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 Q_q(n) = n^q − (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 = 10^8 (~44 million prime values total). Key results:- Unified root count: ω_q(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.