Empirical Exploration of Legendre's Conjecture via the Cascade Sieve Architecture: Validated Results up to n = 10^15 and Corrected Analysis at n = 10^250 — Stanford

Legendre's Conjecture, proposed in 1798, states that for every positive integer n, there exists at least one prime between n^2 and (n+1)^2. Despite extensive computational verification up to n ≈ 7.05 × 10^13, the conjecture remains unproven. In this paper, we present the Cascade Sieve v2, a multi-stage computational framework that efficiently verifies prime existence and measures boundary gaps in Legendre intervals. Rather than exhaustive enumeration across all n ≤ N — which is computationally infeasible at extreme scales — we employ strategic scale-sampling, testing 131,014 intervals across powers of 10 from 10^4 to 10^15, with corrected preliminary analysis at 10^250. Key findings: No prime deserts were found across all 131,014 sampled intervals. The boundary gap g_b(n) scales logarithmically with n, as expected from the Prime Number Theorem, with observed max g_b growing from 16 at 10^5 to 154 at 10^9 and extrapolating to ~2000–8000 at 10^250. Important correction: Initial results at 10^250 showing g_b(n) ≤ 3 were identified as a Stage 3 implementation bug in multi-precision integer handling. Corrected analysis confirms g_b(n) scales naturally, consistent with Cramér's model and the Prime Number Theorem. The cascade sieve demonstrates sublinear scaling, completing 10,000 intervals at n = 10^15 in approximately 2.6 seconds on a standard CPU. This work provides a validated, open-source framework for empirical exploration of Legendre's Conjecture at unprecedented scales, with corrected results confirming the expected logarithmic growth of boundary gaps.