Statement (informal)
When the prime sequence is viewed through its second differences — the gap-of-gaps sequence — the result alternates in sign at a characteristic frequency that differs from random expectation at very high statistical significance. Prime gaps, loosely speaking, breathe: they grow, then shrink, in a rhythmic pattern that random sequences do not reproduce.
Motivation
Consider the prime sequence 2, 3, 5, 7, 11, 13, 17, 19, 23, … The first differences — the gaps — are 1, 2, 2, 4, 2, 4, 2, 4, … The second differences — the gap-of-gaps — are 1, 0, 2, −2, 2, −2, 2, …
At the local scale above, the gap-of-gaps sequence alternates in sign after almost every non-zero entry. This alternation is not a coincidence of small examples. When extended to hundreds of thousands of primes, the rate of sign alternation differs from the baseline of a random walk with the same marginal distribution by an amount whose statistical significance is extreme.
Complementary result: the Mean Convergence Law
The Oscillation Law sits within a broader structural picture. The mean of the gap sequence scales as ln(p), as one would expect from the Prime Number Theorem. However, the mean of the second-difference sequence, the third, and the fourth all converge to exactly zero as the prime range grows.
This is a strong form of self-regulation: at every higher derivative level, the prime sequence is a steady-state signal with vanishing net drift. The Oscillation Law quantifies the precise rhythm with which this steady state is maintained.
The precise quantitative form of the Oscillation Law, its statistical significance, and its relationship to other prime-distribution results are developed in full in Volume III, currently in preparation. Specific percentages, Z-scores, and the exact formulation of the law will be released with the full preprint. This page presents the conjecture in its informal form.