Johnston--Yang's surviving global VK psi decay improves to 0.2043325

The improved envelope

For $L=\log x$ and $r(L)=L^{3/5}/(\log L)^{1/5}$, the claimed envelope holds for every real $x\ge23$. It improves the published decay $0.1853$ and the previous certified same-form decay $0.2043$ without changing any other headline parameter.

Because the amplitude, power, and domain are identical, the larger decay constant makes the new right-hand side strictly smaller at every point of the common range.

The surviving zero-free input

Yang's theorem gives an open Vinogradov--Korobov zero-free region with denominator $51.34$. Replacing it by the closed denominator $c_*=51.3401$ moves the boundary strictly inside that open region and introduces an exact padding of $10^{-4}$.

Minimizing the Johnston--Yang height expression then gives a lower coefficient $Q(c_*)=0.2043529808\ldots$, which is safely above the target decay. No input from the retracted $51.323$ branch is used.

Large-x tail

The proof chooses $L_0=950000000$, $\sigma=0.9999714$, $\theta=0.08917$, and $B_3=0.21617$. Directed interpolation of the source density rows supplies the associated constants, while the fixed trial proves $Q(c_*)r(L)\le\log T\le B_3r(L)$ for every $L\ge L_0$.

The repaired explicit-formula terms are normalized by the target envelope. Each normalized contribution is proved decreasing from $L_0$ onward, and their endpoint sum is below one with reserve about $0.0501$. The native first-density decay is $0.2043373955\ldots$, still strictly above $0.2043325$.

1. Interiorize the open zero-free denominator and derive the coefficient $Q(c_*)$.
2. Choose a source-admissible density tuple and prove the upper and lower height bounds uniformly from $L_0$.
3. Normalize every explicit-formula term by the target envelope and reduce the tail to endpoint inequalities plus monotonicity.

Closing the full range

For $23\le x\le59$, the verifier uses that $\psi(x)$ is constant between prime powers and checks both endpoints of every interval. From $59$ to $e^{58}$ it compares against Johnston--Yang's explicit Buethe bound.

For $58\le L\le L_0$, the target is compared with C-0067's global FKS--Yang bound. An analytic derivative reduction shows that the logarithmic comparison has no interior minimum, so positive endpoint margins cover the entire middle interval.

What the certificate establishes

The directed 256-bit Arb verifier hash-pins the Johnston--Yang source, Yang's thesis, C-0067, C-0070, and a forward use of the published bound. It checks the source interpolation, the open-to-closed padding, the exact $Q(c)$ calculation, every tail derivative, every range splice, and both exact improvements.

It also reruns the repaired published $0.1853$ theorem, including the printed $B_2$ defect, so the comparison baseline is reproduced rather than assumed. A separately pinned independent checker is corroborative; the interval bounds and analytic monotonicity reductions are the proof.