bibkey: cullyhugilljohnston2026truncation authors: Michaela Cully-Hugill and Daniel R. Johnston year: 2026 title: On the error term in the explicit formula of Riemann-von Mangoldt II doi: null url: https://arxiv.org/abs/2402.04272v3 claim: The primary selected-height theorem supplies an explicit complete signed degree-one tail allowance at each actual source; its cutoff selection does not control the corresponding joint Robin finite head. strata_touched: [] license: citation-only triage: anchor

Selected-height explicit formula and the original signed suffix

The primary is Cully-Hugill–Johnston, arXiv:2402.04272v3, Theorem 1.2 on printed p.2 and Table 1 on printed p.3. The versioned PDF has 19 pages and SHA-256 9bf784f264970a5bb82e54547c2dbd68b1a4cc6a69ee679b8fc2159d006a565c. The inspected statement and numerical row are literature suppliers; the full proof and the table-generating computations are not independently reproduced here. No Lean verification is claimed.

Exact supplier contract

The source uses the weak prime-power function $\psi (x)={\sum }_{n\le x}\Lambda (n)$, with every nontrivial zero counted with multiplicity. For fixed $0<\alpha \le 1/2$ and $0\le \omega \le 1$, Theorem 1.2 supplies constants $M,x_M$ such that for every $x\ge x_M$ and

$$\max \{51,{\log }^{2}x\}<Y<\frac{x^{\alpha }-2}{4},$$

there exists $T^{\ast }\in [Y,2Y]$ with

$$\psi (x)=x-\sum _{|\mathrm{Im}\rho |\le T^{\ast }}{m}_{\rho }\frac{x^{\rho }}{\rho }+e(x,T^{\ast }),|e(x,T^{\ast })|\le \frac{Mx(\log x{)}^{1-\omega }}{Y}.$$

The Table 1 row used for the original-source application is $\alpha =1/2$, $\omega =1$, $\log x_M=10^{13}$ and $M=19.81$. Thus its source threshold is $x\ge \exp (10^{13})$, not $x\ge 10^{13}$. This row is attributed to the authors; quoting it is not an independent certificate of their computations.

The height is selected by the theorem for each actual $x$. The contract does not apply to every prescribed height in $[Y,2Y]$, and separate pointwise applications do not select a common height for different sources. The source’s Theorem 1.3 and Corollary 3.3 provide a distinct common-height interval formulation; they are not silently substituted into the pointwise application.

Preserve the complete conjugate-pair endpoint

Let $P_x(T)=2\operatorname{Re}\sum_{0<\operatorname{Im}\rho\le T} m_\rho x^\rho/\rho$ and $q(x)=\log (2\pi )+\frac{1}{2}\log (1-x^{-2})$. Let ${\Lambda }_{\mathbb{R}}(x)$ equal $\log p$ at an actual integer prime power $x=p^k$, and zero otherwise. The classical complete explicit formula with symmetric height ordering gives

$$P_x(\infty )=x-\psi (x)-q(x)+\frac{1}{2}{\Lambda }_{\mathbb{R}}(x).$$

The source’s selected-height statement therefore directly supplies the complete real paired tail $H_x(T^{\ast })=P_x(\infty )-P_x(T^{\ast })$:

$$\left|H_x(T^{\ast })+q(x)-\frac{1}{2}{\Lambda }_{\mathbb{R}}(x)\right|\le \frac{Mx(\log x{)}^{1-\omega }}{Y}.$$

The half-prime-power atom matches weak $\psi$ to the symmetric endpoint. Actual zero real parts and multiplicities remain in $x^{\rho }/\rho$; no critical-line replacement is made. The statement concerns the real conjugate-pair tail, not ordinary convergence of the complex positive-ordinate half-sum at a prime power. No separation from prime powers is assumed: the full weak-$\psi$ error is paid by the primary theorem in its stated domain.

Consumer and remaining boundary

The original-core application uses $x=A=\log N$, while retaining the original integrated response at $B=\log u$ with $N=pu$. Choosing $Y=(\sqrt{A}-4)/4$ satisfies the supplier’s strict domain after its threshold and selects $T^{\ast }\asymp \sqrt{A}$. Combining this direct degree-one supplier with the existing complete degree-two suffix contract J1–J6 pays the actual signed compensated suffix at those same selected cutoffs. It does not reproduce J1–J6 as a new theorem or estimate each zero of the degree-one infinite tail by absolute value.

The pointwise theorem supplies no favorable sign for the corresponding joint finite head, no free payment of $K(A)-\psi (A)$, and no common cutoff chosen independently of the sources. The original actual complete-head condition CR23, PG15/J and RH remain unproved. The application is supplier-relative, without a literature-originality claim or a new generic explicit-formula theorem.