Outcome: completed

Result: Axler's compact prime-counting denominator coefficient improves from 70.935 to 44.053 on the identical range x>=29.53, with a pinned low-li bridge, all 164 FKS Table 4 rows, and the C-0051 asymptotic tail.

The verifier proves denominator positivity, a 16384-cell Arb cover
from \(\log x=3.385\) to \(10\), exact rational logarithmic-integral
majorants, a Table 7 handoff, all 163 Table 4 intervals, and the
asymptotic tail from \(\log x=3000\).

The unique active Table 4 interval is \([2100,2200]\). Its exact
requirement is

\[
44.05242639511077\ldots,
\]

leaving reserve \(0.000573604889227\ldots\) below the proposed
coefficient. The least exact table margin is
\(2.45085\cdot10^{-17}\), and the tail margin at \(L=3000\) exceeds
\(1.8130\cdot10^{-13}\).

Independent source parsing and SymPy/mpmath reconstruction pass.

Checks:

```sh
uv run --frozen python \
  scratch/adjacent-unconditional--axler-compact-pi-denominator-44-053/verify.py
uv run --frozen python \
  scratch/adjacent-unconditional--axler-compact-pi-denominator-44-053/independent_check.py
```

