A Hexagonal Test of the Lemke Oliver–Soundararajan Bias at Modulus 9

Every prime p > 3 lies in the six-element unit group S = {1, 2, 4, 5, 7, 8} ⊂ ℤ₉, which carries a natural hexagonal structure under the doubling map σ(x) = 2x (mod 9). We use this structure to test the conjectures of Lemke Oliver and Soundararajan (2016) on consecutive primes at q = 9, a modulus absent from their numerical comparisons and from those of Ash, Beltis, Gross and Sinnott (2011). First, we prove that secant transitions on the hexagon are exactly the pairs with a ≡ b (mod 3), and that the nine remaining pairs split into three additive levels, each containing two adjacent pairs and one diametric pair. Second, on all 50,847,532 primes from 5 to 10⁹, the ordered transition counts satisfy the prime-power symmetry of their Conjecture 1.6: within each of the ten predicted classes the three counts agree to within 0.3%, while counts across classes range from about 0.99 to 2.03 million; the same holds in 100,000-prime windows at eight scales up to 10¹⁸ and at q = 27. To our knowledge, this is the first numerical test of Conjecture 1.6 at an odd prime-power modulus. A hexagonal corollary follows: adjacent and diametric transitions have the same per-pair frequency (6.860% versus 6.856% up to 10⁹), although individual cross-class pairs differ from one another by up to 6%. Third, against a density-matched Cramér-type random model, the secant frequency shows a prime-specific deficit (z ≈ −8.06 in the first 100,000 primes) that keeps its sign in eight disjoint windows from 10⁷ to 10¹⁸ (sign test p ≈ 0.008; Stouffer-combined z ≈ −16.52). Because secant means same class modulo 3, this deficit is the q = 9 form of the bias against repeated residues that Lemke Oliver and Soundararajan documented at q = 3. Their leading-order formula tracks the observed secant frequency far more closely than the random model, the gap shrinking from 0.84 to 0.08 percentage points between 10⁷ and 10¹⁸. All claims are labeled as proved, conjectural, or empirically observed. The record includes the full manuscript (PDF) and the eight Python verification scripts that reproduce every number in the paper (also listed in Appendix A).