Number-Theoretic Derivation of Sliding Window Coverage Parameter W for Xi Sequences

The X _i , sequence, as a linear derived sequence of odd primes, its sliding window coverage isan important characterization of the uniformity of prime distribution. Aiming at the key problem that the value of the core parameter-sequence set size W_in existing verifcation methodsacks number-theoretic theoretical support and relies on empirical grading, this paper focuses onthe number-theoretic analysis and strict formula derivation of W, constructing a W calculationsystem with both theoretical rigor and engineering practicality. Firstly, based on the PrimeNumber Theorem (PNT) and the asymptotic distribution theorem of prime gaps, the intrin.sic number-theoretic correlation between W and prime density, maximum prime gap is strictlydemonstrated, establishing a rigorous logical chain of "prime distribution characteristics -X _i sequence coverage requirements - W constraints". Secondly, a conservative coeffcient is introduced through the analysis of prime gap fuctuations to complete the strict derivation of the coreformula for W; boundary constraints are added in combination with the prime distribution char-acteristics of small N and extremely large N scenarios to form a complete number-theoreticallyprovable calculation method, and the probability guarantee and engineering verifcation of cov.erage effectiveness are provided. Finally, a sliding window coverage verification framework for X _i ; sequences based on this method is constructed, and the effectiveness of the method is verifedthrough experiments in multi-scale N scenarios. Theoretical analysis shows that the derived Wcan strictly cover the X _i , sequence coverage requirements corresponding to any prime gap within [1, N] in engineering application scenarios, and the derivation process relies entirely on corenumber-theoretic theorems without empirical assumptions. Experimental results demonstratethat the proposed method can effectively ensure coverage effectiveness and has good engineeringpracticality. The core value of this paper lies in providing a strict number-theoretic theoreticalbasis and standardized calculation scheme for W setting, as well as a promotable theoreticalparadigm for parameter optimization of prime-derived sequences.