test(a2a_proxy): add parseIdleTimeoutEnv coverage

parseIdleTimeoutEnv was the only untested pure helper in a2a_proxy.go.
All other helpers (isSystemCaller, isUpstreamBusyError,
isUpstreamDeadStatus, normalizeA2APayload, detectPlatformInDocker)
are already covered in a2a_proxy_test.go. No other files in handlers/
had genuinely untested pure functions after this.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

workspace-server/internal/handlers/a2a_proxy_pure_test.go

@@ -0,0 +1,59 @@
+package handlers
+
+import (
+	"testing"
+	"time"
+)
+
+// =============================================================================
+// parseIdleTimeoutEnv
+// =============================================================================
+
+// parseIdleTimeoutEnv is the only untested pure helper in a2a_proxy.go.
+// Covered in the full proxy path via idleTimeoutDuration package var, but
+// the unit itself had no direct tests until this file.
+
+func TestParseIdleTimeoutEnv_Valid(t *testing.T) {
+	tests := []struct {
+		input string
+		want  time.Duration
+	}{
+		{"30", 30 * time.Second},
+		{"60", 60 * time.Second},
+		{"300", 5 * time.Minute},
+		{"3600", 1 * time.Hour},
+	}
+	for _, tt := range tests {
+		got := parseIdleTimeoutEnv(tt.input)
+		if got != tt.want {
+			t.Errorf("parseIdleTimeoutEnv(%q) = %v; want %v", tt.input, got, tt.want)
+		}
+	}
+}
+
+func TestParseIdleTimeoutEnv_Empty(t *testing.T) {
+	got := parseIdleTimeoutEnv("")
+	if got != defaultIdleTimeoutDuration {
+		t.Errorf("parseIdleTimeoutEnv(%q) = %v; want defaultIdleTimeoutDuration (%v)", "", got, defaultIdleTimeoutDuration)
+	}
+}
+
+func TestParseIdleTimeoutEnv_InvalidInteger(t *testing.T) {
+	cases := []string{"abc", "12.5", "1e3"}
+	for _, v := range cases {
+		got := parseIdleTimeoutEnv(v)
+		if got != defaultIdleTimeoutDuration {
+			t.Errorf("parseIdleTimeoutEnv(%q) = %v; want defaultIdleTimeoutDuration", v, got)
+		}
+	}
+}
+
+func TestParseIdleTimeoutEnv_NonPositive(t *testing.T) {
+	cases := []string{"0", "-1", "-30"}
+	for _, v := range cases {
+		got := parseIdleTimeoutEnv(v)
+		if got != defaultIdleTimeoutDuration {
+			t.Errorf("parseIdleTimeoutEnv(%q) = %v; want defaultIdleTimeoutDuration", v, got)
+		}
+	}
+}