A uniform modular partition of prime candidates for a pair of affine-correlated quadratic polynomials based on an index pre-sieve

We show that an index pre-sieve, defined through congruence conditions modulo a finite set of primes P, induces an explicit modular partition of the indices n ≥ 0 into arithmetic progressions r + MZ, where M is the product of the primes in P. Each class has natural density 1/M.

Specializing to the affine-correlated quadratic polynomials

d(n) = n² + n + 5 and N(n) = 6n² + 6n + 31,

we show that fixing an index filter automatically induces a constant modular signature for the values of d(n) and N(n) modulo each prime in P. We also introduce a natural notion of admissibility of a filter or class: a class is either totally excluded (always producing values divisible by some prime in P), or it is entirely composed of P-rough candidates.

All results concerning modular partitions, signatures, and admissibility are unconditional and follow from the Chinese Remainder Theorem. In a final section, we establish a conceptual bridge with the Bateman–Horn conjecture: assuming it, every admissible class contains infinitely many prime values, with a uniform asymptotic form shared by all admissible classes, while the associated constants may depend on the class.