The Lockstep Law A Parameter-Free Hardy–Littlewood Predictor for Consecutive Prime Correlations

This paper develops a parameter-free Hardy–Littlewood framework for predicting correlations between consecutive prime configurations.

For an even shift (h), a prime (p) is called an (h)-starter when (p+h) is also prime. The lockstep statistic asks how often the immediate successor (q=\operatorname{nextprime}(p)) is itself an (h)-starter. A naive independence model predicts the rate using the ordinary Hardy–Littlewood singular series (\mathfrak S(h)), but direct computation shows substantial and highly structured deviations from that prediction.

The central result of this work is a finite-range predictive law that removes the empirical successor-gap distribution used in earlier versions of the model. Because (p+h) is already prime, the successor gap (g=q-p) satisfies the exact bound (g\le h). The condition that (p+g) is the immediate next prime can therefore be represented by finite inclusion–exclusion over the interior offsets (2,4,\ldots,g-2). Applying Hardy–Littlewood singular-series densities to every resulting constellation yields a predictor depending only on the shift (h) and the numerical range ([A,X]), with no fitted correction factors and no measured successor-gap weights.

For each possible successor gap (g), the model combines the prime constellation required for lockstep with the complete interval-avoidance condition. The resulting prediction is a finite sum of singular-series terms weighted by logarithmic integrals (I_k(A,X)=\int_A^X(\log t)^{-k},dt).

Prospective blind tests provide the principal empirical validation. In fresh full-inclusion–exclusion tests, the model predicted nine nonzero lockstep corrections before their prime data were measured, achieving approximately (0.443%) mean absolute relative error and (0.267%) median absolute error. An exact-zero prediction for (h=4) was also confirmed with zero observed successes. A second blind set reproduced a deliberately nonmonotonic ordering across (h=46,52,58,64,68), with mean absolute relative error (0.310%); the prediction for (h=64) differed from observation by only (0.021%).

The framework also yields a fixed-(h) asymptotic law. Since the probability of an interior prime between (p) and (p+h) is (O_h(1/\ln X)), the successor-gap distribution concentrates on (g=h). Conditional on the Hardy–Littlewood conjecture,

\frac{\mathfrak S({0,h,2h})}{\mathfrak S(h)^2}.
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This produces a sharp arithmetic dichotomy: if (3\nmid h), the progression ({0,h,2h}) is inadmissible modulo (3), giving (\kappa_h^{\infty}=0). If (3\mid h), the limiting correction is determined explicitly by the distinct odd prime divisors of (h).

A detailed (h=32) diagnostic separates the remaining finite-range discrepancy into distinct layers. The full inclusion–exclusion model predicts the successor-gap distribution with high accuracy, including a mean-gap error of only about (0.15%), while raw three-point Hardy–Littlewood counts have approximately (0.23%) mean absolute error. The dominant residual appears upon introducing the fourth correlated prime: raw four-point counts show approximately (1.23%) mean absolute error and fully consecutive lockstep counts approximately (2.07%). This isolates the principal unresolved component as a finite-(X) higher-order Hardy–Littlewood remainder rather than a failure of the successor-gap construction.

The work therefore converts the original lockstep deficit from an empirical correlation into a parameter-free predictive framework connecting Hardy–Littlewood singular series, finite inclusion–exclusion, consecutive prime gaps, arithmetic admissibility, and higher-order prime correlations. Remaining open problems concern analytic lower-order corrections to four-point densities, efficient evaluation for large (h), higher-order lockstep chains, and the asymptotic structure of the finite-range remainder.