A Fractional $3n+1$ Conjecture

💡 Research Summary

The paper introduces a new real‑valued iteration function Δ defined by
 Δ(x) = x/2 if frac(x) < ½,
 Δ(x) = (3x + 1)/2 if frac(x) ≥ ½,
where frac(x) denotes the fractional part of x. This definition mirrors the classic Collatz map on integers but is extended to the whole real line. Starting from any seed u₀ ∈ ℝ, the Δ‑sequence is the orbit u₀, Δ(u₀), Δ²(u₀), … .

Empirical exploration suggests that every Δ‑orbit eventually exhibits one of two behaviours: it either converges to zero, or it falls into a periodic cycle of length 29 when only the integer parts of the terms are observed. The integer‑part cycle is
1, 2, 4, 7, 11, 18, 9, 4, 7, 3, 5, 9, 4, 7, 11, 18, 9, 4, 7, 3, 6, 3, 1, 2, 4, 7, 3, 6, 3, …
which repeats indefinitely. The corresponding real values in the cycle are shown in the paper’s examples (e.g., starting from u₀ = 27 the orbit reaches the cycle after 28 steps).

The central formal result, Theorem 1, states that Conjecture 2 (the eventual convergence to zero or the 29‑cycle) holds for all seeds in the interval