Selberg Zeta Functions, Cuspidal Accelerations, and Existence of Strict Transfer Operator Approaches

The paper gives a complete, algorithmic, and rigorously verified construction of strict transfer operator approaches, including branch reduction, identity elimination, cuspidal acceleration, and a proof of Properties 1–5 for an explicitly defined structure tuple S with enlarged index set  and intervals Îa built via a normalization qξ (Theorem 8.1 and Sections 8.2–8.6). By contrast, the candidate solution skips required preprocessing (admissibility, finite ramification, non-collapsing/identity elimination), sets  = A and Îa = Ia (which conflicts with the paper’s  and Îa), and gives only heuristic arguments for Properties 2–5 (hyperbolicity, uniqueness of coding length, counting representatives, and complex neighborhoods) instead of the paper’s precise proofs. In particular, leaving possible identity branches intact would violate Properties 2–3, and the construction of neighborhoods in Property 5 requires a careful global dependency analysis absent from the candidate outline.