Asymptotic bounds from Models 1, 2, and 2* for the prime-counting function

Let ${\pi}^1(n)$ denote the estimate for the prime-counting function $\pi(n)$ derived from Model 1, let ${\pi}^2(n)$ denote the estimate derived from Model 2, and let ${\pi}^{*}(n)$ denote the estimate derived from Model 2*.

Model bounds conjecture. As $n\to\infty$, the estimates satisfy

$${\pi}^2(n)\leq\pi(n)\leq{\pi}^{*}(n)\leq{\pi}^1(n).$$

These bounds are suggested by inspection of the limited data in Table 4 and by the preceding analysis of the models' underestimates and overestimates. Their validity as $n$ increases remains unproved; the paper also notes that missing contributions from partitions and factorizations of lengths greater than three may reduce the accuracy of Models 2 and 2*.