A parametric family of affine-related quadratic polynomials: index pre-sieve, modular partitions, and Bateman–Horn classes

This paper extends the index pre-sieve and the CRT-based modular partition previously introduced for a specific pair of affine-related quadratic polynomials to a full parametric family. It shows that all structural properties—density of the filtered index set, modular partition into arithmetic progressions, constancy of modular signatures, and admissibility dichotomy—hold uniformly for all parameters. Under the Bateman–Horn conjecture, each admissible congruence class contains infinitely many prime values. The work provides a unified structural framework for organizing prime candidates by modular signature across an infinite family of quadratic polynomials. Note.This paper is the third contribution in a series. The first introduces the index pre-sieve and its density interpretation, while the second develops the induced CRT-based modular partition. The present work provides a parametric generalization of these results.