Nontrivial Cycles of the Collatz Conjecture: The Smooth Model and the Diophantine Bridge

Part 1 of a two-paper series on the Collatz conjecture. We study nontrivial cycles in a smooth model and build an exact Diophantine bridge Real→Integer. We prove the smooth model has no nontrivial cycles; the only discrete fixed point compatible with the original map is 1. For an odd→odd block σ with parameters (a,b), an integer fixed point exists iff D|S(σ) and n=S(σ)/D, where D=2^b−3^a and S(σ)=2^b C(σ). Using a telescoping identity and a gcd argument we exclude all orders except the trivial p_i=2 (yielding n=1). Together with the 3‑adic contraction of T(n)=(3n+1)/2ᵛ²⁽³ⁿ⁺¹⁾, potential cycles reduce to a finite congruence checklist. The companion paper “Excluding Infinite Growth” extends the method to show every trajectory enters S={n | 3n+1=2^k} and reaches 1.