Abstract
Background: The Collatz conjecture, proposed in 1937, asserts that iteration of the map T(n) = n/2 if n even, 3n + 1 if n odd, reaches 1 for every positive integer n. Verified computationally to n < 268, it remains unproven. This study analyzes the conjecture through binary structure, relating the fractional part {log2 n} to zero density z(n) and v2(3n + 1). Methods: We derive a recurrence relation for fractional parts in binary expansions and analyze block structures using linear systems. For sparse binary numbers (z(n) ≥ n/2), we prove strict trajectory decrease in O(log n) steps. Results: Theorem 1 establishes fractional part recurrence. Theorem 2 proves ≥ 50% zero density in 3n when 1 − {α} > 0.55. Theorem 3 shows strict decrease for sparse binaries. Theorem 4 verifies the conjecture for the explicit subclass {an = ∑i n =0 γi2i | n > 1000, z(an) ≥ n/2}, comprising ∼ 2n/2 numbers of length n. Conclusions: The fractional part approach yields new structural insights, confirming the conjecture for a significant explicit subclass while the general case remains open.