## 7 Conclusions  
Spectral methods (such as SM, KL, and WH) are well-consolidated, fundamental, and effective tools for the correct and precise analysis of electromagnetic diffraction problems with one propagation constant, although they are not immediately applicable to problems with multiple propagation constants.  
In this study, we propose a comprehensive theoretical package in the spectral domain with all necessary mathematical tools that for the first time extends the possibilities of spectral analysis to electromagnetic problems involving wedges immersed in an arbitrary linear medium, and is additionally extendable to multiple penetrable angular regions. The theory is presented in an exhaustive way, showing the theoretical background, implementation, and validation. The methodology is based on transverse equations for layered angular structures, the characteristic Green function procedure, the Wiener-Hopf technique, and our novel direct Fredholm factorization method, which reduces GWHEs with multiple propagation constants to integral representations in a unique complex plane. Validation through examples is applied, starting from demonstrating the effectiveness of direct Fredholm factorization applied to GWHEs in the scattering from a PEC wedge immersed in an isotropic medium and ending with the validation of functional equations of angular regions in arbitrary linear media with the analysis of a PEC half-plane immersed in particular anisotropic media. While numerically implementing the method, we observed that one of the main difficulties resides in the correct estimation of kernel functions for the presence of multivalued functions that need particular attention in their definition and calculation.  
The proposed equations are interpreted using a network formalism in order to provide a systematic perspective, particularly for the analysis of complex scattering problems where the complexity of the geometry is broken into subdomains of canonical shapes in which the angular regions are immersed in/made of arbitrarily linear media.  
This work presents significant advancements in the spectral analysis of electromagnetic problems from different mathematical, physical, and engineering perspectives: (1) the first spectral method capable of handling scattering in arbitrary linear media with multiple propagation constants; (2) direct Fredholm factorization, a novel solution procedure for GWHEs, particularly those with multiple propagation constants; (3) a network interpretation of spectral functional equations and related integral representations for angular regions filled by arbitrary linear media (4) computation of the field at each point within the angular region, avoiding spectral analytical extension; and (5) improved quality of approximate spectral solutions from FTEs by reimposing the GWHEs of the problem, which we name iteration.  
This theoretical package has been validated and is ready for future applications.  
[PERSON] and [PERSON] collaborated in the conceptualization, methodology, mathematics, formal analysis, investigation, validation, writing--original draft preparation, writing--review and editing, project administration, and funding acquisition. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Italian Ministry of Universities and Research under PRIN Grant 2017 NT5W7Z GREEN TAGS and Next-Generation EU-PNRR M4C2-Inv 1.4-National Centre for HPC, Big Data, and Quantum Computing (HPC)--Multiscale Modeling and Engineering App.  
Not applicable.  
The original contributions presented in this study are included in the article and the Appendix A; further inquiries can be directed to the corresponding authors. The data presented in this study were obtained by means of an in-house software code implementing the proposed method.  
The authors declare no conflicts of interest.