Axler's compact prime-counting denominator coefficient improves from 70.935 to 44.053

What is proved

Let $L=\log x$. The new denominator is larger than Axler's published denominator because the subtracted coefficient is smaller, so its reciprocal gives a strictly smaller upper bound wherever both denominators are positive.

Denominator and li comparison

Set $s_A(t)=1-t-t^2-3t^3-At^4$ with $A=44.053$. After dividing by $x/L$, the gap between the proposed reciprocal denominator and $\operatorname{li}(x)$ is expressed through $s_A(1/L)$ and the exponential integral.

The certificate proves both the new and published denominators are positive at the lower endpoint and remain positive thereafter. It then covers $3.385\le L\le10$ by 16384 directed rational cells.

Rational majorants and the finite bridge

For $L\ge10$, a short rational majorant for $e^{-L}\operatorname{Ei}(L)$ has positive initial gap and positive differential forcing. Exact multiplication shows that the proposed reciprocal denominator dominates this majorant.

This proves that the proposed right-hand side exceeds $\operatorname{li}(x)$. Buethe's strict computation $\pi(x)<\operatorname{li}(x)$ through $10^{19}$ then supplies the prime-counting inequality over the corresponding range.

Published numerical envelopes

A higher-order rational majorant handles $L\ge40$. Fiori--Kadiri--Swidinsky Table 7 covers the initial high range, and the independently source-checked 164-row Table 4 transcript covers the numerical intervals through $L=3000$.

On each interval the proof computes the exact coefficient required by that row. The unique largest requirement occurs on $2100\le L\le2200$ and is strictly below $44.053$.

Asymptotic closure and certificate

For $L\ge3000$, exact polynomial arithmetic gives a simple positive lower bound for the reciprocal-denominator gap. A pinned global absolute estimate for $\pi(x)-\operatorname{Li}(x)$ closes the tail, with the source identity $\operatorname{Li}(x)=\operatorname{li}(x)-\operatorname{li}(2)$ retained.

At the handoff the certified margin exceeds $1.8130\times10^{-13}$ and the comparison improves afterward. The pinned certificate checks the imported tail dependency before checking every local gate.