Four-polynomial generation conjecture for the natural numbers

Let $p_1,p_2,p_3,p_4$ be the four polynomials constructed in the paper, evaluated at $x,y,z\in\mathbb{N}^*$. The relevant generated parameter is a natural number $q\geq1$. Four-polynomial generation conjecture. The polynomials $p_1,p_2,p_3,p_4$, for $x,y,z\in\mathbb{N}^*$, generate every natural number $q\geq1$. The paper reports computational evidence for coverage up to $10^9$, while explicitly noting that a formal proof remains open. The relation between this parameterization and coverage of integers congruent to $1\pmod{4}$ is part of the proposed approach to the Erdős–Straus conjecture.