## 2 Materials and Methods  
The effect of BAs on selected properties of haptic luvisol soil [26] was investigated in a three-year field experiment. The study was conducted on a private farm in Korzenica, located in the Podkarpackie Province of southeastern Poland (Figure S1). The BAs utilized in the experiment originated from burning forest and agricultural biomass in a fluidized bed furnace. Specifically, the BAs were generated from approximately 70% forest biomass and 30% agricultural biomass. The forest biomass consisted of a balanced mix of deciduous and coniferous trees (50/50), while the agricultural biomass included such components as cereal straw, sunflower husks, and willow. Throughout the entire experiment, BAs with consistent properties were employed. The characteristics of the BAs used in the experiment are detailed in Table S1. The BAs exhibited an alkaline reaction (pH = 12.83) and high salinity (EC = 8.81 mS cm\({}^{-1}\)). Among the analyzed elements, K had the highest concentration, followed by Ca, Mg, P, and Na. The average concentrations of these components in the BAs were 165,617, 145,081, 13,512, and 9244 mg kg\({}^{-1}\), respectively (Table S1). The soil used in the experiment was characterized by a silt texture, acidic reaction, and low salinity. The average values for hydrolytic acidity (HAC), total exchangeable bases (TEB), and cation exchange capacity (CEC) were 1.23, 5.18, and 6.41 cmol kg\({}^{-1}\), respectively, with a base saturation (BS) value of 80.78%. The concentrations of the elements determined in the soil before the experiment were ranked in the following order of decreasing values: K \(>\) Ca \(>\) Mg \(>\) Na, with average concentrations of 1778, 1295, 826, and 212 mg kg\({}^{-1}\), respectively (Table 1). The variants of the experiment differed in the dose of K fertilization; this component accounted for the largest proportion of the BAs used in the experiment, which is related to the biomass combustion technology [27]. The study analyzed the effects of five BAs doses applied at rates ranging from 0.5 to 2.5 Mg ha\({}^{-1}\) (BAs_0.5 to BAs_2.5), corresponding to K\({}_{2}\)O amounts from 100 to 500 kg ha\({}^{-1}\), respectively [28; 29]. The results were compared to two control groups: control (without K fertilization) and NPK (fertilized with K from mineral fertilizers, specifically 60% potassium salt applied at a rate of 175 kg ha\({}^{-1}\)). A detailed description of the experimental design and conditions is provided in the Supplementary Materials. The effect of BAs was analyzed under the cultivation of three crop species: winter oilseed rape (_Brassica napus_ L. var. napus) cv. Mandril (from Syngenta), spring barley (_Hordeum vulgare_ L.) of the RGT Planet variety (breeder RAGT 2n; brewery type), and potatoes (_Solanum tuberosum_ L. cv. Sagitta, mid-early, fry type; breeder HZPC Holland B.V., The Netherlands) [30; 31]. The weather conditions in each year of the experiment are detailed in Table S2. \begin{table}
\begin{tabular}{l c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & **Unit** & \\ \hline Sand (2.0–0.05 mm) & & 17.0 \(\pm\) 2.0 \\ Silt (0.05–0.002 mm) & \% & 75.0 \(\pm\) 4.0 \\ Clay (\textless{}0.002 mm) & & 8.0 \(\pm\) 2.0 \\ Granulometric group & - & Silt (Si) \\ \hline pH KCl & - & 4.98 \(\pm\) 0.16 \\ pH H\({}_{2}\)O & - & 5.95 \(\pm\) 0.05 \\ \hline EC & \(\upmu\)S cm\({}^{-1}\) & 68.0 \(\pm\) 15.5 \\ \hline HAC & & 1.23 \(\pm\) 0.16 \\ TEC & \(\textless{}\textEvery autumn, after harvesting the crops, soil samples were collected from the 0-30 cm layer. The samples were air-dried for two weeks, crushed using a PULVERISERRE 8 soil deagglomerator from Fritsch GmbH, and sifted through a laboratory sieve with a mesh diameter of 2 mm. The following properties were determined in the prepared soil samples: pH was measured using the potentiometric method in a 1 mol dm\({}^{-3}\) KCl solution and in water. HAC and TEB were determined using the Kappen method. Based on HAC and TEB, CEC was calculated according to the formula: CEC = HAC + TEB.