Certified upper bounds in Turán's pure power sum problem (Erdős #519): certificates and checkers

Exact certificates and checkers for rigorous upper bounds on R_n in Turán's pure power sum problem (Erdős problem #519), for 13 values of n from 300 to 1 000 000. Each certificate stores an explicit n-tuple of complex numbers (z_1 = 1 exactly) through the exact dyadic coefficients of its polynomial. The bound max_{1<=k<=n} |z_1^k + ... + z_n^k| <= c is checked for every k by two independent programs (Arb ball arithmetic, and pure integer arithmetic for n <= 5000). Examples: R_3000 <= 0.687169029139, R_100000 <= 0.686409189526, R_1000000 <= 0.686368028005. Files: turan-power-sum-certificates-1.0.0.zip (code, certificates for n <= 20000, logs, README) and certs_large.zip (certificates for n = 30000 to 1000000; their SHA-256 sums are in results/certs/SHA256SUMS.txt inside the first file). These are bounds for specific n and do not by themselves bound limsup R_n. Not peer reviewed. Made with substantial help from AI models (Claude Opus 5.5 and Claude Sonnet 5.5, Anthropic); see README.md. GitHub: https://github.com/Bakeas/turan-power-sum-certificates