Defying the Degree Barrier: Arithmetic Shielding and Bimodal Effective Degree in the Titan Polynomial Family

This deposit contains the complete dataset, source code, and figures accompanying the paper "Defying the Degree Barrier: Arithmetic Shielding and Bimodal Effective Degree in the Titan Polynomial Family" by Ruqing Chen. We study the Titan polynomial family Qq (n) = nq − (n−1) q for 18 prime exponents q ∈ 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 43, 47, 53, 61, 71, 83, 167, with exhaustive primality testing up to n = 10⁸ (~44 million prime values total). Key results: - Unified root count: ωq (p) = gcd (q, p−1) − 1 for all primes p. - Arithmetic shielding: S (f) grows logarithmically, counteracting the 1/d degree barrier. - Sophie Germain bifurcation: S (f) splits into two tracks; penalty factor (q+2) / (2q) → 1/2. - Bateman–Horn fit: all 18 datasets match the heuristic to within 0. 3–3. 1%. - A degree-166 polynomial (q = 167) produces 14. 2× more primes than a generic polynomial of the same degree. Contents: - data/ — Summary tables (CSV): prime counts, singular series, penalty factors, counts at 50 values of N. - figures/ — All 5 paper figures in PDF vector format. - paper/ — LaTeX source and compiled PDF. - scripts/ — Python code to reproduce all computations and figures.