An improvement of the Krasikov–Lagarias lower bound for the 3x+1 problem: π_a(x) ≥ Δ·x^0.8892

Abstract

Let π_a(x) be the number of integers n ≤ x whose 3x+1 trajectory reaches a. Krasikov and Lagarias (Acta Arith. 109, 2003) proved π_a(x) ≥ Δ·x^0.84 for a ≢ 0 (mod 3) via a feasible solution of their nonlinear program L_11^NT(λ); that exponent has remained the published record (cited as such in arXiv:2512.13760, Dec. 2025). This note reports feasible solutions of L_k^NT(λ) for k = 12,…,16, obtained by a damped power iteration of the min-linear operator underlying the program, with the k = 16 certificate verified in exact integer arithmetic (rational λ = 9261173/5000000, certified rational lower bounds for the transcendental weights, exact slack ≥ 7.84·10⁻¹¹ over 14,348,907 constraints). By Theorem 2.2 of Krasikov–Lagarias this yields π_a(x) ≥ Δ·x^γ with γ = log₂(9261173/5000000) = 0.889266839…, for all a ≢ 0 (mod 3) and x ≥ x₀(a). The computation reproduces the published table (k = 2,…,11, including the C_max column) to seven digits and is cross-validated by an independent formulation on two machines. The deposit includes the note (PDF + LaTeX source), the scripts (certificate construction, exact verification, independent cross-check) and the certificate data, allowing full reproduction. The result is a density lower bound only; no claim is made about the Collatz conjecture. Resumen: cota inferior de densidad π_a(x) ≥ Δ·x^0,889266839 para el problema 3x+1, cinco niveles por encima del k = 11 de Krasikov–Lagarias (récord 0,84 desde 2003), con el certificado final verificado en aritmética entera exacta. Sello de prioridad del Sistema Liñán–Collatz (SLC).