A Fractional 3n+1 Conjecture

The paper states and claims to prove that for every real u0 in [0,100], the Δ-trajectory either converges to 0 or eventually loops on a length‑29 cycle (Theorem 1) . Its proof shows that on an interval I one has Δ^29(x) = (3^17 x + B)/2^29, hence Δ^29 is a contraction with a unique fixed point x0, and then asserts that any seed eventually “enters the cycle” after mapping into I . This conflates asymptotic attraction (lim i→∞ Δ^{29 i}(x) = x0) with eventual periodicity, which is false for irrational seeds. Indeed, Δ maps irrationals to irrationals, and any periodic point of this piecewise affine map must be rational; hence positive irrational seeds can never be eventually periodic. The paper quantifies over all reals in [0,100], so its theorem as stated is wrong. The candidate solution correctly identifies this logical flaw and the rationality/irrationality barrier, and also (independently) confirms the existence of the 29‑cycle (though its reported numerator differs from the paper’s; the paper’s x0 = 616136875/(2^29−3^17) is consistent with its own computation) .