Discovery of Length-7 Prime Chains under the Collatz Map f(x) = (3x+1)/2

This archive documents the discovery and mathematical analysis of a rare length-7 prime chain identified within the dynamics of the Collatz conjecture's increasing map, f(x) = (3x+1)/2.
While the Collatz conjecture typically deals with convergence to 1, this research focuses on "Generalized Cunningham Chains" where every iteration yields a prime number.
The primary finding is a chain of seven consecutive primes starting from x₀ = 89,599.

Key Research Highlights:

• The 89,599 Chain: A detailed trace of the seven-step primality sequence.
• Efficient Sieving Strategy: A targeted search method to avoid small prime divisors and identify viable candidates.
• Modular Congruence Analysis: Exploration of x ≡ 199 mod 200 as a necessary condition for sustaining "99" endings across iterations.