𝗦𝗼𝗽𝗵𝗶𝗲 𝗚𝗲𝗿𝗺𝗮𝗶𝗻 𝗣𝗿𝗶𝗺𝗲𝘀: 𝗕𝗮𝗿𝗿𝗶𝗲𝗿-𝗦𝗶𝗱𝗲 𝗢𝗯𝘀𝗲𝗿𝘃𝗮𝘁𝗶𝗼𝗻𝘀
I am publishing a new report on Sophie Germain primes. This work uses the Rei-AIOS toolkit to observe mathematical barriers.
I do not claim to solve the Sophie Germain prime conjecture. I do not claim to move toward a proof of infinity. This paper focuses on observations and formal witnesses of existing barriers.
Here are the four main findings:
Numerical convergence: The ratio of empirical counts to Hardy-Littlewood predictions decreases steadily from N = 10³ to 10⁸. This matches expected asymptotic patterns but is not a proof.
Spectral statistics: Sophie Germain prime gaps show Poisson-like distribution. This is different from Riemann zeros, which follow GUE-like statistics.
Lean 4 formal witnesses: I created an axiom-free set of theorems in Lean 4. These theorems show that single features like "is_prime(n)" cannot detect Sophie Germain primality. You need the full conjunction to pass the barrier.
Record correction: An online audit corrected an internal error. Selberg identified the parity problem in 1949, not the 1960s.
This research follows a strict discipline. We use Bellman-Ford encoding to map feasibility questions. We use Lean 4 to provide verifiable, axiom-free records of finite arithmetic facts.
The goal is barrier-side description. We describe where the current methods stop.
Optional learning community: https://t.me/GyaanSetuAi