From Modular Exclusion to ppb Precision: Asymptotic Rational Scaling in Aligned Ten‑Number Windows

The project investigates the distribution of primes in aligned consecutive 10-number intervals, specifically \(I_k = [10k + 2,\,10k + 11]\). It proves a Restricted Modular Exclusion Law: in these aligned intervals it is impossible to have exactly four primes (maximum 3). It proposes the Asymptotic Rational Law: the prime-sum / non-prime-sum ratio converges to \(R = n_p/(10 - n_p)\) for \(n_p \in \{1,2,3\}\) with error \(O(1/k)\). Multi-scale empirical validation covers 53,441 intervals between \(10^4\) and \(10^{13}\), showing systematic convergence. Observed precision improves from hundreds of ppm to ppb levels at large scales, supporting the asymptotic hypothesis. Variance analyses show superlinear decay of the CV (rate \(\beta \approx 1.4\)), indicating a deterministic limit. The combinatorial proof uses residue classes (mod 2, 3, 5, 30) and depends critically on the alignment \(n \equiv 2 \pmod{10}\). Central limitation: results do not generalize to arbitrary intervals; counterexamples (quadruplets) exist for other alignments. Importance: reveals modularly imposed local asymptotic laws that complement the global view of Number Theory. Practical contribution: provides a well-defined domain for theoretical investigation (Diophantine analysis, L‑functions) and guides future computational validations.