fix(backend): enforce gapless 1..N reindexing normalization across research areas CRUD

- Implement normalizeResearchAreasOrder inside transactions for Create, Update, and Delete operations
- Prevent duplicate or corrupted display_order overlaps when inserting/moving items
- Clean up existing database display_order sequence
- Pass all unit tests and E2E test gates cleanly

backend/internal/router/router_test.go

@@ -358,15 +358,21 @@ func TestResearchAreasDisplayOrderShift(t *testing.T) {
 	defer db.Close()
 	r := router.SetupRouter(db, testAdminToken)
 
-	// Fetch initial research areas
+	// 1. 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
+	initCount := len(initAreas)
 	require.NotEmpty(t, initAreas)
 
-	// Target display_order = 2 (should shift existing 2..N to 3..N+1)
+	// Verify initial areas are contiguous 1..N
+	for idx, a := range initAreas {
+		assert.Equal(t, idx+1, a.DisplayOrder)
+	}
+
+	// 2. Create at target display_order = 2 (should shift existing 2..N to 3..N+1 and normalize 1..N+1)
 	newArea := models.ResearchArea{
 		NameEn:       "Quantum Internet Protocols",
 		NameKr:       ptrString("양자 인터넷 프로토콜"),
@@ -380,22 +386,52 @@ func TestResearchAreasDisplayOrderShift(t *testing.T) {
 	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
+	// Fetch updated list and assert display_order shifted correctly and gapless
+	wAfterCreate := doRequest(r, "GET", "/api/v1/research-areas", nil, "")
+	assert.Equal(t, http.StatusOK, wAfterCreate.Code)
+	var afterCreateResp APIResponse[[]models.ResearchArea]
+	require.NoError(t, json.Unmarshal(wAfterCreate.Body.Bytes(), &afterCreateResp))
+	afterCreateAreas := afterCreateResp.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)
+	assert.Equal(t, initCount+1, len(afterCreateAreas))
+	assert.Equal(t, "Quantum Internet Protocols", afterCreateAreas[1].NameEn)
+	for idx, a := range afterCreateAreas {
+		assert.Equal(t, idx+1, a.DisplayOrder, "Item %d should have display_order %d", idx, idx+1)
+	}
 
-	// Clean up
+	// 3. Update display_order of the created item from position 2 to position 5
+	updateArea := models.ResearchArea{
+		NameEn:       "Quantum Internet Protocols Updated",
+		DisplayOrder: 5,
+	}
+	wUpdate := doRequest(r, "PUT", "/api/v1/research-areas/"+strconvItoa(createdID), updateArea, testAdminToken)
+	assert.Equal(t, http.StatusOK, wUpdate.Code)
+
+	wAfterUpdate := doRequest(r, "GET", "/api/v1/research-areas", nil, "")
+	var afterUpdateResp APIResponse[[]models.ResearchArea]
+	require.NoError(t, json.Unmarshal(wAfterUpdate.Body.Bytes(), &afterUpdateResp))
+	afterUpdateAreas := afterUpdateResp.Data
+
+	assert.Equal(t, initCount+1, len(afterUpdateAreas))
+	assert.Equal(t, "Quantum Internet Protocols Updated", afterUpdateAreas[4].NameEn)
+	assert.Equal(t, 5, afterUpdateAreas[4].DisplayOrder)
+	for idx, a := range afterUpdateAreas {
+		assert.Equal(t, idx+1, a.DisplayOrder, "Item %d should have contiguous display_order %d after update", idx, idx+1)
+	}
+
+	// 4. Delete the item and verify gapless 1..N normalization back to initCount
 	wDel := doRequest(r, "DELETE", "/api/v1/research-areas/"+strconvItoa(createdID), nil, testAdminToken)
 	assert.Equal(t, http.StatusNoContent, wDel.Code)
+
+	wAfterDel := doRequest(r, "GET", "/api/v1/research-areas", nil, "")
+	var afterDelResp APIResponse[[]models.ResearchArea]
+	require.NoError(t, json.Unmarshal(wAfterDel.Body.Bytes(), &afterDelResp))
+	afterDelAreas := afterDelResp.Data
+
+	assert.Equal(t, initCount, len(afterDelAreas))
+	for idx, a := range afterDelAreas {
+		assert.Equal(t, idx+1, a.DisplayOrder, "Item %d should have contiguous display_order %d after delete", idx, idx+1)
+	}
 }
 
 func TestMembersAndAlumniCRUD(t *testing.T) {