# Surviving Johnston--Yang global absolute-psi decay 0.2043325

## Theorem

For every real \(x\geq23\), put

\[
L=\log x,\qquad
r(L)=\frac{L^{3/5}}{(\log L)^{1/5}}.
\]

Then

\[
\boxed{
|\psi(x)-x|
<
0.026\,xL^{1.801}\exp(-0.2043325\,r(L)).
}
\tag{1}
\]

Johnston--Yang, JMAA **527** (2023), article 127460, Theorem 1.4,
prove (1) with decay \(0.1853\). C-0070 proves the same amplitude,
logarithmic power, and range with decay \(0.2043\). The exact
improvements are

\[
0.2043325-0.1853
=\frac{7613}{400000}
=0.0190325,
\tag{2}
\]

\[
0.2043325-0.2043
=\frac{13}{400000}
=0.0000325.
\tag{3}
\]

Thus (1) is strictly smaller than both prior envelopes at every point
of their common domain.

## Pinned inputs

The directed verifier hash-pins:

1. Johnston--Yang arXiv:2204.01980v2, including Theorem 1.4, the
   explicit-formula lemma, the zero-density interpolation rule, and
   the selected Table 3 rows;
2. Yang's 2025 thesis, Theorem 1.3, which proves the open global
   Vinogradov--Korobov region

   \[
   \zeta(\sigma+it)\neq0
   \quad(t\geq3),\qquad
   \sigma>
   1-\frac1{
   51.34(\log t)^{2/3}(\log\log t)^{1/3}};
   \tag{4}
   \]

3. C-0067, which gives for every \(x>2\)

   \[
   \frac{|\psi(x)-x|}{x}
   <
   9.2202181L^{3/2}e^{-0.88178\sqrt L};
   \tag{5}
   \]

4. C-0070 and its verifier, establishing the immediate same-form
   record \(0.2043\); and
5. Tao--Trudgian arXiv:2503.20170v4 as a pinned forward use of
   Johnston--Yang's published \(0.1853\) envelope.

Published analytic lemmas are treated as pinned inputs rather than
reproved from first principles.

## Closed zero-free input

Set

\[
c_*=51.3401.
\tag{6}
\]

The exact padding

\[
c_*-51.34=\frac1{10000}>0
\tag{7}
\]

places the closed \(c_*\)-boundary strictly inside the open region
(4). For

\[
\nu_c(t)=
\frac1{c(\log t)^{2/3}(\log\log t)^{1/3}},
\qquad
T=\min_{t\geq H}t x^{\nu_c(t)},
\tag{8}
\]

where

\[
H=3000175332800,
\tag{9}
\]

write \(\log t=\theta r(L)\). Direct substitution gives

\[
\frac{\log(t x^{\nu_c(t)})}{r(L)}
=
\theta+
\frac{1}{c\theta^{2/3}}
\left(
\frac{\log L}{\log(\theta r(L))}
\right)^{1/3}.
\tag{10}
\]

The minimization repaired in C-0070 yields

\[
\log T\geq Q(c)r(L),
\tag{11}
\]

\[
Q(c)=
\left(\frac5{3c^3}\right)^{1/5}
\left[
\left(\frac32\right)^{2/5}
+
\left(\frac23\right)^{3/5}
\right].
\tag{12}
\]

At \(c=c_*\), 256-bit Arb arithmetic gives

\[
Q(c_*)=
0.204352980832328860089304386414397\ldots.
\tag{13}
\]

The verifier derives (12) in two algebraically equivalent forms and
proves their interval difference is below \(10^{-60}\).

## Repaired tail and parameters

The Johnston--Yang Appendix A calculation requires two repairs already
made explicit in C-0070:

1. the printed \(B_2=0.18525\) with
   \(\sigma=0.9999932\) has native density decay only
   \(0.1852466408<0.1853\);
2. \(T\geq e^{B_2r}\) gives the explicit-formula remainder
   \(4.3128L^{0.6}e^{-B_2r}\), not the printed rational expression.

The source integration endpoint is handled by taking zero-free cutoffs
\(U\downarrow T\). This includes a possible zero at height \(T\) and
gives the valid limiting bound \(2N(\sigma,T)/T\).

Use

\[
L_0=950000000,\qquad
\sigma=0.9999714,\qquad
B_3=0.21617,
\tag{14}
\]

\[
C_1=17.418,\qquad C_2=2.077.
\tag{15}
\]

Johnston--Yang's directed interpolation between the
\(\sigma=0.999\) and \(1\) rows justifies these two density constants.
At \(L_0\), the fixed trial \(\theta=0.08917\) gives upper coefficient

\[
0.216165145820258281067409660194977\ldots<B_3.
\tag{16}
\]

The directed reserve is

\[
B_3-0.2161651458202582810\ldots
=4.8541797417189\ldots\cdot10^{-6}.
\tag{17}
\]

For fixed \(\theta\), the nonconstant factor in (10) decreases from
\(L_0\) onward by an exact derivative-sign reduction. Hence

\[
Q(c_*)r(L)\leq\log T\leq B_3r(L)
\quad(L\geq L_0).
\tag{18}
\]

Johnston--Yang's explicit-formula conditions hold uniformly from
\(L_0\). Their endpoint margins are

\[
Qr-\log(1.8L)
=27139.3025077678480\ldots>0,
\tag{19}
\]

\[
\frac{L}{35}-B_3r-\log(80/18)
=27114124.4884791275\ldots>0.
\tag{20}
\]

The verifier proves both margins increase thereafter.

## Density and explicit-formula terms

Put

\[
p=5-2\sigma=3.0000572,\qquad
a=\frac{5-8\sigma}{3}<0.
\tag{21}
\]

The repaired Johnston--Yang argument gives

\[
\begin{aligned}
s_1={}&
\frac{e^{-L/2}\log^2(H/2\pi)}{2\pi}\\
&+e^{-(1-\sigma)L}
\left(
\frac{B_3^2r^2}{2\pi}
-\frac{\log^2(H/2\pi)}{2\pi}
+1.8642
\right),\\
s_2={}&
2C_1e^{aQr}(Qr)^p
+2C_2e^{-Qr}(Qr)^2,\\
s_3={}&
4.3128L^{0.6}e^{-Qr}.
\end{aligned}
\tag{22}
\]

The native first-density decay is

\[
-aQ
=Q\frac{8\sigma-5}{3}
=0.204337395511657381141574908799860\ldots
>0.2043325.
\tag{23}
\]

Its directed reserve over the target is

\[
0.000004895511657381141574908799860\ldots.
\tag{24}
\]

The power condition also remains strict:

\[
\frac35p=1.80003432<1.801.
\tag{25}
\]

Normalize (22) by

\[
D(L)=0.026L^{1.801}e^{-0.2043325r(L)}.
\tag{26}
\]

The verifier proves every term of
\((s_1+s_2+s_3)/D(L)\) decreases for \(L\geq L_0\):

1. \(u(L)=r(L)q(L)/L\), where
   \(q(L)=3/5-1/(5\log L)\), decreases;
2. \(r(L)q(L)\) increases;
3. the normalized \(e^{-L/2}\) term decreases;
4. the second \(s_1\) term has directed derivative margin

   \[
   0.000011722996057078078829613503499\ldots>0;
   \tag{27}
   \]

5. both density terms and the explicit-formula remainder decrease
   after normalization by (23)--(25) and \(Q>0.2043325\).

At \(L=L_0\),

\[
\frac{s_1+s_2+s_3}{D(L_0)}
=
0.949873252885752217170749759629008\ldots<1.
\tag{28}
\]

The directed tail reserve is

\[
1-\frac{s_1+s_2+s_3}{D(L_0)}
=0.050126747114247782829250240370991\ldots.
\tag{29}
\]

The normalized first density term contributes
\(0.949872727541\ldots\); the remaining four terms together contribute
less than \(5.3\cdot10^{-7}\). Equations (18)--(29) prove (1) for
\(L\geq L_0\).

## Finite and middle ranges

For \(23\leq x\leq59\), \(\psi(x)\) is constant between consecutive
prime powers. The verifier proves the target increases on this range,
checks both endpoint errors on every prime-power interval, and obtains

\[
0.126811416960766941361118221026861\ldots>0.
\tag{30}
\]

For \(59<x\leq e^{58}\), Johnston--Yang's explicit Buethe bound gives
the comparison. At \(x=59\), the endpoint log-margin and derivative
margin are

\[
0.889968914239182871709327013444693\ldots>0,
\tag{31}
\]

\[
0.401400021433924593869405143388011\ldots>0.
\tag{32}
\]

The derivative remains positive, and
\(e^{58}<2.169\cdot10^{25}\), so the source range is complete.

For \(58\leq L\leq L_0\), compare (1) with C-0067. The logarithm of
the target divided by (5) is

\[
F(L)=
\log\frac{0.026}{9.2202181}
+0.301\log L
+0.88178\sqrt L
-0.2043325r(L).
\tag{33}
\]

Writing \(F'(L)=G(L)/L\), the verifier proves that the auxiliary
factor controlling \(G\) decreases. Consequently \(F\) can have an
interior maximum but no interior minimum. Its endpoint margins are

\[
F(58)
=0.301837071311214203822670793556104\ldots>0,
\tag{34}
\]

\[
F(950000000)
=20.7959452015482776112685417333861\ldots>0.
\tag{35}
\]

The independent checker locates the harmless interior maximum near

\[
L=153743265.839237463379778319447\ldots.
\tag{36}
\]

Equations (30)--(36) close every \(x\geq23\).

## Reproduction of the published theorem

The directed verifier separately reruns the repaired
Johnston--Yang tail at Ford's denominator \(57.54\), with

\[
\sigma=0.9999932,\qquad
B_3=0.20680,\qquad
L_0=2.8\cdot10^{10},\qquad
\theta=0.082.
\]

It exposes the printed \(B_2\) defect

\[
0.1853-
0.18525\frac{8\sigma-5}{3}
=0.0000533592>0,
\]

and reconstructs the valid source theorem with native decay

\[
0.1908381568236808908\ldots>0.1853.
\]

The repaired endpoint ratio is below \(10^{-2000}\). This reproduces
the named published baseline before recording either repository
improvement.

## Independent check

The independent checker uses 80-digit `mpmath`, separately coded
formulas, exact source hashes, a 20,001-point geometric bridge scan, a
numerical stationary-point solve, a numerical exact-\(T\) minimizer,
and 400 expanding tail samples. It reproduces:

- the finite, Buethe, and C-0067 margins;
- \(Q(51.3401)\);
- the exact-\(T\) coefficient
  \(0.2161651456898443277\ldots\) at \(L_0\);
- native decay \(0.2043373955116573\ldots\);
- endpoint tail ratio \(0.9498732528857522\ldots\);
- the printed \(B_2\) defect; and
- the repaired published theorem.

The independent path is corroborative. The Arb intervals and analytic
monotonicity reductions are the proof.

## Scope

No stronger zero-free denominator is used. In particular, the
retracted C-0080/C-0093 branch is not a dependency and contributes no
numeric input. This is a pure reoptimization of the surviving
\(c_*=51.3401\) architecture.

The continuous scout places the feasible boundary near
\(0.20433295\), but this is not an optimality theorem. The certified
decimal \(0.2043325\) deliberately retains visible margin rather than
rounding the numerical boundary inward.

```text
exact_map: PASS -- Yang open global VK denominator 51.34 -> exact closed interiorization 51.3401 -> Q(c*)=0.204352980832... -> reoptimized sigma=0.9999714, L0=950000000, theta=0.08917, B3=0.21617 -> repaired Johnston--Yang density and explicit-formula tail -> C-0067/Buethe/finite splices -> same 0.026 amplitude, 1.801 power, and x>=23 range with decay 0.2043325
counterfeit: PASS -- reuse the retracted 51.323 branch; round the continuous boundary; change sigma without rechecking s1; optimize the upper-T coefficient without preserving the published audit; or certify only the asymptotic native exponent
counterfeit_failure: PASS -- the verifier pins C-0067 and C-0070, uses only the surviving 51.3401 denominator, separates the new and published upper-T trials, reruns the repaired source theorem, and certifies every finite, Buethe, C-0067, explicit-formula, tail, and derivative condition
success_delta: PASS -- Johnston--Yang's named decay rises from 0.1853 to 0.2043325, and the immediate verifier-pinned record C-0070 rises from 0.2043 to 0.2043325, with unchanged amplitude, logarithmic power, and range
failure_delta: PASS -- any source drift, nonpositive splice or derivative margin, tail ratio at least one, native decay at most the target, or range failure leaves C-0070's 0.2043 theorem unchanged
stop_condition: PASS -- a located same-form all-x>=23 theorem with decay at least 0.2043325; source or dependency mismatch; use of the retracted branch; nonpositive directed margin; tail ratio at least one; or failure to preserve 0.026, 1.801, and x>=23
```
