Explicit positive lower density for the Collatz Conjecture
Lech Mazur proved that a positive proportion of integers $n$ reach $1$ under the Collatz map in at most $10.46\log n$ steps. We give a quantitative refinement of his construction. For every $N$, at least $\lfloor N/2^{2^{2^{163}}} \rfloor$ odd positive integers $n<N$ reach $1$ within $10.46\log n$ ordinary steps and $3.4881\log n$ accelerated steps. We also obtain results for paths that include any prescribed integer $t$ not divisible by $3$. AxiomProver was used to obtain and formalize this result in Lean.
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