## 2 Methodology  
The first five schemes included in this comparative study are those presented by [PERSON] et al. [8], [PERSON] et al. [9], [PERSON] et al. [12], [PERSON] et al. [13], and [PERSON] et al. [14]. These schemes were selected based on the diversity of their origin and their application in modeling the C6 series. In particular, the schemes reported by [PERSON] et al. [8], [PERSON] et al. [12], [PERSON] et al. [13], and [PERSON] et al. [14] have been employed under industrial conditions, with pressures ranging from 20 to 30 bar, using different catalysts depending on the study. Additionally, the first three derive their schemes from established mechanisms in the literature, whereas [PERSON] et al. [14] and [PERSON] et al. [9] base theirs on experimental data. In the case of [PERSON] et al. [14], the data originate from industrial-scale experiments, while those of [PERSON] et al. [9] are obtained at the laboratory scale. This selection allows for the evaluation of schemes based on both theoretical and experimental mechanisms, as well as the comparison of the results of those applied at industrial and laboratory scales. Each scheme was analyzed for linear independence using a stoichiometric coefficient matrix, where rows represent chemical components and columns correspond to the isomerization reactions of the C6 series. Linear independence was assessed by examining linear combinations of the matrix's columns (or vectors). Nontrivial solutions were used to identify and discard redundant reactions. This linear combination is expressed as follows [18]:  
\[C_{1}v_{1}+C_{2}v_{2}+\ldots+C_{j}v_{j}=0 \tag{1}\]  
where \(C_{j}\) are real numbers (scalar coefficients) and \(v_{j}\) are vectors representing the reactions of the scheme. As a result of this analysis, it was determined that the reaction schemes proposed by [PERSON] et al. [13] and [PERSON] et al. [14] contained a linearly dependent reaction. Consequently, both schemes were modified by removing the 2-MP \(\leftrightarrow\) 3-MP reaction, leading to the development of Schemes 6 and 7 in the comparative study. Scheme 8, by contrast, is a proposed approach that is not based on any specific reaction mechanism. Instead, it explores a possible combination for the isomerization process that is not considered in the other schemes, while simultaneously incorporating linear independence as a criterion for its formulation. The eight schemes used for the comparative analysis of the C6 fraction are summarized in Table 1.  
Each isomerization process has distinct characteristics, meaning that industrial [8; 12; 13; 14] and laboratory-scale [6; 9; 10] processes operate under specific conditions (e.g., number and size of reactors, type of catalysts, phase condition, temperature, and pressure). Therefore, establishing a kinetic reaction set and adjusting kinetic parameters are essential for predicting or fitting the observed process data. Here, the plant data reported by [PERSON] et al. [15] are considered to evaluate the predictive capability of the previously established schemes. This isomerization process was carried out in three packed-bed reactors and utilized a feed primarily composed of C\({}_{5}\)/C\({}_{6}\) paraffins. Each reactor was modeled with a volume of 13.8 m\({}^{3}\) and employed a Pt/Al\({}_{2}\)O\({}_{3}\)-CCl\({}_{4}\) catalyst with a bulk density of \(\rho_{B}=650.91\) kg/m\({}^{3}\) particle density of \(\rho_{B}=865.00\)kg/m\({}^{3}\)[19], as considered by [PERSON] et al. [17]. The feed stream entered the first reactor at 147.28 \({}^{\circ}\)C and exited at 160.50 \({}^{\circ}\)C, then entered the second reactor and was discharged at 169.77 \({}^{\circ}\)C. Subsequently, the flow was cooled to 145.45 \({}^{\circ}\)C before entering the third reactor, where the outlet temperature reached 148 \({}^{\circ}\)C.  
The model considers four isomerization reactions of nC6 in the independent schemes and five in the dependent schemes.