We study the generalized Collatz maps Tₘ defined by: • Tₘ(n) = n / 2 when n is even • Tₘ(n) = (m n + 1) / 2 when n is odd with m an odd integer greater or equal to 3. For the family Tₘ, we establish several structural results. First, we prove the existence of an explicit infinite family of cycles when m = 2ᵏ − 1. Second, we derive the necessary cycle constraint mᵏ = 2ᴸ for any cycle containing k odd elements. Third, we obtain a symbolic‑dynamics growth law involving the empirical odd‑step frequency pᵣ(m), and we show computationally that m = 3 is the unique parameter for which the growth rate is close to zero. For the classical map T₃, we develop a complete analytic framework based on the Partition Function Identity (PFI). This identity expresses each cycle element nⱼ in terms of a partition sum Z(j) and the excess parameter p, defined by p = S / k − log₂ 3, where S is the total height of the cycle. From the PFI we derive: • the Orbit Recursion, governing the evolution of Z(j) • the Orbital Product Identity, relating the product of the terms 3 + 1 / nⱼ to 3ᵏ 2ᵏᵖ • the universal height bound p ≤ log₂(10 / 9) • the Orbital Sandwich, giving two‑sided bounds on the minimum element nₘᵢₙ • the Rho‑Interval, which forces p into a shrinking interval depending on nₘᵢₙ • the Minimum Residue Theorem, proving that nₘᵢₙ must satisfy nₘᵢₙ ≡ 3 mod 4 A modular cascade further eliminates additional residue classes modulo 16 and 32, leaving only four possibilities for nₘᵢₙ modulo 32. The cycle problem for T₃ is then reduced to a finite verification. Using height stability modulo 2¹⁷ and the Z‑recursion, we show that for every odd residue r satisfying r ≡ 3 mod 4 and 1 < r < 2¹⁷, the induced orbit produces an element strictly below r within a bounded number of steps. This contradicts minimality and excludes all non‑trivial cycles. We conclude that the map T₃ admits no non‑trivial cycle. The question of whether all trajectories eventually reach 1 (absence of divergence) remains open.
Franck Coppi (Sun,) studied this question.