Prime gap structure, scale by scale · OEIS Explorer · Sleek view
Every prime sits between two gaps: the distance back to the prime before it, and forward to the prime after. The Second Ratio compares them — r = (gap after − gap before) / (gap after + gap before).
It is 0 when the two gaps are equal (a "balanced" prime), swings toward +1 when the gap after is much larger, and toward −1 when it is much smaller — but it never quite reaches either end. Because it is a ratio, the overall size of the primes cancels out of the formula.
Below: how often each value of r shows up, measured over 15,000 consecutive primes at four scales. If the Second Ratio were truly scale-independent, the four shapes would be identical. They are not — watch the tall ±⅓ spikes erode and the edges fill in as the primes grow.
Each comb is one neighborhood of the number line. The dashed lines mark ±⅓, the most common Second Ratios. Reading top to bottom — from primes near a hundred thousand down to primes near ten million — the ⅓ spikes shrink by about a third and the rare extreme values (toward ±1) steadily fill in. The pattern is real, but it slowly loosens as primes grow — and where it is ultimately heading is an unsolved problem you can watch from your laptop.