---
id: C-0051
type: claim
title: "Fiori--Kadiri--Swidinsky's global pi-minus-Li amplitude improves from 9.2211 to 9.221056"
status: proved-unconditional
family: adjacent-unconditional
statement: >
  For every real x>=2, the absolute value of pi(x)-Li(x) is at most
  9.221056*x*sqrt(log x)*exp(-0.84768363*sqrt(log x)). This replaces
  the amplitude 9.2211 in Fiori--Kadiri--Swidinsky Corollary 22 while
  preserving its absolute-value formula, exponent, and full range.
verifier: canon/witnesses/C-0051/verify.py
depends: []
supersedes: []
sources:
  - literature/2026-07-19-fks-pi-li-amplitude-9-221056-search.md
  - experiments/fks_pi_li_amplitude_9221056/verify.py
  - scratch/adjacent-unconditional--fks-corollary22-pi-li-amplitude-9-221056/RESULT.md
tags: [prime-counting, logarithmic-integral, explicit-bound, pnt-error, fiori-kadiri-swidinsky, interval-arithmetic]
caveats: >
  This is an exact normalization sharpening of the globally
  admissible A_pi=121.107 proved in the source. It does not lower
  that R-normalized amplitude or change the asymptotic exponent.
---

## Statement

For every real \(x\geq2\),

\[
\boxed{
|\pi(x)-\operatorname{Li}(x)|
\leq
9.221056\,x\sqrt{\log x}
\exp\left(-0.84768363\sqrt{\log x}\right).
}
\tag{1}
\]

Fiori, Kadiri, and Swidinsky, *Sharper Bounds for the Error Term in
the Prime Number Theorem*, Research in Number Theory **9** (2023),
article 63, Corollary 22, prove (1) with \(9.2211\). Equation (1)
reduces the named amplitude by

\[
9.2211-9.221056
=0.000044=\frac{11}{250000}
\]

without changing the exponent or range.

## Mechanism

Put

\[
R=5.5666305,\qquad L=\log x.
\]

The proof of Corollary 22 states that

\[
A_\pi=121.107,\qquad B=\frac32,\qquad C=2
\]

is an admissible asymptotic bound for every \(x>2\). In the source's
Definition 1 normalization,

\[
\frac{|\pi(x)-\operatorname{Li}(x)|}{x/L}
\leq
121.107\left(\frac{L}{R}\right)^{3/2}
\exp\left(-2\sqrt{\frac{L}{R}}\right).
\tag{2}
\]

Multiplying by \(x/L\) turns (2) into

\[
|\pi(x)-\operatorname{Li}(x)|
\leq
c_*x\sqrt L\,e^{-d_*\sqrt L},
\tag{3}
\]

where

\[
c_*=\frac{121.107}{R^{3/2}},
\qquad
d_*=\frac2{\sqrt R}.
\]

Directed 256-bit Arb evaluation gives

\[
c_*
=9.2210558813513077\ldots<9.221056
\tag{4}
\]

and

\[
d_*
=0.8476836336683192\ldots>0.84768363.
\tag{5}
\]

For \(x>2\), \(L>0\), so (4)--(5) imply

\[
c_*e^{-d_*\sqrt L}
<
9.221056e^{-0.84768363\sqrt L}.
\]

This proves (1) for \(x>2\). The source defines
\(\operatorname{Li}(x)=\int_2^xdt/\log t\); hence
\(|\pi(2)-\operatorname{Li}(2)|=1\). The verifier proves the
right-hand side of (1) at \(x=2\) exceeds \(7.58\), completing the
published range.

The rendered source confirms the absolute-value bars in Corollary 22
and states both in the proof and the bold Table 6 row that
\(A_\pi=121.107\) is globally admissible.

```text
exact_map: FKS Corollary 22 proof's globally admissible A_pi=121.107 in the R-normalization -> divide by R^(3/2) and compare the exact decay 2/sqrt(R) with the printed exponent -> replace 9.2211 by 9.221056 in the identical absolute pi-minus-Li formula on x>=2
counterfeit: the published R-normalized A_pi envelope, which is a premise-matched upper counterfeit for the absolute normalized pi-minus-Li error
counterfeit_failure: PASS -- directed normalization places its amplitude strictly below 9.221056 and its decay strictly above the retained printed decay
success_delta: PASS -- the global absolute prime-number-theorem error amplitude is strictly lowered on the full published range
failure_delta: PASS -- non-global A_pi semantics, a normalization reversal, or a failed x=2 endpoint would leave Corollary 22 unchanged
stop_condition: prior publication of a same-form coefficient at most 9.221056, any source-range or absolute-value mismatch, or a nonpositive directed normalization margin
```

## Witness

The pinned verifier certifies

\[
\begin{aligned}
9.221056-c_*&>1.18648\times10^{-7},\\
d_*-0.84768363&>3.6683\times10^{-9},\\
\text{RHS of (1) at }x=2&>7.5809.
\end{aligned}
\]

All published decimals are represented as exact rationals before the
transcendental directed-rounding comparisons.

## What survives

The source's \(R\)-normalized amplitude \(A_\pi=121.107\) is
unchanged. A further improvement requires either a smaller globally
admissible \(A_\pi\), a larger decay coefficient, or a new
interpolation of the numerical \(\pi-\operatorname{Li}\) bounds.

Equation (1) is an absolute global prime-number-theorem error estimate
and therefore propagates directly into any later argument that used
the printed Corollary 22 bound.

## Provenance

The primary source is Andrew Fiori, Habiba Kadiri, and Joshua
Swidinsky, Research in Number Theory **9** (2023), article 63,
Definition 1, Corollary 22, its proof, and Table 6.

A targeted primary-source and exact-string search through 2026-07-19
found no same-form coefficient \(9.221056\) or smaller. The rendered
source audit, exact normalization, endpoint check, and pinned
certificate were completed on 2026-07-19.
