---
id: C-0067
type: claim
title: "Fiori--Kadiri--Swidinsky's global absolute psi decay improves to 0.88178"
status: proved-unconditional
family: adjacent-unconditional
statement: >
  For every real x>2, |psi(x)-x| is less than
  9.2202181*x*(log x)^(3/2)*exp(-0.88178*sqrt(log x)).
  This retains C-0053's amplitude and range while replacing the
  Fiori--Kadiri--Swidinsky decay 0.8476836 by 0.88178.
verifier: canon/witnesses/C-0067/verify.py
depends: []
supersedes: [C-0053]
sources:
  - literature/2026-07-19-fks-yang-global-psi-decay-88178-search.md
  - experiments/fks_yang_global_psi_decay_88178/verify.py
  - scratch/adjacent-unconditional--fks-yang-global-psi-decay-improvement/PROOF.md
  - scratch/adjacent-unconditional--fks-yang-global-psi-decay-improvement/INDEPENDENT-REVIEW.md
tags: [chebyshev-psi, explicit-bound, pnt-error, zero-free-region, fiori-kadiri-swidinsky, yang, interval-arithmetic]
caveats: >
  The proof uses the exact interiorized denominator R=4.86201 so that
  the closed FKS hypothesis lies strictly inside Yang's open R=4.862
  zero-free region. The decimal 0.88178 is not asserted optimal over
  all handoffs or FKS parameters. The published FKS zero-density
  constants are primary-source inputs rather than independently
  rederived theorems.
---

## Statement

Let

\[
E_\psi(x)=\frac{|\psi(x)-x|}{x},
\qquad L=\log x.
\]

For every real \(x>2\),

\[
\boxed{
E_\psi(x)
<
9.2202181L^{3/2}\exp(-0.88178\sqrt L).
}
\tag{1}
\]

Fiori--Kadiri--Swidinsky, JMAA **527** (2023), article
127426, Corollary 1.4, prove (1) with amplitude \(9.22022\)
and decay \(0.8476836\). C-0053 sharpens the amplitude to
\(9.2202181\) with the original decay. Equation (1) retains that
amplitude and raises the decay by

\[
0.88178-0.8476836
=0.0340964
=\frac{85241}{2500000},
\tag{2}
\]

a relative increase of \(4.0223\%\). Its right side is therefore
strictly smaller than C-0053's for every \(x>2\).

## Mechanism

Yang's 2025 thesis, Theorem 1.1, proves

\[
\zeta(\sigma+it)\ne0
\quad(t\ge2),\qquad
\sigma>1-\frac1{4.862\log t}.
\tag{3}
\]

FKS Theorem 1.2 phrases its input on the closed boundary. Set

\[
R_*=4.86201>4.862.
\tag{4}
\]

For every \(t\ge3\), the closed \(R_*\)-region lies strictly inside
(3), so \(R_*\) is a literal admissible FKS input. It also satisfies
\(R_*<5.573412\).

Use

\[
\log x_0=2008,\quad c=4,\quad K=100000,\quad
c_1=17.4194,\quad c_2=2.9089.
\tag{5}
\]

The pinned verifier evaluates the complete \(99999\)-cell sum from
FKS equations (3.33) and (5.2) with directed 192-bit Arb arithmetic.
It gives

\[
A(2008,R_*)
=306.408402747535241262311165775\ldots.
\tag{6}
\]

The same implementation reconstructs all nine comparison rows in
FKS Remark 5.1 from below. Every gap between the printed admissible
value and the recomputed exact-parameter value lies in
\((0,0.0011)\).

Normalize (6):

\[
C_*=\frac{A(2008,R_*)}{R_*^{3/2}}
=28.5809657377046811451554534\ldots,
\tag{7}
\]

\[
d_*=\frac2{\sqrt{R_*}}
=0.9070308776151560217944122\ldots.
\tag{8}
\]

FKS Theorem 1.2 yields

\[
E_\psi(x)\le C_*L^{3/2}e^{-d_*\sqrt L}
\qquad(L\ge2008).
\tag{9}
\]

For \(0<L\le2008\), FKS Lemma 5.2 gives

\[
E_\psi(x)\le2L^{3/2}e^{-0.8476836\sqrt L}.
\]

After conversion to decay \(0.88178\), its required coefficient is
increasing in \(L\), and at the endpoint it is

\[
2e^{(0.88178-0.8476836)\sqrt{2008}}
=9.216830136364492\ldots
<9.2202181,
\tag{10}
\]

with directed margin greater than \(0.0033\).

For \(L\ge2008\), conversion of (9) to decay \(0.88178\) has a
decreasing required coefficient because \(d_*>0.88178\). At the
endpoint it is

\[
C_*e^{-(d_*-0.88178)\sqrt{2008}}
=9.218673292397481\ldots
<9.2202181,
\tag{11}
\]

with directed margin greater than \(0.0015\). Equations (10)--(11)
prove (1).

```text
exact_map: Yang's open classical zero-free denominator 4.862 -> interiorized closed FKS input R*=4.86201 -> exact FKS (3.33)/(5.2) amplitude at log x0=2008 -> normalize to C*=A/R*^(3/2), d*=2/sqrt(R*) -> splice FKS Lemma 5.2 below log x=2008 -> retain amplitude 9.2202181 and replace global decay 0.8476836 by 0.88178
counterfeit: substitute R=4.862 into the old FKS envelope while retaining A=121.096, or invoke Yang's open theorem as a closed-boundary theorem without interiorization
counterfeit_failure: PASS -- A is recomputed from all 99999 source cells, and R*=4.86201 makes the boundary containment literal
success_delta: PASS -- the named full-range absolute PNT-error decay rises by 4.0223% with unchanged improved amplitude, strictly reducing the bound at every x>2; this upstream global theorem is more consequential than C-0062's isolated prime-interval coefficients
failure_delta: PASS -- a source-row mismatch, nonpositive low or tail splice margin, or Yang/FKS boundary mismatch would leave C-0053 unchanged
stop_condition: prior same-range bound no larger, any source/range/formula mismatch, failure to reproduce the FKS comparison rows, nonpositive directed margin, or failure to strictly dominate C-0053 on every x>2
```

## Witness

The pinned verifier:

1. hash-pins the FKS PDF/text, Yang thesis PDF/text, and C-0053;
2. checks the exact open-to-closed radius padding;
3. reconstructs all nine FKS Remark 5.1 rows;
4. evaluates all \(99999\) terms of the new FKS amplitude;
5. verifies the FKS \(\sigma_2\) range at the handoff;
6. certifies (6)--(8);
7. proves both monotonic endpoint reductions and margins
   (10)--(11); and
8. verifies the exact amplitude and decay improvements.

An independent checker uses NumPy vectorization, `math.fsum`, and
80-digit `mpmath` rather than importing the Arb implementation. It
reproduces all source rows, (6)--(8), and both splice margins. A
separate read-only hostile review found no mathematical or verifier
correctness defects.

## What survives

The value \(0.88178\) is not claimed optimal. Optimizing the handoff,
the FKS parameters, or a later stronger zero-free region can raise
the decay further.

The theorem changes neither Yang's zero-free coefficient nor the
published FKS zero-density constants. It propagates those inputs into
a stronger global absolute prime-number-theorem error envelope, which
can replace C-0053 in downstream estimates for \(\theta\), \(\pi\),
weighted von Mangoldt sums, and prime intervals.

## Provenance

Primary sources:

- Andrew Fiori, Habiba Kadiri, and Joshua Swidinsky, *Sharper Bounds
  for the Chebyshev Function \(\psi(x)\)*, JMAA **527** (2023),
  article 127426, Theorem 1.2, Corollary 1.4, Lemma 5.2,
  Proposition 3.14, equations (3.24)--(3.33), and (5.1)--(5.2).
- Andrew Yang, *Explicit zero-free regions for the Riemann
  zeta-function*, UNSW PhD thesis (2025), DOI
  `10.26190/unsworks/31825`, Theorem 1.1.
- C-0053 for the immediately preceding full-range amplitude
  sharpening.

A targeted exact-string, primary-source, title, author, and
forward-use search through 2026-07-19 found no same-form decay
\(0.88178\), no stronger same-range pair, and no prior published
Yang-to-FKS recomputation. The source audit, directed certificate,
independent reconstruction, and hostile review were completed on
2026-07-19.
