Constant-factor bounds for Hamiltonian paths in tournaments

This record archives a bilingual research manuscript on constant-factor bounds for the maximum number of directed Hamiltonian paths in an n-vertex tournament. It includes English and Chinese PDF manuscripts, editable Markdown and LaTeX sources, component proof-audit reports, exact finite diagnostic records, and Python verification modules. Let H(T) count vertex permutations forming a directed Hamiltonian path, let P(n) be its maximum over n-vertex tournaments, and put μ_n = n! / 2^(n−1). The manuscript presents a proof that there are absolute constants K and n₀ such that, for all sufficiently large n, (L − K/n) μ_n ≤ P(n) ≤ (C_* + K/n) μ_n, where L = cosh(1)/cos(1) and C_* = (3π⁴ + 4π² − 32)/(π⁴ + 4π² − 32). This manuscript does not claim an exact finite formula for P(n), a sharp global identity with leading constant L, effective numerical values of K or n₀, or an extremal classification. Research status: this is a research manuscript/preprint. It has undergone an AI-assisted internal proof reconstruction and cross-check, but has not undergone external human peer review or formal proof-assistant verification. Finite computations in the archive are diagnostics, not replacements for the all-order proof. AI-use disclosure: AI assistants were used substantively for proof reconstruction and cross-checking, translation, typesetting, and finite diagnostic programming. The named author accepts responsibility for the record’s contents. AI systems are not listed as authors or creators. Repository and version control: https://github.com/LStar404/tournament-hamiltonian-paths