Comparison of Gauss's and Legendre's formula in approximating the prime counting function along with results analysis.

Closed-form nth-prime estimator built on invariant-normalization logic, with deterministic refinement and exact benchmarks across a contract grid spanning n = 10^2 to 10^24.

Unified constructive and non-constructive proofs of three number theory problems: Twin Prime, Collatz, and the Prime Number Theorem. Based on 6n±1 primes and composite removal, with density-based complements. 双子素数予想・コラッツ予想・素数定理を構成的＋非構成的に統合。6n±1型素数構成と除去関数、密度補完による論理統合。