Histogram of prime differences g = p_{n+1} - p_n highlighting primorial spikes
The spacing between consecutive prime numbers $g_n = p_{n+1} - p_n$ displays pronounced structural resonance rather than uniform random distribution. As predicted by the First Hardy-Littlewood Conjecture, prime gaps occurring at multiples of 6 ($g = 6, 12, 18, \dots$) appear with significantly higher frequency than non-multiples of 6.
This spectral concentration stems directly from primorial modular constraints ($2 \times 3 = 6$). Because all primes $p > 3$ are confined to residue classes $1 \pmod 6$ and $5 \pmod 6$, prime gaps must be even. Gaps divisible by 6 avoid prime factor collisions across both $p=2$ and $p=3$, doubling their relative combinatorial density.
This interactive histogram computes gap frequencies across adjustable search domains. Columns corresponding to $g \equiv 0 \pmod 6$ are highlighted in cyan, illustrating the persistent primorial resonance spikes governing local prime spacing.