Spectral form factor in the Hadamard-Gutzwiller model: orbit pairs contributing in the third order

The paper proves partner-orbit existence, uniqueness, and explicit action-difference bounds in the three two-encounter configurations (aas, ppi, api) for geodesic flow on X = Γ\PSL(2,R), using Poincaré sections, the Anosov closing lemmas, the connecting lemma, and time reversal. The stated constants and error terms match those in the candidate solution: for aas, 20ε-closeness and the bound |(T′−T)/2 − ln((1+u1 s1)(1+u2 s2))| ≤ ε^2(21e^{−T1}+30e^{−T2}+12e^{−T3}+19e^{−T4}) with uniqueness for ε < ε*/40 (Theorem 4.1) ; for ppi, 19ε and ≤ 54ε^4 + 25ε^2∑ e^{−Ti} with uniqueness for ε < ε*/38 (Theorem 4.3) ; and for api, 19ε and ≤ 54ε^4 + 25ε^2(e^{−T1}+e^{−T3}+e^{−T4}) with uniqueness for ε < ε*/38 (Theorem 4.5) . The single-encounter building block (Theorem 3.1) gives the ln(1+us) action jump with an exponentially small error and ≲8ε shadowing (unique if ε < ε*/18) . The candidate’s algebraic identities in SL(2,R), use of a_t–c_u, b_s conjugation, closing/connecting lemmas, and expansivity-based uniqueness align with the paper’s toolkit (definitions and lemmas in §2) . Minor presentational differences (e.g., quoting 10ε per single surgery vs the paper’s 7–8ε) do not affect correctness. Overall, the proofs are essentially the same in structure and estimates.