Analytic and Probabilistic Foundations of the Lozan Method Gap Based Prime Neighborhood Scores via the Hardy Littlewood Heuristic Part 4

We develop analytic and probabilistic foundations for the Lozan method, a gap-based approach to quantifying local prime neighborhood structure. Given a finite set of offsets (gaps) DDD, the Lozan functional LD(n)=∑d∈D(χ(n+d)+χ(n−d))L_D(n) = \sum_{d \in D} \big( \chi(n + d) + \chi(n - d) \big)LD(n)=d∈D∑(χ(n+d)+χ(n−d)) counts prime occurrences in a symmetric neighborhood of nnn, where χ\chiχ denotes the prime indicator function. Motivated by experimental findings that certain gaps, notably multiples of 666, exhibit enhanced neighborhood contributions, we study E[LD(n)]\mathbb{E}[L_D(n)]E[LD(n)] under the Hardy–Littlewood prime kkk-tuple heuristic. We derive explicit formulas linking expected gap contributions to the singular series for prime pairs, obtaining the gap-dependent multiplier σ(d)=∏p∣dp>2p−1p−2.\sigma(d) = \prod_{\substack{p \mid d \\ p > 2}} \frac{p - 1}{p - 2}.σ(d)=p∣dp>2∏p−2p−1. This yields a principled explanation of small-prime exclusion effects and provides a theoretical bridge between wheel-sieve constructions and quantitative neighborhood scoring. We further formulate gap selection as a constrained optimization problem, establish monotonicity and dominance properties of σ(d)\sigma(d)σ(d), and give explicit optimal configurations under bounded-gap regimes. Finally, under explicit Cramér-type independence assumptions, we obtain variance bounds and concentration results for LD(n)L_D(n)LD(n), clarifying conditions under which the Lozan score is expected to exhibit stable behavior.