nil || nextUTC == nil {
+		t.Fatal("computeNextRun returned nil")
+	}
+
+	// With empty TZ (default Asia/Shanghai), 9:00 CST already passed (it's 10:00 CST),
+	// so next run should be tomorrow 9:00 CST = today 01:00 UTC + 24h.
+	// With UTC, 9:00 UTC hasn't happened yet (it's 02:00 UTC), so next run = today 9:00 UTC.
+	// They must differ.
+	if *nextEmpty == *nextUTC {
+		t.Errorf("empty TZ and UTC TZ produced same next run time, default TZ is not working")
+	}
+}
+
+// TestComputeNextRun_ServiceDefaultTZ verifies that SetDefaultTimezone()
+// takes effect when schedule.TZ is empty (3-tier fallback).
+func TestComputeNextRun_ServiceDefaultTZ(t *testing.T) {
+	tmpDir := t.TempDir()
+	storePath := filepath.Join(tmpDir, "jobs.json")
+	cs := NewCronService(storePath, nil)
+
+	// Set service-level default to US/Eastern
+	cs.SetDefaultTimezone("US/Eastern")
+
+	now := time.Date(2026, 3, 4, 0, 0, 0, 0, time.UTC)
+	nowMS := now.UnixMilli()
+
+	// Schedule with empty TZ should use service default (US/Eastern), not Asia/Shanghai
+	scheduleEmpty := &CronSchedule{Kind: "cron", Expr: "0 9 * * *", TZ: ""}
+	scheduleCST := &CronSchedule{Kind: "cron", Expr: "0 9 * * *", TZ: "Asia/Shanghai"}
+
+	nextEmpty := cs.computeNextRun(scheduleEmpty, nowMS)
+	nextCST := cs.computeNextRun(scheduleCST, nowMS)
+
+	if nextEmpty == nil || nextCST == nil {
+		t.Fatal("computeNextRun returned nil")
+	}
+
+	// US/Eastern 9:00 != Asia/Shanghai 9:00, so they must differ
+	if *nextEmpty == *nextCST {
+		t.Errorf("service default TZ (US/Eastern) and Asia/Shanghai produced same time; SetDefaultTimezone not working")
+	}
+
+	// Schedule with explicit TZ should override service default
+	scheduleExplicit := &CronSchedule{Kind: "cron", Expr: "0 9 * * *", TZ: "UTC"}
+	nextExplicit := cs.computeNextRun(scheduleExplicit, nowMS)
+	if nextExplicit == nil {
+		t.Fatal("computeNextRun returned nil")
+	}
+	nextUTC := time.UnixMilli(*nextExplicit).UTC()
+	if nextUTC.Hour() != 9 {
+		t.Errorf("explicit TZ=UTC: next run hour = %d, want 9", nextUTC.Hour())
+	}
+}