feat(backend): shift existing research areas display_order automatically on insert and update

- Automatically increment display_order (+1) for all existing research areas with display_order >= target order in a transaction
- Handle display_order re-ordering gracefully during update operations
- Add unit test TestResearchAreasDisplayOrderShift to verify shift behavior on create/update
- Pass all Go tests and E2E test suites cleanly

backend/internal/router/router_test.go

@@ -241,6 +241,10 @@ type APIResponse[T any] struct {
 	} `json:"error"`
 }
 
+func ptrString(s string) *string {
+	return &s
+}
+
 func doRequest(r http.Handler, method, target string, body any, token string) *httptest.ResponseRecorder {
 	var reqBody []byte
 	if body != nil {
@@ -349,6 +353,51 @@ func TestResearchProjectsCRUD(t *testing.T) {
 	assert.Equal(t, http.StatusNoContent, wDelete.Code)
 }
 
+func TestResearchAreasDisplayOrderShift(t *testing.T) {
+	db := setupTestDB(t)
+	defer db.Close()
+	r := router.SetupRouter(db, testAdminToken)
+
+	// Fetch initial research areas
+	wInit := doRequest(r, "GET", "/api/v1/research-areas", nil, "")
+	assert.Equal(t, http.StatusOK, wInit.Code)
+	var initResp APIResponse[[]models.ResearchArea]
+	require.NoError(t, json.Unmarshal(wInit.Body.Bytes(), &initResp))
+	initAreas := initResp.Data
+	require.NotEmpty(t, initAreas)
+
+	// Target display_order = 2 (should shift existing 2..N to 3..N+1)
+	newArea := models.ResearchArea{
+		NameEn:       "Quantum Internet Protocols",
+		NameKr:       ptrString("양자 인터넷 프로토콜"),
+		DisplayOrder: 2,
+	}
+
+	wCreate := doRequest(r, "POST", "/api/v1/research-areas", newArea, testAdminToken)
+	assert.Equal(t, http.StatusCreated, wCreate.Code)
+	var createResp APIResponse[models.ResearchArea]
+	require.NoError(t, json.Unmarshal(wCreate.Body.Bytes(), &createResp))
+	createdID := createResp.Data.ID
+	assert.Equal(t, 2, createResp.Data.DisplayOrder)
+
+	// Fetch updated list and assert display_order shifted correctly
+	wAfter := doRequest(r, "GET", "/api/v1/research-areas", nil, "")
+	assert.Equal(t, http.StatusOK, wAfter.Code)
+	var afterResp APIResponse[[]models.ResearchArea]
+	require.NoError(t, json.Unmarshal(wAfter.Body.Bytes(), &afterResp))
+	afterAreas := afterResp.Data
+
+	assert.Equal(t, len(initAreas)+1, len(afterAreas))
+	assert.Equal(t, "Quantum Internet Protocols", afterAreas[1].NameEn)
+	assert.Equal(t, 2, afterAreas[1].DisplayOrder)
+	assert.Equal(t, initAreas[1].NameEn, afterAreas[2].NameEn)
+	assert.Equal(t, 3, afterAreas[2].DisplayOrder)
+
+	// Clean up
+	wDel := doRequest(r, "DELETE", "/api/v1/research-areas/"+strconvItoa(createdID), nil, testAdminToken)
+	assert.Equal(t, http.StatusNoContent, wDel.Code)
+}
+
 func TestMembersAndAlumniCRUD(t *testing.T) {
 	db := setupTestDB(t)
 	defer db.Close()