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certificate C-0098

Fiori--Kadiri--Swidinsky's global absolute psi decay improves to 0.8817882

For every real $x>2$, the global absolute Chebyshev-$\psi$ bound holds with amplitude $9.2202181$ and decay $0.8817882$. This keeps the amplitude and full range of the preceding certified bound while increasing its decay from $0.88178$, and it improves the published Fiori-Kadiri-Swidinsky decay $0.8476836$ in the same displayed form.

\[|\psi(x)-x|<9.2202181x(\log x)^{3/2}e^{-0.8817882\sqrt{\log x}}\]
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What is proved

Writing $L=\log x$, the theorem gives one explicit inequality for the complete range $x>2$. Because the amplitude is unchanged from the preceding $0.88178$ result, the larger decay makes the right-hand side strictly smaller at every point in the range.

\[\frac{|\psi(x)-x|}{x}<9.2202181L^{3/2}e^{-0.8817882\sqrt L}\quad(x>2)\]

Supply an admissible zero-free input

The upstream classical zero-free theorem gives an open region with denominator $4.8568$. Fiori-Kadiri-Swidinsky formulate their input on a closed boundary, so the proof uses the slightly larger exact denominator $R_*=4.85681$.

That closed region lies strictly inside the proved open region. The padding is therefore a logical interiorization, not a numerical improvement of the zero-free theorem.

\[R_*=4.85681>4.8568\]
\[\zeta(\sigma+it)\ne0\quad\text{for }\sigma>1-\frac{1}{4.8568\log t}\]

Recompute the asymptotic envelope

At the handoff $L=2008$, the certificate recomputes all $99\,999$ cells in the Fiori-Kadiri-Swidinsky amplitude formulas using $R_*$. Normalization produces a tail coefficient $C_*$ and decay $d_*$.

Since $d_*>0.8817882$, conversion to the target decay becomes easier as $L$ increases. It is enough to check the tail at the handoff.

\[C_*=28.6643229697\ldots,\qquad d_*=\frac{2}{\sqrt{R_*}}=0.9075163092\ldots\]
\[E_\psi(x)\le C_*L^{3/2}e^{-d_*\sqrt L}\quad(L\ge2008)\]

Splice the low and high ranges

For $0<L\le2008$, the published low-range envelope has coefficient $2$ and decay $0.8476836$. Converting it to the target decay requires the largest coefficient at $L=2008$.

For $L\ge2008$, the recomputed asymptotic envelope requires its largest converted coefficient at the same endpoint. Directed arithmetic proves positive margins on both sides, so the two bounds splice into one all-range inequality.

\[d_{\rm low}=0.8476836+\frac{\log(9.2202181/2)}{\sqrt{2008}}=0.8817882015416\ldots\]
\[9.2202181-C_*e^{-(d_*-0.8817882)\sqrt{2008}}>0.1702763\]

Role of the certificate

The verifier hash-pins and reruns the C-0096 zero-free certificate and the C-0067 Fiori-Kadiri-Swidinsky reconstruction, including their source checks. It then restores directed precision, checks the open-to-closed padding, recomputes every amplitude cell, and certifies both splice margins.

The proof therefore covers the absolute value and every real $x>2$ rather than only the asymptotic range.

Pinned certificate

The certificate reruns the pinned zero-free and prior psi-bound dependencies, interiorizes the open zero-free radius, recomputes the complete Fiori-Kadiri-Swidinsky amplitude at the new input, and proves positive directed margins for both sides of the $\log x=2008$ splice.

uv run --frozen python canon/witnesses/C-0098/verify.py
  • canon/witnesses/C-0098/verify.py
  • canon/witnesses/C-0098/PIN.md
  • canon/witnesses/C-0096/verify.py
  • canon/witnesses/C-0067/verify.py
  • canon/claims/C-0096-global-classical-zeta-zero-free-denominator-4-8568.md
  • canon/claims/C-0067-fks-yang-global-psi-decay-0-88178.md

Scope

The bound holds for every real $x>2$ with amplitude $9.2202181$. The active low-range splice fixes the same-amplitude decay cap at $0.8817882015416\ldots$.

Sources

  • Canonical claimcanon/claims/C-0098-fks-c0096-global-psi-decay-0-8817882.md
  • Certificate pincanon/witnesses/C-0098/PIN.md
  • Zero-free dependencycanon/claims/C-0096-global-classical-zeta-zero-free-denominator-4-8568.md
  • Prior global psi boundcanon/claims/C-0067-fks-yang-global-psi-decay-0-88178.md