% Local unrefereed draft; no submission, deposit or posting has been made. % Copyright 2026 Ricardo Pinho. CC BY 4.0. \documentclass[11pt]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,booktabs} \usepackage[expansion=false]{microtype} \usepackage[hyphens]{url} \usepackage[hidelinks]{hyperref} \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newcommand{\CC}{\mathbb C} \newcommand{\RR}{\mathbb R} \title[A step-profile certificate for Turan's constant]{A 256-step certificate for Tur\'an's pure power-sum constant} \author{Ricardo Pinho} \address{Independent researcher} \thanks{ORCID: \href{https://orcid.org/0009-0006-2744-2599}{0009-0006-2744-2599}. Licensed under CC BY 4.0.} \date{22 October 2026; local unrefereed draft} \keywords{Tur\'an power sums, coefficient cancellation, asymptotic construction, rigorous numerics} \begin{document} \begin{abstract} We give an exact-input certificate for the upper bound $C_{42}\le0.6888863712$ in the pure power-sum problem. The construction refines the step profiles of Griego, R\"ohrig and Plata to 256 middle blocks, without changing their coefficient-cancellation mechanism. It improves the strongest public proposal located in our 22 October 2026 audit, $0.688970$, by $0.0000836288$. An unmodified public interval checker and a separately written Arb checker verify the strict finite inequalities. The asymptotic implication is proved below; no explicit finite threshold, optimality, human peer review or formal verification is claimed. All material remains local and unsubmitted. \end{abstract} \maketitle \section{Statement and prior constructions} Define \[ R_n=\min_{\substack{z_1,\ldots,z_n\in\CC\\\max_j|z_j|=1}} \max_{1\le k\le n}\left|\sum_{j=1}^n z_j^k\right|, \qquad C_{42}=\limsup_{n\to\infty}R_n. \] This is constant 42a in Tao and contributors' table~\cite{table}. The condition is maximum modulus one; the other roots may lie inside the unit disk. \begin{theorem}\label{thm:main} The pure power-sum constant satisfies \[ C_{42}\le\frac{215276991}{312500000}=0.6888863712. \] \end{theorem} The table lists Harcos's $0.69368$ and Griego's proposed $0.6906538$~\cite{biro,griego}. Open PR~184 proposes R\"ohrig's eight-step bound $0.688983$~\cite{roehrig}. A separate public repository by Kerrigan Plata proposes $0.688970$ using six nonuniform middle levels~\cite{plata}. We reproduced both latter certificates before optimizing. Plata's value is the strongest public upper proposal located, and is the comparison used here; both proposals explicitly lack expert peer review. The local audit is bounded, not an exhaustive priority claim. We retain the generating function, free final block and Newton-identity argument of these works. The changes are a finer middle profile and new rational inputs. No new power-sum reduction theorem is needed. \section{A finite-step criterion} Put $\tau=1/3$ and $B=2/3$. Let \[ \tau=t_00,\quad s=1-\alpha, \] and choose middle values $\eta_0,\ldots,\eta_{M-1}$. Set $\delta_0=\eta_0-s$ and $\delta_j=\eta_j-\eta_{j-1}$ for $j>0$. Define \[ I_\alpha(x)=\int_0^x\frac{v^{\alpha-1}}{1-v}\,dv, \quad K=I_\alpha(\tau),\quad D=I_a(\tau)>0. \] For $t+r<1$ put \[ J_\alpha(t,r)=\int_0^{1-t-r}\frac{v^{\alpha-1}}{1-v} \log\frac{(1-t-v)(1-r-v)}{tr}\,dv, \] and put $J_\alpha(t,r)=0$ otherwise. Positive real powers use real logarithms. This also equals the integral of $(1-u-v)^{\alpha-1}/(uv)$ over $u\ge t$, $v\ge r$, $u+v\le1$. Finally set \begin{equation}\label{eq:Y} Y=1+sK-\sum_j\delta_j\bigl(I_\alpha(1-t_j)-K\bigr) +\frac12\sum_{i,j}\delta_i\delta_jJ_\alpha(t_i,t_j). \end{equation} \begin{proposition}\label{prop:criterion} If $|s|\lfloor nt_j\rfloor}, \qquad S_m=s+w_m. \] Every such value is one of the $\eta_j$. Write $\beta_l=(\alpha)_l/l!$ and $P_m=n\beta_{n-m}/m$ on $F$. These numbers are nonzero because $a>0$. Set \begin{align*} Y_n={}&n\beta_n-n\sum_{m\in M_n}\frac{w_m\beta_{n-m}}m +\frac n2\sum_{\substack{u,v\in M_n\\u+v\le n}} \frac{w_uw_v\beta_{n-u-v}}{uv}+s\sum_{m\in F}P_m,\\ S_m={}&\frac{Y_n}{\sum_{r\in F}|P_r|} \frac{\overline{P_m}}{|P_m|}\qquad(m\in F). \end{align*} Consider coefficients $b_l$ defined by \[ \mathcal B(z)=\frac1{1-z} \exp\left(-\sum_{m=1}^n\frac{S_mz^m}{m}\right) =\sum_{l\ge0}b_lz^l. \] Relative to $(1-z)^{-\alpha}$ every correction index exceeds $A$. Since $3(A+1)>n$, at most two corrections contribute to $b_n$. A final correction plus any other correction has index at least $n+1$. Expanding the exponential therefore gives exactly $nb_n=Y_n-\sum_{m\in F}S_mP_m=0$. Let $y_2,\ldots,y_n$ be the roots of $Z^{n-1}+b_1Z^{n-2}+\cdots+b_{n-1}$, with multiplicity, and let $y_1=1$. The logarithmic derivative of $\mathcal B$ and Newton's identities give the same triangular recurrence for $S_k$ and $Q_k=\sum_j y_j^k$ through $k=n$; at the last step use $b_n=0$. Hence $Q_k=S_k$ for $1\le k\le n$. Dividing all roots by $\Lambda=\max_j|y_j|\ge1$ cannot increase these power-sum moduli. It remains to bound the final powers. The gamma-ratio asymptotic is \[ \beta_l=\frac{l^{\alpha-1}}{\Gamma(\alpha)}(1+O(l^{-1})), \qquad \sum_{l\le L}|\beta_l|=O((L+1)^a). \] Away from $l=0$, ordinary Riemann sums give \[ \Gamma(\alpha)n^{-\alpha}Y_n\longrightarrow Y, \qquad |\Gamma(\alpha)|n^{-a}\sum_{m\in F}|P_m| \longrightarrow D. \] For completeness, discard first $l<\varepsilon n$ in each sum. In a single sum $m\asymp n$. In a double sum group by $l=n-u-v$: there are at most $n$ pairs and $uv\gg n^2$. After the relevant normalization the discarded contribution is $O(\varepsilon^a+n^{-a})$ in both cases. This permits $\varepsilon\downarrow0$, including the singular endpoint. The finite number of step discontinuities and the integer floors do not affect the limit. Thus $|Y_n|/\sum_F|P_m|\to|Y|/D0. \] Outward enclosures, shortened here, give \begin{align*} D&\in[0.892184209107241336566750826518, 0.892184209107241336566750826519],\\ |Y|/D&\in[0.688886370983465645822767803129, 0.688886370983465645822767803130],\\ C^2D^2-|Y|^2&\in[2.37472575330897264258, 2.37472575330897264259]\,10^{-10}. \end{align*} Proposition~\ref{prop:criterion} proves Theorem~\ref{thm:main}. Two independently authored implementations accept these inputs. The first is R\"ohrig's unchanged public \texttt{mpmath.iv} checker at 100 decimal places, using explicit geometric series tails for $I$ and $J$. The second, written by an independent agent from the mathematical notes without reading either search code or the first checker, uses 768-bit Arb balls. It computes $I$ through \[ I_\alpha(x)=\frac{x^\alpha}{\alpha} {}_2F_1(1,\alpha;\alpha+1;x) \] and $J$ through a separately bounded logarithmic series with 400 terms. No optimization status, sampled integral or floating-point maximum is accepted as a certificate. Corrupted profiles, malformed thresholds and the too-strong radius $0.6888863709$ are rejected. \section{Reproduction and limits} The local package includes pinned public source code, both checkers, exact data and transcripts. From its root, with Python, mpmath and python-flint available, run \begin{verbatim} python verify_primary.py certificates/c42_256.json python independent/check_bounds.py \ certificates/c42_256.json --bits 768 --terms 400 \end{verbatim} The proof gives existence of an eventual threshold, not its numerical value. The optimizer's final status was a line-search warning with tiny floating constraint residuals; only the subsequent strict interval checks justify the result. Neither the profile nor its resolution is claimed optimal. The independent review is a separate AI-agent review, not independent human expert review or proof-assistant formalization. Final source checks and the exact search commands are recorded in the local \texttt{PRIOR\_ART.md} and \texttt{REPORT.md}. \section*{Acknowledgments and assistance} The method is due to Griego and its step-profile refinements to R\"ohrig and Plata. Analysis and proof drafting by AI agents under my direction; I reviewed the results at summary level. Search, code, verification and exposition were developed with OpenAI Codex assistance; the second checker was written in a separate agent context. No submission, upload or contact occurred. \begin{thebibliography}{9} \bibitem{table} T. Tao and contributors, \emph{Optimization constants in mathematics, 42a}, \url{https://teorth.github.io/optimizationproblems/constants/42a.html}. \bibitem{biro} A. Bir\'o, \emph{An upper estimate in Tur\'an's pure power sum problem}, Indag. Math. (N.S.) \textbf{11} (2000), 499--508. \bibitem{griego} S. Griego, \emph{An improved asymptotic certificate for Tur\'an's pure power sum constant}, v1.0.0, \url{https://github.com/sebastian-griego/turan-c42-certificate/tree/v1.0.0}. \bibitem{roehrig} A. R\"ohrig, \emph{Verification certificates for C42 and C51}, 9 September 2026, commit ed6387399cf6de02eaa4cf10c43787633496619a, \url{https://github.com/AndreRoehrig/optimization-constants-certificates}. \bibitem{plata} K. Plata, \emph{Proposed six-level bound for Turan's pure power-sum constant}, public review draft, \url{https://github.com/krplatz/turan-c42-six-level-review}. \end{thebibliography} \end{document}