% >>> [agent-claude 2026-10-23 thakur-conjectures harvest] \documentclass[11pt]{amsart} \usepackage[margin=1.1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage[T1]{fontenc} \usepackage{microtype} \usepackage{booktabs,tabularx} \usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{claimed}[theorem]{Claimed result} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{evidence}[theorem]{Computational evidence} \newcommand{\F}{\mathbb F} \newcommand{\Fq}{\mathbb F_q} \newcommand{\Q}{\mathbb Q} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \newcommand{\C}{\mathbb C} \newcommand{\Kbar}{\overline{K}} \newcommand{\pit}{\tilde\pi} \newcommand{\Ap}{A_{+}} \newcommand{\val}{\operatorname{val}} \newcommand{\dlog}{\operatorname{dlog}} \newcommand{\ev}{\operatorname{ev}} % >>> [agent-codex 2026-10-27 query-report-front-matter] % \title[Thakur's conjectures in function field arithmetic]{FRONTIER DUMP FOR Taylor Dupuy on 2026-10-23 on Thakur's conjectures in function field arithmetic} % \thanks{Generated with the agent skills of % \href{https://github.com/tdupu/mathcity}{mathcity}, following the % \href{https://github.com/tdupu/mathcity/blob/main/subdomains/latex/skills/using-latexpowers/SKILL.md}{\texttt{using-latexpowers} skill}. % The generation details are recorded in the % \href{https://github.com/tdupu/dumps/blob/main/ai/ai-usage.md}{master AI-usage statement} % of the \texttt{tdupu/dumps} repository, which may be private.} % \date{October 23, 2026} \title[Thakur's conjectures in function field arithmetic]{Thakur's Conjectures in Function Field Arithmetic: A Report On Queries to Frontier Agents} % \date{Created: 2026-10-23; revised: 2026-10-27} \date{23 October 2026 (created); 27 October 2026 (revised)} % <<< [agent-codex 2026-10-27 query-report-front-matter] \begin{document} \begin{abstract} % >>> [agent-codex 2026-10-27 query-report-abstract] % This manuscript arose from an AI-assisted investigation initiated with % the \texttt{frontier-dump} skill during Taylor Dupuy's talk \emph{How to Use % AI in Mathematical Research Effectively} at the University of Rochester % Number Theory Seminar on 23 October 2026. The resulting research reports % were synthesized with \texttt{harvest-dumps} and developed into this % manuscript with \texttt{using-latexpowers}. This report was created on 23 October 2026 from six AI research reports generated with the \texttt{frontier-dump} skill on Dinesh Thakur's conjectures in function field arithmetic. Taylor Dupuy initiated those queries during his talk \emph{How to Use AI in Mathematical Research Effectively} at the University of Rochester Number Theory Seminar on that date. The six reports were synthesized with \texttt{harvest-dumps} and developed into this manuscript with \texttt{using-latexpowers}. % <<< [agent-codex 2026-10-27 query-report-abstract] We survey the current status of several conjectures of Dinesh Thakur and his coauthors on the arithmetic of $\Fq[t]$: the degree hypotheses for power sums, the characterisation of zeta-like multizeta values in genus zero and in higher genus, the relations among multizeta values, the distribution of zeros of Goss zeta functions, and the automata-theoretic transcendence questions for special values. For each conjecture we give the precise statement and its status in the literature. The six computational research dumps report several results and refutations. Among them, the recursion predicted by Thakur's second power-sum hypothesis fails for an explicit family of exponents at thirteen non-prime values of $q$ (a computer-assisted result); the shuffle relations, even with Frobenius and Carlitz's Eulerian evaluations, do not generate all relations among multizeta values; a closed formula for a $v$-adic valuation difference conjectured by Thakur holds only after a sign change (and, for $d\ge4$, halving), and the corrected formula is proved, with computer assistance, for degrees $d\le 11$; and the $v$-adic Newton polygon of a Goss zeta function over $\F_2[t]$ at a prime of degree two has a slope of multiplicity one, which appears to contradict a result claimed in a 2023 preprint (this conflict needs human verification). The shuffle result and an assembly of published theorems describing all relations are elementary, and we claim no novelty for them. We also write out a proof of a theorem of Wen and Yao on $v$-adic arithmetic gamma values. Computational evidence is labelled as such throughout, and the open problems are listed. This manuscript is AI-generated by Claude Code (model \texttt{claude-opus-5-5}) from AI-generated research dumps; its proofs received only AI review, and no human has checked them. \end{abstract} \maketitle \begin{center} \textbf{AI Generated}\qquad\textbf{PRINTED BY: Taylor Dupuy} \end{center} \tableofcontents %====================================================================== \section{Introduction}\label{sec:intro} Thakur's work on $A=\Fq[t]$ has produced a large and precise set of conjectures, many of them found by computation: on the degrees of power sums of monic polynomials, on which multizeta values are rational multiples of a single zeta value, on the relations among multizeta values, on the valuations of $v$-adic power sums and hence the zeros of Goss's $v$-adic zeta functions, and on the algebraicity of special values via automata. Some of these conjectures have since been proved, some only in part, and some have turned out to be false. This paper records their status as of October 2026 and adds new partial results, corrections and counterexamples. A single example shows how prime-field intuition can fail. For $q$ prime, Thakur proved that the degrees $s_d(k)=-\deg\sum_{a}a^{-k}$ (sum over monic $a$ of degree $d$) satisfy the recursion $s_d(k)=s_{d-1}(s_1(k))+s_1(k)$. For $q=4$ and $k=75$ one computes $s_1(75)=96$ and $s_1(96)=384$, so the recursion predicts $s_2(75)=480$, while in fact $s_2(75)=348$ (Example~\ref{ex:q4k75}; this example is Thakur's). Section~\ref{sec:powersums} shows that the failure is systematic. \subsection*{Audience and conventions} We write for readers who know the Carlitz module, its period $\pit$, the Carlitz zeta values, Thakur's multizeta values and Goss's zeta functions. We recall definitions and fix notation in Section~\ref{sec:notation}, but we do not develop the theory. Every statement carries one of the following statuses. \begin{itemize} \item \emph{Imported} results are cited at the head of the statement. When we did not read the cited passage ourselves, we say how it was seen (abstract only, or as reported by another source). \item Results announced in \emph{unrefereed preprints} are stated as \emph{claimed results} or in prose, never as theorems. This applies to \cite{ChenOno2026}, \cite{Hu26}, \cite{KMU23}, \cite{CWY24} and \cite{DVPC14}. \item Results \emph{proved here} have complete proofs in this paper. Some are new, some are elementary assemblies of published theorems, and some write out arguments that the literature omits; we say which. \item \emph{Computer-assisted} results are finite statements certified by an exact computation, and the certificate is described. \item \emph{Computational evidence} (the environment of that name) records finite numerical observations. It never establishes the general claim. \item \emph{Conjectures} are attributed. New conjectures are marked as such. \end{itemize} Results from the six AI-generated research records that this paper harvests are cited as \cite{DataPS,DataZL,DataHG,DataMZ,DataGZ,DataAT}; certificates name the scripts and data files there. Each section begins with the refutations and corrections it contains and then turns to positive results. Diophantine approximation exponents and continued fractions are outside the scope of this paper. \subsection*{Summary} Table~\ref{tab:status} summarises the status of each conjecture. \begin{table}[ht] \small \begin{tabularx}{\textwidth}{@{}p{3.3cm}Xp{4.2cm}@{}} \toprule Conjecture & Status in the literature & New here\\ \midrule H1, H2 (power sums) \cite{Thakur2009PowerSums} & Recursion for $q$ prime: \cite{Thakur2009PowerSums}; H1, H2 for $q=p$ claimed in the preprint \cite{ChenOno2026} & The H2 recursion fails at $13$ non-prime $q$ (Thm.~\ref{thm:H2-fails}); H1 open (evidence only)\\ Zeta-like MZVs, $\Fq[t]$ \cite{LRT14} & Weight $\le q^2$ settled \cite{LeNgoDac22}; families proved \cite{Chen17}; tails Eulerian \cite{CPY19} & Props.~\ref{prop:zl-depth2-min}, \ref{prop:zl-weight-bound}; completeness above $q^2$ open\\ Zeta-like, higher genus \cite{LRT20} & Both numbered conjectures proved \cite{CNP23} & Erratum in a display of \cite{CNP23} (Rem.~\ref{rem:rteg2})\\ All MZV relations & Todd's conjecture: $B^*$ form proved \cite{CCM23}; $B$ form claimed in the preprint \cite{Hu26} & Shuffle-type relations do not suffice (Thm.~\ref{thm:not-shuffle}, elementary)\\ Goss zeros, $v$-adic \cite{Thakur2013Vadic} & Ordinary curves: claimed in the preprint \cite{KMU23} & Lemmas \ref{lem:LB}, \ref{lem:L}; Thakur's formulas proved for $d\le 11$ with one correction (Thm.~\ref{thm:thakur-vadic}); apparent conflict with \cite[Thm.~1.10]{KMU23}\\ Automata, $v$-adic $\Gamma$ \cite{WenYao02} & Open beyond $\deg v=1$ and residue $0$; rational arguments claimed in the preprint \cite{CWY24} & Proof of \cite[Thm.~8]{WenYao02} written out (Thm.~\ref{thm:wen-yao-A})\\ \bottomrule \end{tabularx} \caption{Status of the conjectures discussed in this paper.} \label{tab:status} \end{table} %====================================================================== \section{Notation and standing facts}\label{sec:notation} Throughout, $q=p^f$ is a power of the prime $p$, $A=\Fq[t]$, $K=\Fq(t)$, and $K_\infty=\Fq((1/t))$ is the completion at the infinite place, with completed algebraic closure $\C_\infty$. We write $\Ap$ for the monic elements of $A$ and $A_{+,d}$ for those of degree $d$. Put $[n]=t^{q^n}-t$, $D_n=\prod_{i=1}^{n}[i]^{q^{n-i}}$ (so $D_0=1$ and $D_n$ is the product of the elements of $A_{+,n}$), and $\ell_n=\prod_{i=1}^{n}(t-t^{q^i})$. A finite place is a monic irreducible $v\in A$; we write $D=\deg v$ and reserve $d$ for the degree of summation in power sums. $K_v$ is the completion of $K$ at $v$, and $\val_v$ its normalised valuation. \begin{definition}\label{def:powersums} Let $k\in\Z$ and $d\ge0$. \begin{enumerate} \item The \emph{power sum} is $S_d(k)=\sum_{a\in A_{+,d}}a^{-k}\in K$, and $s_d(k)=-\deg S_d(k)=\val_\infty S_d(k)$. \item For a finite place $v$, the \emph{$v$-adic power sum} is $S_{d,v}(k)=\sum_{a\in A_{+,d},\,v\nmid a}a^{-k}\in K$, and $v_d(k)=\val_v S_{d,v}(k)$ (with $v_d(k)=\infty$ if $S_{d,v}(k)=0$). \end{enumerate} \end{definition} This is the notation of \cite{Thakur2009PowerSums,Thakur2013Vadic}; note that $S_d(k)$ has exponent $-k$. \begin{definition}\label{def:mzv} Let $\mathfrak s=(s_1,\dots,s_r)$ be a tuple of positive integers. \begin{enumerate} \item The \emph{weight} of $\mathfrak s$ is $w(\mathfrak s)=s_1+\dots+s_r$ and its \emph{depth} is $r$. \item Thakur's \emph{multizeta value} is $\zeta(\mathfrak s)=\sum_{d_1>d_2>\dots>d_r\ge0} S_{d_1}(s_1)\cdots S_{d_r}(s_r)\in K_\infty$. \item $\zeta(\mathfrak s)$ (or $\mathfrak s$) is \emph{zeta-like} if $\zeta(\mathfrak s)/\zeta(w(\mathfrak s))\in K$. \item $\zeta(\mathfrak s)$ is \emph{Eulerian} if $\zeta(\mathfrak s)/\pit^{\,w(\mathfrak s)}\in K$. \end{enumerate} \end{definition} Sections~\ref{sec:highergenus} and~\ref{sec:goss} also use a more general base ring. \begin{definition}\label{def:general-A} Let $X$ be a smooth, projective, geometrically irreducible curve over $\Fq$ of genus $g$, and let $\infty$ be a closed point of $X$ of degree $d_\infty$. \begin{enumerate} \item The \emph{base ring} is the ring $A$ of functions on $X$ regular away from $\infty$. Its fraction field is $K$, and $K_\infty$ is the completion at $\infty$. \item The \emph{class number} of $A$ is the order of its ideal class group, and $H\subset K_\infty$ is the Hilbert class field of $A$ in the sense of \cite{CNP23} (the maximal unramified abelian extension of $K$ in which $\infty$ splits completely). \item Suppose $d_\infty=1$. Fix a uniformiser of $K_\infty$ and the sign function $\operatorname{sgn}\colon K_\infty^\times\to\Fq^\times$ it determines \cite[\S1.2]{CNP23}. An element $a\in A$ is \emph{monic} if $\operatorname{sgn}(a)=1$, and $A_{+,d}$ is the set of monic $a$ with $\deg a=-\val_\infty(a)=d$. The power sums $S_d(k)$ and the multizeta values $\zeta_A(\mathfrak s)$ are defined by the formulas of Definitions~\ref{def:powersums} and~\ref{def:mzv} with these $A_{+,d}$, and $\zeta_A(\mathfrak s)$ is \emph{zeta-like} if $\zeta_A(\mathfrak s)/\zeta_A(w(\mathfrak s))\in K$. \item With $d_\infty=1$, let $\pit_A$ be the period of the standard sign-normalised rank-one Drinfeld module attached to $A$. Following \cite{LRT20} and \cite{CNP23}, $\zeta_A(\mathfrak s)$ is \emph{Eulerian} if $\zeta_A(\mathfrak s)/\pit_A^{\,w(\mathfrak s)}$ is algebraic over $K$. \item The \emph{$p$-rank} of $X$ is the $\F_p$-dimension of the $p$-torsion of its Jacobian over $\overline{\F}_q$. $X$ is \emph{ordinary} if its $p$-rank equals $g$; an elliptic curve is \emph{supersingular} if its $p$-rank is $0$. \end{enumerate} \end{definition} For $A=\Fq[t]$ the Carlitz module is the sign-normalised module, and the two notions of Eulerian in Definitions~\ref{def:mzv} and~\ref{def:general-A} agree: by Chang's Euler dichotomy \cite[Cor.~2.3.3]{Chang14}, a multizeta value $Z$ of weight $w$ either has $Z/\pit^{\,w}\in K$ or is algebraically independent from $\pit$. When $A$ has class number one, $H=K$. Lara Rodr\'iguez and Thakur state that, apart from the rings $\Fq[t]$, exactly four base rings with $d_\infty=1$ have class number one \cite[\S3]{LRT20}:\footnote{Kramer-Miller and Upton \cite[\S1.1]{KMU23} speak of ``exactly eight such $A$ in all characteristics''; we did not reconcile the two counts, which may use different conventions for $\infty$.} \begin{enumerate} \item[(i)] $\F_2[x,y]/(y^2+y=x^3+x+1)$; \item[(ii)] $\F_3[x,y]/(y^2=x^3-x-1)$; \item[(iii)] $\F_4[x,y]/(y^2+y=x^3+w)$, where $w^2+w+1=0$; \item[(iv)] $\F_2[x,y]/(y^2+y=x^5+x^3+1)$. \end{enumerate} In each case $\infty$ is the unique point at infinity of the affine model. These are the curves used in the computations of Section~\ref{sec:goss}. The first three curves have genus $1$ and the last has genus $2$. The three elliptic curves are supersingular (in characteristic $2$ the Weierstrass equation has $a_1=0$, and in characteristic $3$ it has $a_2=0$), and the genus-$2$ curve (iv), an Artin--Schreier curve $y^2+y=f(x)$, has $p$-rank $0$. We refer to these rings as (i)--(iv) throughout. We use two standard facts without further comment. Every $\zeta(\mathfrak s)$ is nonzero \cite{Thakur2009Relations}. By Carlitz, $\zeta(n)/\pit^{\,n}\in K$ when $(q-1)\mid n$ \cite[\S5]{Thakur2004Book}, and by Yu's theorem it is transcendental otherwise \cite{Yu91}. Hence a zeta-like value of weight divisible by $q-1$ is Eulerian. Conversely, an Eulerian value is zeta-like, and its weight is divisible by $q-1$, because $\pit^{\,n}\in K_\infty$ only when $(q-1)\mid n$. %====================================================================== \section{Power sums: Thakur's hypotheses H1 and H2}\label{sec:powersums} \subsection{Statements} For $k\ge0$ and $d\ge0$, Carlitz's generating-function theorem \cite[2.3]{Thakur2009PowerSums} expresses $S_d(k+1)$ as a finite sum over decompositions $k=\sum_{i=0}^{d}k_iq^i$ with $k_i\ge0$. By Lucas's theorem only decompositions in which the $k_i$ add without carry in base $p$ contribute. The term of such a decomposition has degree $b(0,d)+\sum_i k_i\,b(i,d)$, where \begin{equation}\label{eq:bid} b(i,d)=-\Bigl[\frac{q^{d+1}-q^{i+1}}{q-1}+iq^i\Bigr] \qquad(0\le i\le d), \end{equation} as in \cite[2.3.1(4)]{Thakur2009PowerSums}. \begin{remark}\label{rem:bid-sign} In a plain-text extraction of \cite[(4)]{ChenOno2026} the term $iq^i$ appears outside the bracket, which one might take as an alternative convention. In fact the bracketed form \eqref{eq:bid} is the one that matches the exact degrees in Section~\ref{sec:ps-results}; the discrepancy may be an extraction artefact, and the typeset preprint was not checked. \end{remark} \begin{conjecture}[Hypothesis H1, {\cite[2.3.1]{Thakur2009PowerSums}}]\label{conj:H1} For every $q$, $d$ and $k$, among the carry-free decompositions of $k$ in Carlitz's formula for $S_d(k+1)$ there is exactly one whose term has maximal degree. \end{conjecture} Under H1, $s_d(k+1)=-(b(0,d)+\sum k_ib(i,d))$ for the maximising decomposition. The second hypothesis comes from the expansion $S_{d}(k)=-\sum_{j\ge1,\,(q-1)\mid j}\binom{-k}{j}t^{-(k+j)}S_{d-1}(k+j)$ \cite[3.2]{Thakur2009PowerSums}. \begin{conjecture}[Hypothesis H2, {\cite[3.2]{Thakur2009PowerSums}}]\label{conj:H2} Let $q$ be prime and $k,d\ge1$. Then the minimum of $s_{d-1}(s_1(k)+j)+s_1(k)+j$ over $j\ge0$ with $(q-1)\mid j$ and $\binom{s_1(k)+j-1}{k-1}\not\equiv0\pmod p$ is attained exactly once, at $j=0$. Consequently \begin{equation}\label{eq:recursion} s_d(k)=s_{d-1}(s_1(k))+s_1(k). \end{equation} \end{conjecture} \begin{theorem}[{\cite[Thm.~1]{Thakur2009PowerSums}}]\label{thm:thakur-recursion} If $q$ is prime, then \eqref{eq:recursion} holds for all $k\in\Z$ and $d\ge1$. \end{theorem} Thakur also states a third hypothesis. \begin{conjecture}[Hypothesis H3, {\cite[A.10]{Thakur2009PowerSums}}]\label{conj:H3} For every $q$, $d\ge1$ and $k\ge1$ with $p\nmid k$, $s_d(k)1$ the unit lands at position $e+fi$, each base-$p$ digit holds at most $p-1$ units, and positions $f$ apart are coupled. Equal-cost exchanges therefore exist (the tie of Remark~\ref{rem:A8tie} is one). A proof of H1 for $q=p^f$ must show that no equal-cost exchange is available at the optimum. \end{remark} %====================================================================== \section{Zeta-like multizeta values over \texorpdfstring{$\Fq[t]$}{Fq[t]}}\label{sec:zetalike} \subsection{The conjectures} Since $\zeta(p\mathfrak s)=\zeta(\mathfrak s)^p$, one restricts to \emph{primitive} tuples, those $\mathfrak s$ whose entries are not all divisible by $p$ \cite[\S2]{LRT14}. Lara Rodr\'iguez and Thakur \cite{LRT14} conjectured a description of zeta-like tuples of depth $r\ge2$. We list their conjectures, numbered as in \cite{LRT14}. \begin{conjecture}[{\cite[Conj.~4.1--4.5]{LRT14}}]\label{conj:LRT14} Let $\mathfrak s=(s_1,\dots,s_r)$ with $r\ge2$. \begin{enumerate} \item[(4.1)] If $\mathfrak s$ is zeta-like, then (1) $s_i\le s_{i+1}$ and $(q-1)s_i\le s_{i+1}\le(q^2-1)s_i$ for $1\le i1$ occur only in weights $p^m(q^k-1)$, and primitive ones only in weights $q(q-1)$ or $q^n-1$ if $q>2$, and $2^n-1$ or $2^n$ if $q=2$. (2) If $q=p$, the weight of a zeta-like, non-Eulerian tuple of depth $r>1$ is $p^m$ times a number whose base-$p$ digits contain no $0$ and at most one $1$. (3) The smallest weight of a zeta-like value of depth $r$ is $q^{r-1}$. (4) If $q>2$, the smallest weight of an Eulerian value of depth $r$ is $q^r-1$, $(q-1)q$ or $q-1$ according as $r>2$, $r=2$ or $r=1$. (5) The weight $q^k$ is not the weight of a zeta-like primitive tuple of depth $r$ if $k>r>3$, or if $k>3$ and $r\in\{2,3\}$. \item[(4.4)] The primitive zeta-like tuples of depth $2$ and weight at most $q^2$ are exactly $(i,j(q-1))$ with $1\le i\le q$ and $i\le j\le\lfloor(q^2-i)/(q-1)\rfloor$. \item[(4.5)] For $q=2$, the primitive zeta-like tuples of depth $2$ are exactly $(1,1)$, $(1,3)$, $(3,5)$, $(2^n-1,2^n)$ and $(2^n,2^{n+1}+2^n-1)$ for $n\ge1$. \end{enumerate} \end{conjecture} In addition, \cite[Conj.~4.6]{LRT14} gives three families of arbitrary depth with explicit ratios; the first is the case $A=\Fq[t]$ of the family in Theorem~\ref{thm:cnp} below, and we refer to \cite{LRT14} for the constants. Without the word ``primitive'', (4.5) would be false: for $q=2$ the tuple $(1,1)$ is zeta-like, hence so is $(2,2)$, since $\zeta(2,2)/\zeta(4)=(\zeta(1,1)/\zeta(2))^2$. A related completeness statement is the list of primitive Eulerian multizeta values in \cite[\S6.2]{CPY19}, which \cite{CPY19} certify only up to finite weight bounds. \subsection{What is known} It is natural to expect the characterisation to be wide open. In fact much of it is settled; Table~\ref{tab:zl-status} summarises. \begin{theorem}[{\cite[Cor.~4.4.3]{CPY19}}]\label{thm:cpy-tail} If $\zeta(s_1,\dots,s_r)$ is zeta-like, then each of $\zeta(s_2,\dots,s_r)$, $\zeta(s_3,\dots,s_r)$, \dots, $\zeta(s_r)$ is Eulerian. In particular $s_2,\dots,s_r$ are divisible by $q-1$. \end{theorem} \begin{table}[ht] \small \begin{tabularx}{\textwidth}{@{}lX@{}} \toprule Statement & Status\\ \midrule 4.1(2), tail half & Proved: Theorem~\ref{thm:cpy-tail}\\ 4.4; all zeta-like values of weight $\le q^2$, any depth & Proved by H.~H.~Le and T.~Ngo~Dac \cite{LeNgoDac22} (we saw only the abstract)\\ 4.6, all three families with their constants & Proved by H.-J.~Chen \cite[Thms.~4.4, 4.5, 5.1, 5.2]{Chen17} (as read in \cite{DataZL}); \cite{DataZL} also states, unverified, that this covers the families of \cite{KuanLin16}\\ 4.1(1); 4.1(2), prefix half & Open\\ 4.2, 4.3 beyond weight $q^2$ & Open\\ 4.5 & Open (certified up to weight $256$ in \cite{CPY19})\\ Completeness of \cite[\S6.2]{CPY19} & Open\\ \bottomrule \end{tabularx} \caption{Status of Conjecture~\ref{conj:LRT14}. We did not read \cite{KuanLin16} or \cite{ChenKuan18}, which may settle further depth-two cases.} \label{tab:zl-status} \end{table} \begin{remark}\label{rem:zl-attribution} Two attributions are easily confused. The classification of zeta-like values of weight at most $q^2$ is due to H.~H.~Le and T.~Ngo~Dac \cite{LeNgoDac22}. It is not due to Im--Kim--Le--Ngo~Dac--Pham, whose paper \cite{IKLNDP24} concerns the Zagier--Hoffman conjectures. Likewise, arXiv:2205.09929, which proves Thakur's basis conjecture, is by Chang, Chen and Mishiba \cite{CCM23}. It is not a single-author paper of Chen. \end{remark} \begin{remark}\label{rem:chen-constant} The constant in Chen's version of the second family \cite[Thm.~4.4]{Chen17} looks different from the constant in \cite[4.6(2)]{LRT14}. In fact they coincide: the exponent of $[j]$ telescopes to $q^{n+2-j}$, and $\ell_i=(-1)^iL_i$ in the two normalisations. The equality was also confirmed numerically. One might also expect 4.6(2) to need $q>2$, because \cite[Thm.~4.4]{Chen17} assumes it. The identity holds numerically for $q=2$, $n=0,1,2$, and \cite[Thm.~5.2]{Chen17} covers $q=2$. \end{remark} \subsection{Two elementary results} \begin{proposition}\label{prop:zl-depth2-min} For every $q$, the smallest weight of a zeta-like tuple of depth $2$ is $q$, and $(1,q-1)$ is zeta-like. \end{proposition} This is the case $r=2$ of Conjecture~\ref{conj:LRT14}(4.3)(3). \begin{proof} If $(s_1,s_2)$ is zeta-like, then $(q-1)\mid s_2$ by Theorem~\ref{thm:cpy-tail}, so $s_2\ge q-1$ and the weight is at least $1+(q-1)=q$. For the second claim, take $n=0$, $s_1=1$, $s_2=0$ in \cite[Thm.~3.1(2)]{LRT14}. It gives $a=1$, $b=q-1$ and $\zeta(1,q-1)=\zeta(q)/\ell_1$, with $\ell_1=t-t^q\in K^\times$. \end{proof} \begin{proposition}\label{prop:zl-weight-bound} A zeta-like tuple of depth $r$ has weight at least $1+(r-1)(q-1)$. \end{proposition} \begin{proof} By Theorem~\ref{thm:cpy-tail}, each of $s_2,\dots,s_r$ is a positive multiple of $q-1$, and $s_1\ge1$. \end{proof} For $r\ge3$ this is far below the conjectured $q^{r-1}$. If $q>2$ and $r\ge4$, then Conjecture~\ref{conj:LRT14}(4.3)(4) for the Eulerian tail of depth $r-1\ge3$ would give weight at least $1+q^{r-1}-1=q^{r-1}$, so (4.3)(3) would follow. For $r=3$ this argument gives only $1+q(q-1)d_2>\dots}S_{d_1}(s_1) \cdots S_{d_r}(s_r)$ of one weight $w$ that vanishes for every $d$; summing over $d$ turns such a family into an element of $V_w$ lying in $R_w$. The binary relation $R_1(d)=S_d(q)+D_1S_{d+1}(1,q-1)$ is of this kind \cite[\S3]{Todd18}, and its sum over $d$ is \eqref{eq:R1}. Todd defines operators $B_U$ and $C_V$, indexed by tuples $U,V$, on such families; for instance $B_U(R)(d)=S_d(U)\sum_{jcd$ and $r\equiv n\pmod D$, then $v_d(k_n)=q^nM+J$, where $(M,J)$ is the lexicographically least pair (first $M$, then $J$) with $T_{M,J}(r)\ne0$, and $v_d(k_n)=\infty$ if there is none. \end{lemma} \begin{proof} Since $v$ is separable, $t\mapsto\omega+\varepsilon$ embeds $K_v$ into $\F_{q^D}((\varepsilon))$ with $\val_v$ equal to the $\varepsilon$-adic order. The coefficients of $a$ lie in $\Fq$, so $a(\omega+\varepsilon)^{q^n}=a(\omega^{q^n}+\varepsilon^{q^n}) =a(\omega^{q^r}+\varepsilon^{q^n})$, which is a unit because $\omega^{q^r}$ is a root of $v$ and $v\nmid a$. Therefore \[ S_{d,v}(k_n)\mapsto\sum_a a(\omega+\varepsilon)^c\, a(\omega^{q^r}+\varepsilon^{q^n})^{-b} =G_r(\varepsilon,\varepsilon^{q^n}) =\sum_{M,J}T_{M,J}(r)\,\varepsilon^{J+q^nM}. \] Here $0\le J\le cdcd$. In Lemma~\ref{lem:L} the coefficients $T_{M,J}(r)$ depend on $n$ only through $r$, so the least pair $(M_r,J_r)$ does too, which gives the first claim. If $q^nM'_r+J'_r=q^nM_r+J_r$ for two distinct $n$ in one class, subtracting gives $M'_r=M_r$ and then $J'_r=J_r$. \end{proof} \subsection{Refutations and corrections} \subsubsection*{A slope of multiplicity one} It is natural to expect, from Claimed result~\ref{claim:kmu} applied to $\Fq[t]$ (genus $0$, $\infty$ of degree $1$, and ordinary), that at a place $v$ of degree $d_v=2$ every nonzero slope of $\zeta_v(x,y)$ on the identity component has even multiplicity. We find a slope of multiplicity one. \begin{proposition}[computer-assisted]\label{prop:kmu-slope} Let $q=2$, $v=t^2+t+1$ and $k=27$. Then \[ \bigl(v_d(27)\bigr)_{d=0}^{6}=(0,1,0,6,33,75,198), \] and the Newton polygon of $\sum_dS_{d,v}(27)x^d$ begins with the slope $0$ of multiplicity $2$, followed by the slope $6$ of multiplicity $1$. \end{proposition} \begin{proof} The seven valuations were computed exactly from the power series of $a(\omega+\varepsilon)^{-27}$ in $\F_4[[\varepsilon]]$, where $\omega$ is a root of $v$ (see the proof of Lemma~\ref{lem:L}), by two independent implementations (\cite{DataGZ}, \texttt{code/vadic.py}; and a separate direct summation, \cite{Checks}, \texttt{independent\_check.py}). The points $(0,0)$, $(1,1)$, $(2,0)$ give the slope $0$ with multiplicity $2$. The line through $(2,0)$ and $(3,6)$ is $y=6(x-2)$. The points with $4\le d\le6$ lie strictly above it ($33>12$, $75>18$, $198>24$), and for $d\ge7$ Lemma~\ref{lem:LB} gives $v_d(27)\ge2^{d-2}-1>6(d-2)$. Hence $(3,6)$ is a vertex, the segment from $(2,0)$ to $(3,6)$ is an edge, and the next slope is larger than $6$. \end{proof} The exact values for $d\le11$ in \cite{DataGZ} (\texttt{data/vadic\_np\_certified.json}), combined with Lemma~\ref{lem:LB}, certify the initial slopes $0,0,6,27,42,123$, so the slope $27$ also has multiplicity one; this part rests on a single implementation. The point $(3,6)$ lies below the chord from $(2,0)$ to $(4,33)$; the endpoints of the predicted blocks are matched, and only interior vertices dip. \begin{remark}[needs human verification]\label{rem:kmu-conflict} Under our reading of \cite{KMU23}, Proposition~\ref{prop:kmu-slope} contradicts Claimed result~\ref{claim:kmu}. The exponent $y=-27$ is divisible by $q^{d_v}-1=3$, so the Teichm\"uller factors $\omega_v(a)^y$ are trivial and Goss's series coincides with $\zeta_{A,v}(x,1,-27)$ of \cite[Def.~2.17, Rem.~2.18]{KMU23}. Rescaling $x$ by a unit does not change multiplicities. Claimed result~\ref{claim:kmu} then requires the slope $0$ with multiplicity $g-1+d_v+d_\infty=2$, which is what we find, and every nonzero slope with multiplicity divisible by $d_v=2$, which fails for the slope $6$. Since \cite[Thm.~1.10]{KMU23} follows from \cite[Thm.~1.5]{KMU23} applied to the ring of functions regular off $v$ on $\mathbb P^1$ (genus $0$, with the place at infinity of degree $2$), either that case of \cite[Thm.~1.5]{KMU23} or the comparison \cite[Cor.~2.20]{KMU23} would have to fail. We have not audited the proofs in \cite{KMU23}, a v1 preprint, and the conflict should be checked by a human expert before it is relied on. The same phenomenon occurs (by the same certified method, in a single implementation \cite{DataGZ}) for this $v$ at $k=3,6,12,15,51,54$; for $q=2$, $v=t^3+t+1$ at $k=7,14,21,42$ (at $k=21$ a pair occurs where a triple is required); and for $q=3$, $v=t^2+1$ at $k=8,16,24,40,48$. So it is not tied to $p\mid d_v$. \end{remark} \subsubsection*{Thakur's closed form in \texorpdfstring{\cite[9.1(IV)]{Thakur2013Vadic}}{9.1(IV)}} Let $q=2$ and $v=t^2+t+1$, and put $f(n,d)=v_d(5\cdot2^n-1)-v_d(2^{n+1}-1)$. Thakur conjectures \cite[9.1(IV)]{Thakur2013Vadic} that $f(n,d)=(3d-6)2^n$ for even $d$, $f(n,d)=(3d-15)2^n$ for odd $d>3$, and, for $d=3$, $f(n,3)=2^n$ for odd $n$ and $2^n-1$ for even $n$. It is natural to accept this closed form; in fact the sign is wrong and, for $d\ge4$, the size is doubled. \begin{theorem}[computer-assisted]\label{thm:thakur-vadic} Let $q=2$ and $v=t^2+t+1$. For all $n\ge1$: \begin{enumerate} \item\label{item:tv-35} for $2\le d\le11$, Thakur's conjectured formula \cite[3.5]{Thakur2013Vadic} holds: $v_d(2^n-1)=2^{n+d-2}-(2^n-1)\lfloor d/2\rfloor+(-1)^{d-1}$ for even $n$, and $v_d(2^n-1)=2^{n+d-2}-(2^n-1)\lfloor d/2\rfloor-\lfloor (d+1)/2\rfloor +\lfloor d/2\rfloor$ for odd $n$; \item\label{item:tv-wn} for $3\le d\le11$, $w_n=v_d(3\cdot2^n-1)$ satisfies the recursions of \cite[9(ii)]{Thakur2013Vadic}; \item\label{item:tv-f} for $3\le d\le11$, \[ f(n,d)=\begin{cases} -(3d-6)\,2^{n-1}, & d\text{ even},\\ -(3d-15)\,2^{n-1}, & d\text{ odd},\ d\ge5,\\ -2^n, & d=3,\ n\text{ odd},\\ -(2^n-1), & d=3,\ n\text{ even}. \end{cases} \] \end{enumerate} For $d=12$, the three statements hold for all $n$ with $2^n>12$. \end{theorem} \begin{proof}[Certificate] In each case the conjectured value, and both terms of $f(n,d)$, have the form $2^nM_r+J_r$ with $r=n\bmod2$. For the recursions of 9(ii) this holds because a first-order recursion $x_n=2x_{n-1}+c_{n\bmod2}$ has solutions of the form $2^nC+e_{n\bmod2}$. By Corollary~\ref{cor:L-shape}, with $D=2$ and $(b,c)\in\{(1,1),(3,1),(5,1),(2,1)\}$, it suffices to compute the pairs $(M,J)$ of Lemma~\ref{lem:L} for $r=0,1$, compare them with the formulas at four consecutive values $n\ge n_0$ (where $2^{n_0}>d$), and check the finitely many $n2$ holds for $N\ge1$. The least pairs are $(M,J)=(4,2)$, respectively $(2,2)$ (\cite{DataGZ}, \texttt{code/prove\_s7.py}; second run in \cite{Checks}, \texttt{prove\_s7\_rerun.md}). The direct values at $N=1$ are $14$ and $8$, as predicted. \end{proof} \subsubsection*{Further computational findings} \begin{evidence}\label{ev:kmu-rem16} In this evidence, the next one and Remark~\ref{rem:f4-thakur}, $R_A$ and $R_{\mathrm{line}}$ are taken at $y=k$ for an integer $k\ge1$, so that the coefficients are the power sums $S_d(-k)=\sum a^{k}$. For non-ordinary $X$, Kramer-Miller and Upton expect \cite[Rem.~1.6]{KMU23} that $R_A$ consists of values $s_1,\dots,s_{g-1+d_\infty}$ with $0\le s_i\le\alpha_{r,1}$, together with the values $\alpha_{r,i}(r-1)$, each with multiplicity $d_\infty$, of Claimed result~\ref{claim:kmu-infty}. The computations contradict this. \begin{enumerate} \item For the genus-$2$, $p$-rank-$0$ ring (iv) of Section~\ref{sec:highergenus}, for every odd $k\le511$, $R_A=R_{\mathrm{line}}\sqcup\{\text{the two smallest elements of } R_{\mathrm{line}}\}$. So the second extra value is the second element of $R_{\mathrm{line}}$, which exceeds the expected bound; for $k=3$, the line has $\{1,3\}$ and $A$ has $\{1,1,3,3\}$. \item For the genus-$1$ ring (iii) and $k=181$, the line has $\{21,177\}$ and $A$ has $\{21,21,189\}$, which is not $R_{\mathrm{line}}$ plus one element. The degrees $\deg S_2=320$, $\deg S_3=312$, $S_4=0$ were confirmed by an independent implementation in Singular. The same occurs at $k=437,565,661,693,709,721,757,949$ and persists for $k=181+j\cdot2^m$ with $m\le14$. \item For $q=4$ and $k=1,5,17,65,257$ both sets are empty, although the expected shape requires $g-1+d_\infty=1$ value $s_1$ in $R_A$. \end{enumerate} \end{evidence} \begin{evidence}\label{ev:bockle} Böckle showed that for the ring (i), $R_A$ agrees with $R_{\mathrm{line}}$ except that the smallest element has multiplicity two (as reported in \cite[\S1.1.3]{KMU23}; we could not obtain Böckle's preprint). It is natural to expect the same pattern for the other genus-$1$ rings. The computations confirm it for $q=2$ ($512$ tested $k\le1023$) and $q=3$ ($485$ of the $486$ tested $k\le728$, the exception being $k=1$), but it fails for $q=4$ at the nine values of Evidence~\ref{ev:kmu-rem16}(2). The guess $\deg S^A_{d+1}=\deg S^{\mathrm{line}}_d+\deg S^{\mathrm{line}}_1$ for the genus-$1$ rings also fails, in $9$ of $150$ cases for $q=2$, $51$ of $167$ for $q=3$ and $71$ of $130$ for $q=4$. \end{evidence} \begin{remark}\label{rem:f4-thakur} In the tested range $k\le1023$, the $\F_4$ anomaly of Evidence~\ref{ev:kmu-rem16}(2) occurs at nine values, $k=181,437,565,661,693,709,721,757,949$. They include all eight values $181,437,565,661,693,709,721,949$ (odd, at most $1000$) at which Thakur found the recursion \eqref{eq:recursion} to fail for $q=4$ and the power sums $S_2(-k)$ (Remark~\ref{rem:thakur-nonprime}); the only additional value is $757$. $R_{\mathrm{line}}$ was computed by direct summation \cite{DataGZ}, so the coincidence is not an artefact of using the recursion. We do not know a mechanism; see Question~\ref{q:goss}. \end{remark} \begin{evidence}\label{ev:distinct} For $q=2$ and $v=t^2+t+1$, every tested pair of equal-slope $v$-adic zeros consists of two distinct zeros rather than an inseparable double zero. The zeros are those of $x\mapsto\zeta_v(x,-k)$, and the pairs $(k,\lambda)$ were $(3,0)$, $(3,9/2)$, $(9,0)$, $(9,39/2)$, $(15,33/2)$, $(19,77/2)$, $(5,1/2)$, $(11,1/2)$, $(17,1/2)$, $(17,7/2)$ and $(9,3/2)$. A double zero $z$ would force $f_e(z^2)=f_o(z^2)=0$ for the even and odd parts $f_e,f_o$; in each case either $f_o$ has no root of the relevant valuation, or Newton iteration in $\F_4((\varepsilon))$ separates the roots of $f_e$ and $f_o$. Goss's Conjecture~5 \cite{Goss00} allows finitely many exceptions, so this is evidence against it, not a disproof. \end{evidence} \begin{evidence}\label{ev:trivial} It is natural to expect irregular trivial zeros to be sporadic. In the computations they follow an exact rule. At $\infty$ (with $y=k$ as in Evidence~\ref{ev:kmu-rem16}), a trivial zero had order $2$ exactly when $k$ was among the doubled real parts: $k=1$ in genus $1$, $q=2$; $k=3,9,21,33,69,81,129,261,273,321,513$ in genus $1$, $q=4$; and nine values in genus $2$. For $q=3$ and $k=3^n+1$ the real part is also doubled, but the order is $1$: the pair consists of two distinct zeros. Also $\zeta_A(-k)\ne0$ for all $341$ tested $k$ with $(q-1)\nmid k$, consistent with Thakur's question. Off the identity component ($q=2$, $\deg v=2$) there is one zero of slope $0$ instead of two; for $k\equiv2\pmod3$ the other slopes come in half-integer pairs, while for $k\equiv1\pmod3$ they are mixed. \end{evidence} \begin{remark}\label{rem:goss-literature} The literature already contains four failures of naive expectations, which we carry forward without re-verification: Thakur's observation that the naive Riemann hypothesis needs modification at primes of degree two \cite{Thakur2013Vadic}; Goss's near-trivial zeros and Thakur's irregular trivial zeros (as described in \cite[\S1]{KMU23}); and Böckle's doubled smallest zero (as reported in \cite{KMU23}). The computations re-verified Theorem~\ref{thm:rh-infty} in all $1154$ tested cases, with real parts divisible by $q-1$, and the degree-one $v$-adic Riemann hypothesis claimed in the unrefereed preprint \cite{DVPC14} in every tested case. \end{remark} \subsection{A conjecture and open problems} \begin{conjecture}[new]\label{conj:S} Let $A$ be a base ring of class number one over a prime field $\F_p$, with $d_\infty=1$, whose curve has genus $g\ge1$ and $p$-rank $0$ (so $A$ is one of (i), (ii), (iv)). Then for every integer $k\ge2$, at $y=k$ (power sums $S_d(-k)$), $R_A=R_{\mathrm{line}}\sqcup\{\text{the }g\text{ smallest elements of } R_{\mathrm{line}}\}$. \end{conjecture} It holds on the tested ranges for $(g,q)=(1,2),(1,3),(2,2)$, and it is false for $(g,q)=(1,4)$ (Evidence~\ref{ev:kmu-rem16}), which is why we restrict to prime fields. \begin{question}\label{q:goss} \begin{enumerate} \item What is the correct multiplicity law for $d_v\ge2$? The data suggest that the polygon passes through the endpoints of the blocks predicted by Claimed result~\ref{claim:kmu}, but that slopes inside a block may split. \item The ordinary case with $d=1$ and $g\ge1$ was not tested: there are no ordinary examples of class number one, and non-principal ideals were not implemented. \item Prove Conjecture~\ref{conj:S} and explain the $\F_4$ anomaly. Is it caused by the failure of the line-side recursion \eqref{eq:recursion} for $q=4$ (Remark~\ref{rem:f4-thakur})? \item Test the transition recipes 9(iii)--(v) of \cite{Thakur2013Vadic} for $d\ge4$, and settle the sign in 9(iv) (Remark~\ref{rem:f5}). \item Does Goss's Conjecture~5 hold for almost all $y$? \end{enumerate} \end{question} %====================================================================== \section{Automata and transcendence of special values}\label{sec:automata} \subsection{The criterion and two corrections} For a finite field $\F$ of characteristic $p$, Christol's theorem \cite{CKMFR80} says that $\sum_{n}c_nx^n\in\F[[x]]$ is algebraic over $\F(x)$ if and only if $(c_n)$ is $p$-automatic, if and only if its $Q$-kernel $\{(c_{Q^an+b})_n: a\ge0,\ 0\le bD$, and $\dlog D_{D,v}=-1/[D]-v'/v$. For $m=kD$ with $k\ge1$, $t^{q^m}=\alpha+s^{Q^k}$, so $[kD]=s^{Q^k}-s$ and \[ \frac1{[kD]}=-\frac1s-G_k,\qquad G_k=\frac{s^{Q^k-2}}{1-s^{Q^k-1}}=\sum_{m\ge1}s^{m(Q^k-1)-1}. \] Thus $\dlog D_{kD,v}=G_k-G_{k-1}$ for $k\ge2$. The logarithmic derivative of $F(b)$ is the limit of those of its partial products, and Abel summation, with $b$ bounded and $G_k\to0$, gives \[ L=\frac{F(b)'}{F(b)}\equiv\sum_{k\ge2}\varepsilon_kG_k\pmod{\F_Q(s)}. \] If $F(b)$ were algebraic, $L$ would be algebraic by Step~0, and hence so would $s\sum_{k\ge2}\varepsilon_kG_k$. \emph{Step 5 (an infinite kernel).} We have \[ s\sum_{k\ge2}\varepsilon_kG_k=\sum_{N\ge1}c(N)s^N, \] with $c(N)=\sum_{k\ge2,\,(Q^k-1)\mid N}\varepsilon_k\in\F_p$. For $a\ge1$ the sequence $u_a(N)=c(Q^aN+1)$ lies in the $Q$-kernel of $c$. By Lemma~\ref{lem:divisibility}, $u_a(Q^h-2)=\sum_{k\ge2,\,k\mid\gcd(a,h)} \varepsilon_k$ for $h\ge1$. Let $a>a'\ge2$ with $\varepsilon_a$ and $\varepsilon_{a'}$ nonzero modulo $p$. Let $h$ be the least divisor of $a$ that does not divide $a'$ and has $\varepsilon_h\not\equiv0$; it exists because $a$ itself qualifies, and $h\ge2$. Then \[ u_a(Q^h-2)-u_{a'}(Q^h-2)=\sum_{k\mid h,\ k\nmid a'}\varepsilon_k =\varepsilon_h\not\equiv0\pmod p, \] since every other divisor $k$ of $h$ with $k\nmid a'$ is a divisor of $a$ smaller than $h$, and so has $\varepsilon_k\equiv0$. Thus the $u_a$, for the infinitely many $a$ with $\varepsilon_a\not\equiv0$, are pairwise distinct. The $Q$-kernel of $c$ is infinite, so by Christol's theorem the series is transcendental over $\F_p(s)$, hence over $\F_Q(s)$. Therefore $F(b)$, and with it $\Pi_v(n)$, is transcendental. \end{proof} \begin{remark}\label{rem:residue-r} For residue classes $r\ne0$ the same method stops. Put $\beta_r=\alpha^{q^r}-\alpha\ne0$. For $j\equiv r\pmod D$, $[j]=\beta_r-s+s^{q^j}$ is a $v$-adic unit, and telescoping gives $\dlog F(b^{(r)})\equiv\sum_k\varepsilon_ku_{kD+r}$ with $u_j=\sum_{m\ge1}(-1)^ms^{mq^j}(\beta_r-s)^{-m-1}$. Two steps are missing: (R1) each residue-$r$ series is transcendental when $\varepsilon$ is not eventually $0$, and (R2) different residue classes cannot cancel. Pure $q$-power sums such as $\sum s^{Q^k}$ are algebraic, so transcendence must come from the binomial twist. Two natural shortcuts fail. The shift identity $D_{j+1,v}=D_{j,v}^{\,q}[j+1]/[j+1-D]$ writes the residue-$r$ factor as $G^{q^r}W$, but $\dlog(G^{q^r})=0$, so taking logarithmic derivatives only returns the residue-$r$ series. And the untelescoped series $-\sum_j(n_j-n_{j+D})/[j]$ does not converge $v$-adically, because $1/[j]\not\to0$ at $v$; only the grouping used in Step~4 converges. \end{remark} \begin{evidence}\label{ev:kernel} $\Pi_v(n)$ was computed exactly modulo $s^P$, and the number of distinct $p$-kernel prefixes of length $L$ was counted. For $(q,v,P,L)=(2,t^2+t+1, 8192,64)$, eventually $D$-periodic digits gave $2$ prefixes; residue-$0$ non-periodic digits (Theorem~\ref{thm:wen-yao-A}) and residue-$1$ non-periodic digits (open) both grew to the maximum of about $2^7$. Similar growth appeared for $(3,t^2+1,6561,27)$ and $(2,t^3+t+1,8192,64)$. The logarithmic-derivative formula held to $O(s^{398})$, the telescoped identity held exactly to $O(s^{P-2})$ for $(q,D)=(2,2),(2,3),(3,2)$, and Lemma~\ref{lem:divisibility} had no exceptions for $Q\in\{2,3,4,5,7,8,9,16,27\}$, $a,b\le8$, $c\le12$. As a calibration the algebraic series $\sum s^{Q^k}$ gave bounded counts. Finite windows cannot prove transcendence. \end{evidence} \begin{question}\label{q:at} \begin{enumerate} \item Prove (R1) and (R2), and hence Conjecture~\ref{conj:wen-yao}. \item Give a period-method proof of the theorem of Mend\`es France and Yao \cite{MFY97}. \item Are nonzero $v$-adic multizeta values transcendental when $\deg v\ge2$? \item Settle the $v$-adic geometric gamma conjecture \cite[Rem.~1.4.3]{CWY24}. \item Determine the nature of $\Gamma(0)$ for Thakur's shtuka-theoretic gamma function \cite[\S4.3]{ThakurCRM}. \end{enumerate} \end{question} %====================================================================== \section{Summary of open problems}\label{sec:open} \begin{enumerate} \item H1 for $q=p^f$, $f\ge2$; Conjecture~\ref{conj:H2star}; the family of Theorem~\ref{thm:H2-fails} for every non-prime $q$, and uniqueness of its minimiser beyond $q\le27$; Thakur's negative-exponent examples, which were not re-verified (Section~\ref{sec:powersums}). \item Completeness of the zeta-like lists above weight $q^2$; the constraints 4.1(1), the prefix half of 4.1(2), 4.2, 4.3 and 4.5 (Question~\ref{q:zl}). \item The higher-genus speculations of Question~\ref{q:hg}; a comparison of the numerical identity of Evidence~\ref{ev:hg} with the closed formula for $\alpha_{n,k}$ in \cite{CNP23}. \item The origin of $R_1$, the $\F_2$-rational relations, a refereed account of the $B$ form of Todd's conjecture, and the inclusion of $\pit$ (Question~\ref{q:mz}). The relation rank for $q=5$, $w=5$ was not computed at adequate precision. \item Adjudication of the conflict in Remark~\ref{rem:kmu-conflict}, by a human expert; Conjecture~\ref{conj:S}; Question~\ref{q:goss}; the small-$n$ check for $d=12$ in Theorem~\ref{thm:thakur-vadic}; an independent audit of the code behind Theorems~\ref{thm:H2-fails},~\ref{thm:thakur-vadic} and~\ref{thm:thakur-s7}. \item Conjecture~\ref{conj:wen-yao} beyond residue class $0$, and Question~\ref{q:at}. \end{enumerate} %====================================================================== \section{Software and AI assistance}\label{sec:ai} \subsection*{Methods: generation} This AI-generated article originated in a prompt given during Taylor Dupuy's talk \emph{How to Use AI in Mathematical Research Effectively} at the University of Rochester Number Theory Seminar on 23 October 2026. The prompt used \texttt{frontier-dump} to investigate selected Thakur conjectures, followed by \texttt{harvest-dumps} and \texttt{using-latexpowers} to assemble this manuscript. Diophantine approximation exponents and continued fractions were excluded because another researcher was already working on them. A Claude Code agent \cite{ClaudeCode} using the model \texttt{claude-opus-5-5} wrote the article on 23 October 2026 from six research dumps produced the same day by six Claude Code agents using the same model. Taylor Dupuy did not write or check any part of the mathematics. The research agents worked alone and could not delegate. Their reports are the source of every result here that is not cited to the literature, with two qualifications: the $d=12$ part of Theorem~\ref{thm:thakur-vadic} comes from an addendum that the coordinating agent of the research run added to the Goss record after its script finished; and the agent that assembled this paper wrote or re-derived the proofs of Lemmas~\ref{lem:additive}, \ref{lem:LB} and~\ref{lem:L}, Corollary~\ref{cor:L-shape}, Propositions~\ref{prop:kmu-slope}, \ref{prop:zl-depth2-min} and \ref{prop:zl-weight-bound}, Theorems~\ref{thm:not-shuffle}, \ref{thm:todd-hu} and~\ref{thm:wen-yao-A}, Lemma~\ref{lem:divisibility} and Proposition~\ref{prop:omega} from the research records. Agents are not authors and are not the source of any mathematical claim; responsibility for the content rests with the human who prints it. On 26 October 2026, a Codex desktop agent based on GPT-6 revised this provenance account and the title-page printing credit from Taylor Dupuy's clarification. The exact model identifier was not exposed to the agent. This editorial revision changed no mathematical statement or proof and received no human mathematical review. % >>> [agent-codex 2026-10-27 query-report-edit-record] On 27 October 2026, a Codex desktop agent based on GPT-6 revised the title, creation-date line, and opening abstract paragraph at Taylor Dupuy's request. It also removed a title-footnote link to a private AI-usage ledger. No mathematical statement or proof changed. % <<< [agent-codex 2026-10-27 query-report-edit-record] \subsection*{Methods: synthesis and review} The manuscript was assembled with the \texttt{harvest-dumps} and \texttt{using-latexpowers} skills of \texttt{mathcity} \cite{Mathcity}: a source map, an outline, a filled presentation and an integrated manuscript. Separate Claude Code reviewer agents, which did not write the text they judged, reviewed the outline (accepted after three rounds) and the filled presentation (accepted after three rounds), and a third reviewer checked the integrated manuscript and its compiled PDF; the reports, rounds and verdicts are kept in the directory \texttt{latex/thakur-conjectures/harvest/reviews/} of the repository that holds \cite{Checks}. All review was by AI agents; no human has checked the proofs, the computations or the citations. \subsection*{Methods: computations} The research agents used SageMath~10.8 \cite{SageMath} and Python. Their scripts and data are in the research records \cite{DataPS,DataZL,DataHG,DataMZ,DataGZ,DataAT}. While writing this paper the coordinating agent independently recomputed Example~\ref{ex:q4k75}, the valuations in Proposition~\ref{prop:kmu-slope}, and part \eqref{item:tv-f} of Theorem~\ref{thm:thakur-vadic} for $3\le d\le5$, $2\le n\le4$, with its own Python code. It also reran two of the research agents' scripts: the minimiser behind the uniqueness statement of Theorem~\ref{thm:H2-fails}, and the computation behind Theorem~\ref{thm:thakur-s7}. The certificates in Theorems~\ref{thm:H2-fails}, \ref{thm:thakur-vadic} and~\ref{thm:thakur-s7} otherwise rest on the research agents' code, which has not been independently audited. No proof-assistant verification was done. Several computations required care with $t$-adic precision; see Evidence~\ref{ev:fp-rational}. \subsection*{Methods: citations} Every arXiv identifier cited was checked against the arXiv API, and journal data against Crossref or the publisher's page. Where a pinpoint comes from a research agent's reading of a source that we did not open, the text says so. The preprints \cite{ChenOno2026,Hu26,KMU23,CWY24,DVPC14} are unrefereed. \bibliographystyle{amsalpha} \bibliography{references} \end{document} % <<< [agent-claude 2026-10-23 thakur-conjectures harvest]