Survival Bias in 7-adic Prime Chains: Overcoming the Parity Trap

This archive documents the discovery of rare prime number chains generated by a 7-adic Collatz-like map defined by: f(n) = (8n + k) / 7, where k ∈ {−1, +5, −3, +3, −5, +1} is selected based on n mod 7 to ensure the numerator is divisible by 7 and the result remains odd. A parity-correcting algorithm was introduced to overcome the "parity trap"—a phenomenon that causes early termination of prime chains due to even-number composites. By comparing a minimal-distance strategy with a survival-biased approach that enforces odd parity, the number of long prime chains discovered increased fivefold. Key Research Highlights: - The Discovered Chains: Twenty length-7+ prime chains, including one 9-step prime-preserving sequences - 7-adic Branching Structure: Modular rules ensure integrality and odd parity - Survival Mode Algorithm: Forces odd k-values to avoid parity traps - Efficient Search: Fully parallelized Python implementation using sympy.isprime for deterministic primality testing