1：Experimental Design and Method Selection:

The nanoparticles were synthesized using a co-precipitation method. The effects of dopant concentrations (Sc3+, Er3+, Yb3+) on properties were studied. Theoretical models included the Scherrer equation for crystalline size calculation and the equation IUC = Pn for up-conversion luminescence mechanism analysis.

2：Sample Selection and Data Sources:

Samples were prepared with varying stoichiometric ratios of Scx:TGG (x = 26–52 mol%), Ery:TGG (y =

3：25–25 mol%), and Ybz:

TGG (z = 0–6.0 mol%). Rare-earth oxides and gallium oxide were used as starting materials.

4：0 mol%). Rare-earth oxides and gallium oxide were used as starting materials. List of Experimental Equipment and Materials:

3. List of Experimental Equipment and Materials: Equipment included X-ray diffractometer (XRD-7000, SHIMADZU), scanning electron microscope (SEM, EVOMA-25, ZEISS), energy-dispersive X-ray spectrometer (EDS, X-Max50), spectrofluorometer (Model RF-5301PC, SHIMADZU) with 980-nm laser diode, and vibrating sample magnetometer (VSM, LAKE-SHORE 7410). Materials included rare-earth oxides (analytically pure), gallium oxide (Ga2O3, 99.99%), nitric acid, hydrochloric acid, hydrogen peroxide (H2O2), ammonia solution (NH3·H2O), ammonium bicarbonate (NH4HCO3), deionized water, and alcohol.

5：0). Materials included rare-earth oxides (analytically pure), gallium oxide (Ga2O3, 99%), nitric acid, hydrochloric acid, hydrogen peroxide (H2O2), ammonia solution (NH3·H2O), ammonium bicarbonate (NH4HCO3), deionized water, and alcohol. Experimental Procedures and Operational Workflow:

4. Experimental Procedures and Operational Workflow: Rare-earth oxides were dissolved in acids, with Tb nitrate treated with H2O2 to convert Tb4+ to Tb3+. The pH was adjusted to 2.0 using NH3·H2O. A precipitant solution of NH4HCO3 was prepared. The nitrate solution was added dropwise to the precipitant, stirred, left for 24 h, centrifuged, washed, dried at 80 °C for 48 h, and calcined at 1000–1200 °C for 3 h.

6：3+. The pH was adjusted to 0 using NH3·H2O. A precipitant solution of NH4HCO3 was prepared. The nitrate solution was added dropwise to the precipitant, stirred, left for 24 h, centrifuged, washed, dried at 80 °C for 48 h, and calcined at 1000–1200 °C for 3 h. Data Analysis Methods:

5. Data Analysis Methods: XRD for phase identification and crystalline size (Scherrer equation), SEM for morphology, EDS for chemical composition, photoluminescence spectroscopy for up-conversion emission spectra, and VSM for magnetic properties. Up-conversion mechanisms were analyzed using power-dependent intensity measurements.