Bounded Gaps for High-Degree Polynomial Primes: A Cyclotomic Maynard--Tao Roadmap

We propose a concrete roadmap for establishing bounded gaps among primes of the Titan polynomial family Q_q(n) = n^q − (n−1)^q, exploiting the Arithmetic Shielding property proved in our companion paper (doi:10.5281/zenodo.18582880).

We introduce a Null-Sparse Decomposition of the moduli space and prove that, for this polynomial family, the Bombieri–Vinogradov error term is identically zero on a density-1 set of moduli. The bounded-gaps problem thus reduces to an equidistribution estimate on a sparse set of moduli (those divisible by primes p ≡ 1 mod q).

Three pillars support the roadmap:
(I) Null Error Theorem — BV error vanishes on all null moduli (verified at 83× concentration ratio);
(II) Massive Admissibility — k_max = 2,173 for q = 167, an order of magnitude beyond the Maynard–Tao requirement;
(III) Square-root cancellation in exponential sums — 40/40 numerical tests consistent with the p^{1/2} bound (max ratio 1.93).

Under a Sparse Bombieri–Vinogradov Hypothesis (strictly weaker than classical BV), we prove a conditional bounded-gaps theorem for Titan polynomial primes.

Paper II in the Titan Polynomial series.
Code & Data: https://github.com/Ruqing1963/titan-bounded-gaps