Constructive Symmetry: Hidden Structure in Prime Arithmetic Progressions Revealed by the Base Primorial

Prime distribution is typically regarded as a paradigm of randomness. This work introduces the “base primorial”—the product of consecutive primes from 2 up to O (denoted #O)—as a core constructive tool, revealing a hidden symmetry in prime arithmetic progressions.

When the common difference d of an arithmetic progression is an integer multiple of the base primorial (i.e., d = K·#O, K a natural number), all terms occupy the same residue class modulo every prime ≤ O—a strict translational symmetry. This symmetry is broken by a structural law: let P be the next prime greater than O, then the progression cycles through all residue classes modulo P, so at most P−1 consecutive terms can avoid divisibility by P, imposing an absolute upper bound of P on the length of any prime progression with this difference.

Based on this principle, we propose a complete system of constructive conjectures, supported by systematic verification: for P = 2 to 13, 100 examples were obtained for each, all strictly obeying the symmetry-imposed bound. Notably, when the difference equals the base primorial itself (K=1) and the first term equals P, only 13 such prime progressions were found below 10¹⁰⁰⁰, demonstrating that the symmetry constraint strengthens with scale rather than weakening.

This framework is being extended to consecutive prime progressions and twin prime distribution. Preliminary results suggest that the symmetry-breaking mechanism revealed by the base primorial is effective in a broader class of prime distribution problems.