\documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage[a4paper,margin=27mm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{aliascnt} \usepackage{booktabs,array,longtable} \usepackage{placeins} \usepackage{microtype} \usepackage{xcolor} \usepackage{enumitem} \usepackage{pgfplots} \pgfplotsset{compat=1.18} \usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref} \usepackage[nameinlink,noabbrev]{cleveref} \hypersetup{ pdfauthor={Zhenyu Sun}, pdftitle={Exact Intervals and Arithmetic Structure in Fuchs' Extremal Problem}, pdfsubject={Weighted extrema of monic integer polynomials}, pdfkeywords={Fuchs problem, integer polynomials, Prouhet-Tarry-Escott, exact enumeration}} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newaliascnt{proposition}{theorem} \newtheorem{proposition}[proposition]{Proposition} \aliascntresetthe{proposition} \newaliascnt{lemma}{theorem} \newtheorem{lemma}[lemma]{Lemma} \aliascntresetthe{lemma} \newaliascnt{corollary}{theorem} \newtheorem{corollary}[corollary]{Corollary} \aliascntresetthe{corollary} \theoremstyle{definition} \newaliascnt{definition}{theorem} \newtheorem{definition}[definition]{Definition} \aliascntresetthe{definition} \newaliascnt{question}{theorem} \newtheorem{question}[question]{Question} \aliascntresetthe{question} \theoremstyle{remark} \newaliascnt{remark}{theorem} \newtheorem{remark}[remark]{Remark} \aliascntresetthe{remark} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \newcommand{\R}{\mathbb R} \newcommand{\Q}{\mathbb Q} \newcommand{\dd}{\,d} \newcommand{\lc}{\operatorname{lc}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\cC}{\mathcal C} \newcommand{\cM}{\mathcal M} \newcommand{\cP}{\mathcal P} \newcommand{\cE}{\mathcal E} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} \newcommand{\I}{\mathbf 1} \setlist[itemize]{topsep=3pt,itemsep=2pt} \setlength{\emergencystretch}{2em} \title{Exact Intervals and Arithmetic Structure\protect\\ in Fuchs' Extremal Problem} \author{Zhenyu Sun} \date{} \begin{document} \maketitle \begin{abstract} We determine Fuchs' weighted unit-circle extremum for monic integer polynomials for $0<\lambda\le3$. The proof combines a fractional difference energy with a sharp rearrangement inequality and an explicit gap for nonnegative integer sequences that are not interval indicators. We obtain the finite-degree minimum on $(1,2)$ at every degree and on $(2,3)$ at every degree at least five. Every minimizing sequence on these intervals eventually consists, up to monomial factors, of Dirichlet polynomials and products of two adjacent Dirichlet polynomials, respectively. At integer parameters, Parseval's identity reduces the problem to integer coefficient norms. The lower bound is attained exactly when a multiplicity-free ideal Prouhet--Tarry--Escott pair exists. Classical PTE constructions give exact values for the integers $1$ through $10$ and for $12$, and a two-value alternative at $11$. Subset counting gives a quadratic upper bound in general. The interval formulas and explicit constructions show that $2$, $3$ and $4$ are strict local maxima; the formula on $(2,3)$ has a unique minimum. A Pell construction gives an upper bound on $(4,5)$, and coefficient-norm gaps exclude Dirichlet products uniformly near certain integers. For each fixed degree bound, the extremum, divided by a common gamma factor, is locally the lower envelope of finitely many rational functions. Exact lattice enumeration identifies a nonproduct minimizer on an open interval containing $17/4$ for every degree bound from $61$ to $80$. \end{abstract} \noindent\textbf{2020 Mathematics Subject Classification.} Primary 11C08, 11D72; Secondary 42A05, 11Y50, 30C10. \smallskip \noindent\textbf{Keywords.} Integer polynomials; weighted extremal problem; Prouhet--Tarry--Escott problem; discrete fractional energy; exact lattice enumeration. \section{Introduction and main results} \label{sec:introduction} For a polynomial $P$ and a real parameter $\lambda>-1/2$, put \begin{equation} \label{eq:energy-definition} \cE_\lambda(P)=\frac{1}{2\pi}\int_{-\pi}^{\pi} |1-e^{i\theta}|^{2\lambda}|P(e^{i\theta})|^2\dd\theta. \end{equation} Let $\cP$ be the set of all monic polynomials in $\Z[z]$, including the constant polynomial $1$. For integers $N\ge0$, define \begin{equation} \label{eq:F-definition} F(\lambda)=2\pi\inf_{P\in\cP}\cE_\lambda(P),\qquad F_N(\lambda)=2\pi\min_{\substack{P\in\cP\\\deg P\le N}} \cE_\lambda(P),\qquad \lambda>0. \end{equation} The determination of $F$ is Problem 4.25 in Hayman and Lingham's \emph{Research Problems in Function Theory}, where it is attributed to W.~H.~J. Fuchs \cite{HaymanLingham2019}. The monic integer condition prevents arbitrary rescaling, while the degree remains unrestricted. The problem therefore asks both how small the weighted norm can be and whether a polynomial attains its infimum. We write \begin{equation} \label{eq:C-definition} \cC(\alpha)=\frac{\Gamma(2\alpha+1)}{\Gamma(\alpha+1)^2}, \qquad D_L(z)=1+z+\cdots+z^L\quad(L\in\Z,\ L\ge0). \end{equation} The following formulas determine the extremal function through $\lambda=3$. \begin{theorem}[The first three intervals] \label{thm:main-intervals} One has \begin{align} F(\lambda)&=2\pi\cC(\lambda),&&0<\lambda\le1, \label{eq:main-base}\\ F(1+\alpha)&=4\pi\cC(\alpha),&&0<\alpha\le1, \label{eq:main-second}\\ F(2+\alpha)&=4\pi\frac{\alpha+2}{\alpha+1}\cC(\alpha), &&0<\alpha\le1. \label{eq:main-third} \end{align} The infimum is attained for $0<\lambda\le1$ and at $\lambda=2,3$. It is not attained on $(1,2)\cup(2,3)$. For $0<\lambda<1$, the minimizing polynomials are exactly the monomials $z^s$. At $\lambda=1$, they are exactly $z^sD_L$, with $s,L\ge0$. \end{theorem} The proof on $(2,3)$ uses a quantitative rearrangement inequality. Among nonnegative integer sequences of fixed mass, a block of consecutive ones minimizes the fractional energy, and every other sequence has an explicit positive energy gap. Applied to the positive and negative parts of $(1-z)^2P$, this gap forces both masses to equal two in every sufficiently advanced member of a minimizing sequence. The second difference then factors, up to a monomial, into two binomials. The same estimates also determine the finite-degree minimum. \begin{theorem}[Finite degree and minimizing sequences] \label{thm:main-degree} Let $0<\alpha<1$ and let $c_j(\alpha)$ be the Fourier coefficients of $|1-e^{i\theta}|^{2\alpha}$, as in \cref{eq:fourier}. For every $N\ge0$, \begin{equation} \label{eq:main-degree-second} F_N(1+\alpha)=4\pi\bigl(\cC(\alpha)-c_{N+1}(\alpha)\bigr), \end{equation} with unique minimizer $D_N$. For $N\ge5$, put $M=\floor{(N+1)/2}$. Then \begin{equation} \label{eq:main-degree-third} \frac{F_N(2+\alpha)}{2\pi} =4\cC(\alpha)+2c_1(\alpha)+2c_{2M+1}(\alpha) -4c_M(\alpha)-4c_{M+1}(\alpha). \end{equation} Its minimizers are precisely \[ z^sD_{M-1}D_M,\qquad 0\le s\le N-(2M-1). \] Furthermore, every minimizing sequence for $F(1+\alpha)$ eventually has the form $z^{s_j}D_{L_j}$ with $L_j\to\infty$. Every minimizing sequence for $F(2+\alpha)$ eventually has the form $z^{s_j}D_{L_j-1}D_{L_j}$ with $L_j\to\infty$. \end{theorem} For each fixed degree bound, the minimizers are independent of the parameter within $(0,1)$ and within $(1,2)$. The same holds on $(2,3)$ for every $N\ge5$. The sequence statements also classify all asymptotically optimal polynomials on $(1,2)$ and $(2,3)$. At an integer $m$, set \begin{equation} \label{eq:M-definition} \cM_m=\min\left\{\norm{Q}_2^2: Q\in\Z[z],\ (1-z)^m\mid Q,\ \lc(Q)=(-1)^m\right\}, \end{equation} where $\norm{Q}_2^2$ is the sum of the squares of the coefficients. The moment interpretation of divisibility by $(1-z)^m$ connects the problem with equal sums of like powers. \begin{theorem}[Integer parameters] \label{thm:main-integer} For $m\ge1$, one has $F(m)=2\pi\cM_m$ and \begin{equation} \label{eq:main-integer-bounds} 4\pi m\le F(m)\le 2\pi\min\{2^m,m(m-1)+2\}. \end{equation} Equality in the lower bound holds exactly when there are disjoint $m$-element sets $A,B\subset\Z$ with \[ \sum_{a\in A}a^r=\sum_{b\in B}b^r\qquad(1\le r-1/2$ and $j\in\Z$, \begin{equation} \label{eq:fourier} c_j(\alpha)=\frac1{2\pi}\int_{-\pi}^{\pi} |1-e^{i\theta}|^{2\alpha}e^{-ij\theta}\dd\theta =(-1)^j\frac{\Gamma(2\alpha+1)} {\Gamma(\alpha+j+1)\Gamma(\alpha-j+1)}. \end{equation} Reciprocal gamma factors at poles are interpreted as zero. For $0<\alpha<1$, put $d_j(\alpha)=-c_j(\alpha)$ for $j\ge1$. Then \begin{align} d_j(\alpha)&=\frac{\sin(\pi\alpha)}{\pi} \int_0^1t^{j-\alpha-1}(1-t)^{2\alpha}\dd t, \label{eq:kernel-integral}\\ \frac{d_{j+1}(\alpha)}{d_j(\alpha)} &=\frac{j-\alpha}{j+\alpha+1}, \label{eq:kernel-recurrence}\\ \cC(\alpha)&=2\sum_{j\ge1}d_j(\alpha). \label{eq:kernel-sum} \end{align} The sequences $d_j$ and $\delta_j=d_j-d_{j+1}$ are strictly positive, strictly decreasing and strictly convex. \end{lemma} \begin{proof} The substitution $u=\theta/2$ and the beta integral give $c_0=\cC(\alpha)$. Put $w(\theta)=(2-2\cos\theta)^\alpha$. Away from zero, \[ (w\sin\theta)'=\alpha w+(\alpha+1)w\cos\theta. \] Since $w(\theta)\sin\theta\to0$ as $\theta\to0$ when $\alpha>-1/2$, integration by parts against $e^{-ij\theta}$ gives \[ (\alpha+1-j)c_{j-1}+2\alpha c_j+(\alpha+1+j)c_{j+1}=0. \] At $j=0$ this yields $c_1=-\alpha c_0/(\alpha+1)$. Induction then gives the multiplicative recurrence $(j+\alpha+1)c_{j+1}=(j-\alpha)c_j$ for $j\ge0$. Together with $c_{-j}=c_j$, it proves \cref{eq:fourier}, including the vanishing coefficients at integer parameters. For $0<\alpha<1$, the reflection formula gives \[ d_j=\frac{\Gamma(2\alpha+1)\sin(\pi\alpha)}{\pi} \frac{\Gamma(j-\alpha)}{\Gamma(j+\alpha+1)}, \] and the beta integral gives \cref{eq:kernel-integral}. Successive differences insert a factor $1-t$ in that integral, proving all the strict inequalities. The gamma ratio is $O(j^{-2\alpha-1})$, so the Fourier series converges absolutely. Its value at $\theta=0$ is zero, which proves \cref{eq:kernel-sum}. \end{proof} \begin{lemma}[Difference energy] \label{lem:difference} For $0<\alpha<1$ and every real finite sequence $a$, \begin{equation} \label{eq:difference-energy} \cE_\alpha(a)=\sum_{h\ge1}d_h(\alpha) \sum_{j\in\Z}(a_{j+h}-a_j)^2. \end{equation} At $\alpha=1$, the identity becomes $\cE_1(a)=\sum_j(a_{j+1}-a_j)^2$. \end{lemma} \begin{proof} Expanding the squares in \cref{eq:difference-energy} and using \cref{eq:kernel-sum} gives the Fourier quadratic form $\sum_{i,j}a_i a_jc_{i-j}(\alpha)$. All exchanges of sums are justified by finite support and $\sum d_h<\infty$. At $\alpha=1$ the only nonzero Fourier coefficients are $c_0=2$ and $c_{\pm1}=-1$. \end{proof} \begin{proposition}[The base interval] \label{prop:base} For every nonzero integer polynomial $A$ and $0<\alpha\le1$, $\cE_\alpha(A)\ge\cC(\alpha)$. For $0<\alpha<1$, equality holds exactly for $A=\pm z^s$. At $\alpha=1$, equality holds exactly for $A=\pm z^sD_L$. Consequently \cref{eq:main-base} and its equality assertions hold. \end{proposition} \begin{proof} For each $h\ge1$, a nonzero residue class of the integer sequence modulo $h$ starts at zero, reaches a nonzero integer and returns to zero. Thus $\sum_j(a_{j+h}-a_j)^2\ge2$. Summing with the weights $d_h$ gives the result for $0<\alpha<1$. Equality forces each such squared variation to equal two. Taking $h$ larger than the support diameter gives $2\sum a_j^2=2$, hence a single coefficient $\pm1$. At $\alpha=1$, equality in the unit-step variation means that exactly two jumps have magnitude one and every other jump is zero. The sequence is therefore a single block of ones or minus ones. The leading coefficient condition selects the positive sign. \end{proof} \subsection{Rearrangement and a strict gap for integer masses} For an integer $s\ge1$, define \begin{equation} \label{eq:S-definition} S_s=\cE_\alpha(D_{s-1}) =s\cC(\alpha)-2\sum_{h=1}^{s-1}(s-h)d_h, \qquad S_0=0. \end{equation} Throughout this subsection $0<\alpha<1$ is fixed. The increments are \begin{equation} \label{eq:S-increments} S_{s+1}-S_s=2\sum_{h>s}d_h>0. \end{equation} They strictly decrease, so $S_s$ is strictly increasing and strictly concave as a sequence. In particular, $S_{a+b}\le S_a+S_b$. \begin{lemma}[Integer rearrangement with a gap] \label{lem:mass-gap} Let $p$ be a nonnegative integer sequence of finite support and mass $\sum_jp_j=s\ge1$. Then \begin{equation} \label{eq:mass-minimum} \cE_\alpha(p)\ge S_s, \end{equation} with equality exactly when $p$ is the indicator of an interval of $s$ consecutive integers. If $s\ge2$ and $p$ is not such an indicator, then \begin{equation} \label{eq:mass-defect} \cE_\alpha(p)\ge S_s+2(d_1-d_s). \end{equation} The constant in \cref{eq:mass-defect} is attained by the indicator of $\{0,1,\ldots,s-2,s\}$. \end{lemma} \begin{proof} First let $p=\I_A$, where $A=\{x_1<\cdots2(d_1-d_s). \] This proves \cref{eq:mass-defect} for repeated coefficients and completes the equality classification. \end{proof} \subsection{The second interval and finite-degree attainment} \begin{proposition}[A zero-sum energy] \label{prop:zero-sum} Let $0<\alpha<1$, let $b\ne0$ be an integer sequence with $\sum b_j=0$, and let its support lie in an interval of integer length $H\ge1$, meaning that its support diameter is at most $H$. Then \begin{equation} \label{eq:zero-sum-energy} \cE_\alpha(b)\ge2\bigl(\cC(\alpha)+d_H\bigr). \end{equation} Equality holds exactly for two coefficients $+1,-1$ at distance $H$. If the positive mass $s=\sum_j\max(b_j,0)$ is at least two, then \begin{equation} \label{eq:second-mass-gap} \cE_\alpha(b)>2S_2 =2\cC(\alpha)+\frac{2(1-\alpha)}{1+\alpha}\cC(\alpha). \end{equation} \end{proposition} \begin{proof} Write $b=p-n$ for its nonnegative positive and negative parts. They have disjoint supports and the same mass $s\ge1$. All their cross interactions have nonzero index, so \begin{equation} \label{eq:positive-negative} \cE_\alpha(b)=\cE_\alpha(p)+\cE_\alpha(n) +2\sum_{i,j}d_{|i-j|}p_i n_j. \end{equation} By \cref{prop:base}, the first two terms are at least $\cC(\alpha)$. The cross term is at least $2d_Hs^2$. This proves \cref{eq:zero-sum-energy}, and its equality case follows from \cref{prop:base}. If $s\ge2$, \cref{lem:mass-gap} and strict positivity of the cross term give $\cE_\alpha(b)>2S_s\ge2S_2$. \end{proof} \begin{proof}[Proof of \cref{eq:main-second,eq:main-degree-second}] For a monic $P$ of degree at most $N$, put $B=(1-z)P$. Its coefficient sum is zero and its support diameter is at most $N+1$. By \cref{prop:zero-sum}, $\cE_{1+\alpha}(P)=\cE_\alpha(B)\ge2(c_0+d_{N+1})$. The polynomial $D_N$ attains the bound. Its uniqueness follows because equality requires the two extreme indices $0,N+1$ and the leading coefficient of $B$ is $-1$. Letting $N\to\infty$ proves \cref{eq:main-second} for $0<\alpha<1$ and shows that the infimum is not attained. At $\alpha=1$, a nonzero integer zero-sum sequence takes both positive and negative values and is zero outside a finite interval. Its total unit-step variation is therefore at least four. Each jump is an integer, so its squared variation is also at least four. The polynomial $1+z$ attains four. Thus $F(2)=8\pi$. \end{proof} \begin{corollary}[Exact error and sequence rigidity] \label{cor:second-rigidity} For $0<\alpha<1$, \begin{align} F_N(1+\alpha)-F(1+\alpha)&=4\pi d_{N+1}, \label{eq:second-error}\\ 4\pi d_{N+1}&\sim 4\Gamma(2\alpha+1)\sin(\pi\alpha)(N+1)^{-2\alpha-1}. \end{align} Every minimizing sequence for $F(1+\alpha)$ eventually has the form $z^{s_j}D_{L_j}$, where $L_j\to\infty$. \end{corollary} \begin{proof} The error formula follows from the preceding result, and its asymptotic form follows from the gamma ratio in \cref{lem:fourier}. The positive gap in \cref{eq:second-mass-gap} forces the positive and negative masses of $(1-z)P_j$ to equal one for all sufficiently large $j$. The leading coefficient then gives $(1-z)P_j=z^{s_j}(1-z^{L_j+1})$. Formula \eqref{eq:second-error} implies $L_j\to\infty$. \end{proof} \subsection{The third interval} Fix $0<\alpha<1$. The second difference $(1-z)^2P$ has vanishing zeroth and first moments. The rearrangement gap restricts its positive mass to two whenever its energy is sufficiently close to the infimum. Define \begin{equation} \label{eq:T-gap} T_\alpha=4\cC(\alpha)-2d_1 =\frac{2(\alpha+2)}{\alpha+1}\cC(\alpha),\qquad g_\alpha=\frac{6(1-\alpha)(2-\alpha)} {(\alpha+1)(\alpha+2)(\alpha+3)}\cC(\alpha)>0. \end{equation} \begin{lemma}[Two vanishing moments] \label{lem:two-moments} Let $0<\alpha<1$, and let $B(z)=\sum_{j\ge0}b_jz^j$ be a nonzero integer polynomial with $\sum_j b_j=\sum_jjb_j=0$, and let $s$ be its positive mass. Then $s\ge2$. If $s\ge3$, then \begin{equation} \label{eq:third-mass-gap} \cE_\alpha(b)>T_\alpha+g_\alpha. \end{equation} If $s=2$ and the leading coefficient is $+1$, its polynomial has the form \begin{equation} \label{eq:four-term-factorization} B(z)=z^u(1-z^a)(1-z^b),\qquad u\ge0,\quad a,b\ge1. \end{equation} The possibility $a=b$ is included. \end{lemma} \begin{proof} Mass one would give $B=z^u-z^v$ with $u=v$ by the first-moment condition, contrary to $B\ne0$. Write $B=p-n$ as in \cref{eq:positive-negative}. If both parts were interval indicators of mass $s$, equality of their first moments would force the same starting index. This contradicts their disjoint supports. At least one part therefore satisfies \cref{eq:mass-defect}. The cross term is positive, and hence \[ \cE_\alpha(B)>2S_s+2(d_1-d_s) \ge2S_3+2(d_1-d_3) \] when $s\ge3$. The second inequality uses the increase of $S_s$ and the decrease of $d_s$. Substituting the first three Fourier coefficients gives \[ 2S_3+2(d_1-d_3)-T_\alpha =2c_0-4d_1-4d_2-2d_3=g_\alpha. \] For $s=2$, the positive and negative parts each consist of two points counted with multiplicity, with equal sums. Their supports are disjoint. The pair containing the largest point also contains the smallest: otherwise its sum would be larger than the other sum. The prescribed leading coefficient makes this the positive pair. Subtract its smallest point $u$ from all four indices. The negative indices are $a,b>0$, and the remaining positive index is $a+b$. This proves \cref{eq:four-term-factorization}. \end{proof} For later use write \begin{equation} \label{eq:Iab} I_\alpha(a,b)=\cE_\alpha\bigl((1-z^a)(1-z^b)\bigr) =4c_0+2c_{a+b}+2c_{|a-b|}-4c_a-4c_b. \end{equation} For $a0. \end{equation} For $a=b$, one has $I_\alpha(a,a)=6c_0-2d_{2a}+8d_a>6c_0>T_\alpha$. On the other hand, $I_\alpha(a,a+1)\to T_\alpha$ as $a\to\infty$. \begin{proof}[Proof of \cref{eq:main-third}] Put $B=(1-z)^2P$. Its first two moments vanish. The cases $s\ge3$ and $s=2$ are covered by \cref{lem:two-moments} and \cref{eq:product-global-lower}, respectively, and both give $\cE_\alpha(B)>T_\alpha$. The sequence $P=D_{a-1}D_a$ gives energies tending to $T_\alpha$. This proves the formula and nonattainment for $0<\alpha<1$. At $\alpha=1$, the integer moment lower bound in \cref{thm:integer-reduction} gives $F(3)\ge12\pi$; equality is attained by $D_1D_2=1+2z+2z^2+z^3$, whose third difference has six coefficients of absolute value one. This completes \cref{thm:main-intervals}. \end{proof} \begin{corollary}[Rigidity on the third interval] \label{cor:third-rigidity} Let $0<\alpha<1$. If $P_j$ is a minimizing sequence for $F(2+\alpha)$, then, for all sufficiently large $j$, \[ P_j=z^{s_j}D_{L_j-1}D_{L_j},\qquad L_j\to\infty. \] \end{corollary} \begin{proof} The gap $g_\alpha$ forces positive mass two eventually. In \cref{eq:product-global-lower}, a difference $b-a\ge2$ incurs the fixed gap $2(d_1-d_2)>0$, while diagonal pairs have a fixed positive gap as well. Thus $b=a+1$ eventually. The remaining energy excess is $4d_a+4d_{a+1}-2d_{2a+1}>4d_a$. It tends to zero as $a\to\infty$, and this lower bound shows that the converse also holds. \end{proof} \subsection{The shape of the extremal function} \begin{proposition}[A local maximum and a unique interior minimum] \label{prop:shape} The function $F$ is continuous on $(0,3]$, with \[ F'(2-)=8\pi,\qquad F'(2+)=-4\pi. \] Consequently $2$ is a strict local maximum. On $(2,3)$ the function $F$ has exactly one critical point, a strict minimum. Writing it as $2+\alpha_*$, the number $\alpha_*\in(0,1)$ is the unique solution of \begin{equation} \label{eq:critical-point} 2\psi(2\alpha+1)-2\psi(\alpha+1) -\frac1{(\alpha+1)(\alpha+2)}=0, \end{equation} where $\psi=\Gamma'/\Gamma$. \end{proposition} \begin{proof} The endpoint values agree in \cref{thm:main-intervals}. Since $\cC'(0)=0$ and $\cC'(1)=2$, the stated one-sided derivatives follow directly. The logarithmic derivative of $F(2+\alpha)$ equals the left side of \cref{eq:critical-point}. By the duplication formula it is \[ h(\alpha)=\psi(\alpha+1/2)-\psi(\alpha+1)+2\log2 -\frac1{(\alpha+1)(\alpha+2)}. \] The trigamma function $\psi'$ strictly decreases on the positive real axis, as follows from $\psi'(x)=\sum_{n\ge0}(x+n)^{-2}$. Thus $h'(\alpha)>0$. Finally $h(0)=-1/2$ and $h(1)=5/6$, so $h$ has exactly one zero. \end{proof} The critical point has the numerical approximations \[ \alpha_*=0.1646512225\ldots,\qquad F(2+\alpha_*)=24.2137162258\ldots. \] \Cref{fig:exact-region} plots the exact formulas and enlarges the minimum on the third interval. \begin{figure}[htbp] \centering \begin{tikzpicture} \begin{axis}[width=.56\textwidth,height=5.4cm, xmin=0,xmax=3,ymin=.8,ymax=6.2, xlabel={$\lambda$},ylabel={$F(\lambda)/(2\pi)$}, xtick={0,1,2,3},ytick={1,2,3,4,5,6}, grid=major,grid style={gray!15},tick label style={font=\small}, label style={font=\small}] \addplot[blue!65!black,very thick] coordinates {(0.020000,1.0006395320) (0.040000,1.0024902275) (0.060000,1.0054620377) (0.080000,1.0094785080) (0.100000,1.0144745488) (0.120000,1.0203946384) (0.140000,1.0271913638) (0.160000,1.0348242281) (0.180000,1.0432586689) (0.200000,1.0524652462) (0.220000,1.0624189669) (0.240000,1.0730987190) (0.260000,1.0844867975) (0.280000,1.0965685028) (0.300000,1.1093318014) (0.320000,1.1227670367) (0.340000,1.1368666826) (0.360000,1.1516251321) (0.380000,1.1670385160) (0.400000,1.1831045468) (0.420000,1.1998223844) (0.440000,1.2171925189) (0.460000,1.2352166707) (0.480000,1.2538977027) (0.500000,1.2732395447) (0.520000,1.2932471278) (0.540000,1.3139263273) (0.560000,1.3352839134) (0.580000,1.3573275085) (0.600000,1.3800655502) (0.620000,1.4035072599) (0.640000,1.4276626150) (0.660000,1.4525423266) (0.680000,1.4781578189) (0.700000,1.5045212139) (0.720000,1.5316453169) (0.740000,1.5595436066) (0.760000,1.5882302258) (0.780000,1.6177199753) (0.800000,1.6480283094) (0.820000,1.6791713331) (0.840000,1.7111658010) (0.860000,1.7440291178) (0.880000,1.7777793400) (0.900000,1.8124351791) (0.920000,1.8480160055) (0.940000,1.8845418545) (0.960000,1.9220334322) (0.980000,1.9605121235) (1.000000,2.0000000000) (1.020000,2.0012790639) (1.040000,2.0049804550) (1.060000,2.0109240754) (1.080000,2.0189570160) (1.100000,2.0289490976) (1.120000,2.0407892767) (1.140000,2.0543827276) (1.160000,2.0696484563) (1.180000,2.0865173378) (1.200000,2.1049304925) (1.220000,2.1248379337) (1.240000,2.1461974380) (1.260000,2.1689735950) (1.280000,2.1931370056) (1.300000,2.2186636028) (1.320000,2.2455340734) (1.340000,2.2737333652) (1.360000,2.3032502641) (1.380000,2.3340770319) (1.400000,2.3662090937) (1.420000,2.3996447688) (1.440000,2.4343850378) (1.460000,2.4704333414) (1.480000,2.5077954054) (1.500000,2.5464790895) (1.520000,2.5864942557) (1.540000,2.6278526546) (1.560000,2.6705678268) (1.580000,2.7146550169) (1.600000,2.7601311004) (1.620000,2.8070145197) (1.640000,2.8553252301) (1.660000,2.9050846532) (1.680000,2.9563156379) (1.700000,3.0090424277) (1.720000,3.0632906339) (1.740000,3.1190872133) (1.760000,3.1764604517) (1.780000,3.2354399507) (1.800000,3.2960566188) (1.820000,3.3583426661) (1.840000,3.4223316019) (1.860000,3.4880582356) (1.880000,3.5555586800) (1.900000,3.6248703581) (1.920000,3.6960320110) (1.940000,3.7690837090) (1.960000,3.8440668644) (1.980000,3.9210242470) (2.000000,4.0000000000) (2.020000,3.9633173619) (2.040000,3.9328462772) (2.060000,3.9080222597) (2.080000,3.8883616604) (2.100000,3.8734482772) (2.120000,3.8629225596) (2.140000,3.8564728396) (2.160000,3.8538281600) (2.180000,3.8547523699) (2.200000,3.8590392362) (2.220000,3.8665083712) (2.240000,3.8770018235) (2.260000,3.8903812101) (2.280000,3.9065252912) (2.300000,3.9253279126) (2.320000,3.9466962502) (2.340000,3.9705493094) (2.360000,3.9968166348) (2.380000,4.0254372000) (2.400000,4.0563584463) (2.420000,4.0895354510) (2.440000,4.1249302030) (2.460000,4.1625109725) (2.480000,4.2022517605) (2.500000,4.2441318158) (2.520000,4.2881352133) (2.540000,4.3342504823) (2.560000,4.3824702798) (2.580000,4.4327911036) (2.600000,4.4852130381) (2.620000,4.5397395319) (2.640000,4.5963771996) (2.660000,4.6551356490) (2.680000,4.7160273271) (2.700000,4.7790673852) (2.720000,4.8442735605) (2.740000,4.9116660715) (2.760000,4.9812675265) (2.780000,5.0531028443) (2.800000,5.1271991848) (2.820000,5.2035858893) (2.840000,5.2822944291) (2.860000,5.3633583622) (2.880000,5.4468132971) (2.900000,5.5326968624) (2.920000,5.6210486834) (2.940000,5.7119103631) (2.960000,5.8053254688) (2.980000,5.9013395232) (3.000000,6.0000000000)}; \addplot[only marks,mark=*,mark size=1.7pt,red!65!black] coordinates {(2,4)}; \end{axis} \end{tikzpicture}\hfill \begin{tikzpicture} \begin{axis}[width=.40\textwidth,height=5.4cm, xmin=2,xmax=2.4,ymin=3.84,ymax=4.12, xlabel={$\lambda$},xtick={2,2.2,2.4},ytick={3.85,3.95,4.05}, scaled ticks=false,grid=major,grid style={gray!15}, tick label style={font=\small},label style={font=\small}] \addplot[blue!65!black,very thick] coordinates {(2.000000,4.0000000000) (2.005000,3.9902126491) (2.010000,3.9808433679) (2.015000,3.9718816391) (2.020000,3.9633173619) (2.025000,3.9551408332) (2.030000,3.9473427280) (2.035000,3.9399140825) (2.040000,3.9328462772) (2.045000,3.9261310209) (2.050000,3.9197603365) (2.055000,3.9137265465) (2.060000,3.9080222597) (2.065000,3.9026403588) (2.070000,3.8975739885) (2.075000,3.8928165440) (2.080000,3.8883616604) (2.085000,3.8842032024) (2.090000,3.8803352549) (2.095000,3.8767521139) (2.100000,3.8734482772) (2.105000,3.8704184368) (2.110000,3.8676574707) (2.115000,3.8651604356) (2.120000,3.8629225596) (2.125000,3.8609392354) (2.130000,3.8592060142) (2.135000,3.8577185991) (2.140000,3.8564728396) (2.145000,3.8554647258) (2.150000,3.8546903832) (2.155000,3.8541460676) (2.160000,3.8538281600) (2.165000,3.8537331624) (2.170000,3.8538576932) (2.175000,3.8541984829) (2.180000,3.8547523699) (2.185000,3.8555162972) (2.190000,3.8564873082) (2.195000,3.8576625431) (2.200000,3.8590392362) (2.205000,3.8606147119) (2.210000,3.8623863819) (2.215000,3.8643517425) (2.220000,3.8665083712) (2.225000,3.8688539245) (2.230000,3.8713861349) (2.235000,3.8741028087) (2.240000,3.8770018235) (2.245000,3.8800811257) (2.250000,3.8833387286) (2.255000,3.8867727099) (2.260000,3.8903812101) (2.265000,3.8941624302) (2.270000,3.8981146301) (2.275000,3.9022361265) (2.280000,3.9065252912) (2.285000,3.9109805498) (2.290000,3.9156003796) (2.295000,3.9203833083) (2.300000,3.9253279126) (2.305000,3.9304328165) (2.310000,3.9356966902) (2.315000,3.9411182487) (2.320000,3.9466962502) (2.325000,3.9524294955) (2.330000,3.9583168263) (2.335000,3.9643571241) (2.340000,3.9705493094) (2.345000,3.9768923404) (2.350000,3.9833852120) (2.355000,3.9900269550) (2.360000,3.9968166348) (2.365000,4.0037533509) (2.370000,4.0108362355) (2.375000,4.0180644532) (2.380000,4.0254372000) (2.385000,4.0329537021) (2.390000,4.0406132157) (2.395000,4.0484150259) (2.400000,4.0563584463)}; \addplot[only marks,mark=*,mark size=1.8pt,red!65!black] coordinates {(2.1646512225,3.8537326280)}; \end{axis} \end{tikzpicture} \caption{The exact extremal function on $(0,3]$ (left), and the unique minimum on $(2,3)$ (right). Both curves use the analytic formulas of \cref{thm:main-intervals}.} \label{fig:exact-region} \end{figure} \section{Integer parameters and ideal PTE solutions} \label{sec:integer} \subsection{Parseval reduction and the equality case} At integer parameters the normalized weighted integral is a sum of coefficient squares. Divisibility by a power of $1-z$ then turns the minimization into a problem about equal power sums. For a coefficient polynomial $Q=\sum_jb_jz^j$, write $\norm{Q}_1=\sum_j|b_j|$ and $\norm{Q}_2^2=\sum_jb_j^2$. Divisibility by $(1-z)^m$ is equivalent to \begin{equation} \label{eq:moments} \sum_j j^r b_j=0\qquad(0\le r\deg Q_m$. The coefficients of $Q_m(z)Q_n(z^L)$ are products without overlap, so their squared sum is $\cM_m\cM_n$. The zero at one has order at least $m+n$ and the leading sign is correct. The exact-order assertion follows by applying \cref{lem:integer-mass} at order $m+1$. \end{proof} \subsection{Explicit integer values} The pairs in \cref{tab:pte-small} have equal power sums through degree $m-1$. Their incidence polynomials have the required leading signs. Thus they give $F(m)=4\pi m$ for $1\le m\le6$. \begin{table}[htbp] \centering \caption{Multiplicity-free ideal PTE pairs for the first six parameters. The degree is that of the monic quotient in \cref{eq:pte-quotient}.} \label{tab:pte-small} \small \begin{tabular}{@{}ccll@{}} \toprule $m$ & degree & $A_m$ & $B_m$\\ \midrule 1&0&$\{0\}$&$\{1\}$\\ 2&1&$\{0,3\}$&$\{1,2\}$\\ 3&3&$\{0,4,5\}$&$\{1,2,6\}$\\ 4&7&$\{0,4,7,11\}$&$\{1,2,9,10\}$\\ 5&13&$\{0,4,8,16,17\}$&$\{1,2,10,14,18\}$\\ 6&16&$\{0,5,6,16,17,22\}$&$\{1,2,10,12,20,21\}$\\ \bottomrule \end{tabular} \end{table} For $m=7,8,9,10,12$, the following pairs give the remaining values in \cref{thm:main-integer}: \begin{equation} \label{eq:large-pte} \begin{aligned} A_7&=\{0,18,27,58,64,89,101\},\\ B_7&=\{1,13,38,44,75,84,102\},\\ A_8&=\{0,4,9,23,27,41,46,50\},\\ B_8&=\{1,2,11,20,30,39,48,49\},\\ A_9&=\{0,24,30,83,86,133,157,181,197\},\\ B_9&=\{1,17,41,65,112,115,168,174,198\},\\ A_{10}&=\{0,12,125,213,214,412,413,501,614,626\},\\ B_{10}&=\{5,6,133,182,242,384,444,493,620,621\},\\ A_{12}&=\{0,11,24,65,90,129,173,212,237,278,291,302\},\\ B_{12}&=\{3,5,30,57,104,116,186,198,245,272,297,299\}. \end{aligned} \end{equation} The size-seven pair is recorded in \cite[Section~5.4.2, p.~48]{Caley2012}. The size-eight pair is the expansion of the pure product with lengths $(1,2,3,5,7,8,11,13)$ listed in \cite[Table~1]{Cipu2004}. The size-nine, size-ten and size-twelve pairs are obtained by translation and, where necessary, interchange from the symmetric examples in \cite[equations~(3)--(5)]{CoppersmithEtAl2024}; see also \cite{BorweinLisonekPercival2003}. The accompanying program verifies the moment identities for every displayed pair using integer arithmetic. The quotient degrees for $m=7,8,9,10,12$ are $95,42,189,616,290$, respectively. The size-twelve pair also gives an admissible polynomial at order eleven after changing its sign. Hence $22\le\cM_{11}\le24$. By parity, $\cM_{11}\in\{22,24\}$, proving the stated two-value alternative. The value $22$ is equivalent to the existence of a multiplicity-free ideal pair of size eleven. The symmetric search in \cite[Section~4.1 and Table~2]{CoppersmithEtAl2024} excludes such integer solutions up to height $3500$ in that symmetric class. Here height means $\max_{a\in A}|a|$ in the centered normalization $B=-A$. The general ideal existence question remains open; see also the discussion in \cite{CrootMaoYip2026}. \begin{remark}[Extremizers need not be palindromic] \label{rem:nonpalindromic} The pair $\{0,9,10\},\{1,6,12\}$ has equal first and second moments. It gives the nonpalindromic minimizer at $m=3$ \[ 1+2z+3z^2+4z^3+5z^4+6z^5+6z^6+5z^7+3z^8+z^9. \] Indeed its third difference is $1-z-z^6+z^9+z^{10}-z^{12}$. Consequently palindromicity cannot be imposed as a general search restriction. Palindromicity is therefore not imposed in the coefficient searches reported below. \end{remark} \section{General bounds and lifting operations} \label{sec:bounds} \subsection{A quadratic upper bound} \begin{proposition}[Subset counting] \label{prop:quadratic} For every $m\ge1$, one has $\cM_m\le m(m-1)+2$. \end{proposition} \begin{proof} Put $s=m(m-1)/2+1$ and consider all $s$-element subsets of $\{1,\ldots,H\}$. There are $\binom Hs\asymp_mH^s$ such subsets. The vector of power sums of orders $1,\ldots,m-1$ takes at most \[ \prod_{r=1}^{m-1}(sH^r+1)=O_m(H^{s-1}) \] values. For sufficiently large $H$, two distinct subsets have the same vector. Deleting their common elements gives two nonempty disjoint sets of equal cardinality at most $s$, with the same moments through order $m-1$, including order zero. Their signed incidence polynomial has squared norm at most $2s$ and is divisible by $(1-z)^m$. Its sign can be chosen to meet the leading coefficient condition. \end{proof} This is the classical pigeonhole construction for equal power sums. Starting with subsets makes the resulting coefficients have absolute value one; a bound on the cardinality of a multiset alone would not give the same squared-norm estimate. For comparison, Prouhet's binary construction \[ \prod_{r=0}^{m-1}(1-z^{2^r}) \] has $2^m$ coefficients of absolute value one. Combining these two constructions proves \cref{eq:main-integer-bounds} and completes \cref{thm:main-integer}. \subsection{Dilation and adjacent parameter values} Multiplication by a long Dirichlet polynomial raises the parameter by one and asymptotically doubles the energy. Dilating the exponents of an integer extremizer gives a second comparison, which retains the factor $\cC(\alpha)$. \begin{proposition}[Dirichlet lifting] \label{prop:lifting} For every $\lambda>0$ and integer $r\ge0$, \begin{equation} \label{eq:lifting} F(\lambda+1)\le2F(\lambda),\qquad F(\lambda+r)\le2^rF(\lambda). \end{equation} \end{proposition} \begin{proof} For a fixed monic integer $P$, the polynomial $PD_N$ is admissible and \[ \cE_{\lambda+1}(PD_N) =\frac1{2\pi}\int_{-\pi}^{\pi} |1-e^{i\theta}|^{2\lambda}|P(e^{i\theta})|^2 \bigl(2-2\cos((N+1)\theta)\bigr)\dd\theta. \] The Riemann--Lebesgue lemma gives the limit $2\cE_\lambda(P)$. Take the infimum over $P$ and iterate. \end{proof} \begin{theorem}[Interpolation from integer extrema] \label{thm:interpolation} For every integer $k\ge1$ and $0<\alpha<1$, \begin{equation} \label{eq:interpolation} \frac{\cC(\alpha)}2F(k+1) \le F(k+\alpha)\le\cC(\alpha)F(k). \end{equation} Consequently, \begin{equation} \label{eq:global-growth} 2\pi(k+1)\cC(\alpha)\le F(k+\alpha) \le2\pi\cC(\alpha)\min\{2^k,k(k-1)+2\}. \end{equation} In particular $F(\lambda)=\Omega(\lambda)$ and $F(\lambda)=O(\lambda^2)$ as $\lambda\to\infty$. \end{theorem} \begin{proof} For $Q=(1-z)^kP$, \cref{eq:difference-energy} gives \[ \cE_{k+\alpha}(P) =\sum_{h\ge1}d_h\norm{(1-z^h)Q}_2^2. \] Each polynomial in the sum is admissible at order $k+1$, so its squared norm is at least $\cM_{k+1}$. Summing and using \cref{eq:kernel-sum} proves the lower bound. For the upper bound, choose a minimizer $Q_k$ for $\cM_k$ and, for positive integers $L$, put \[ P_L(z)=Q_k(z^L)/(1-z)^k. \] This is a monic integer polynomial. Its energy equals $\cE_\alpha(Q_k(z^L))$, whose nonconstant Fourier interactions tend to zero as $L\to\infty$. The limit is $\cC(\alpha)\norm{Q_k}_2^2=\cC(\alpha)\cM_k$. The remaining assertions follow from the integer bounds and $1\le\cC(\alpha)\le2$ on $[0,1]$. The latter follows either by the gamma formula or by differentiating it: $\psi$ is increasing and $\cC(0)=1$, $\cC(1)=2$. \end{proof} \section{Dirichlet products and their limitations} \label{sec:products} \subsection{An exact two-factor optimization} The first three interval formulas lead to Dirichlet products as natural candidates at higher parameters. We first optimize the two-factor case, then construct higher-interval bounds and identify intervals where these products are suboptimal. For $\lambda=k+\alpha$, $k\ge1$ and $0<\alpha<1$, a Dirichlet product with positive integer lengths $L_1,\ldots,L_k$ is \begin{equation} \label{eq:product-definition} P_{L_1,\ldots,L_k}(z)=\prod_{i=1}^kD_{L_i-1}(z). \end{equation} It has degree $\sum_i(L_i-1)$, and \begin{equation} \label{eq:product-energy} \cE_{k+\alpha}(P_{L_1,\ldots,L_k}) =\cE_\alpha\left(\prod_{i=1}^k(1-z^{L_i})\right). \end{equation} \begin{theorem}[Two-factor minimum at a degree bound] \label{thm:two-products} Let $0<\alpha<1$ and $N\ge1$. Put $M=\floor{(N+1)/2}$. Among the positive integer pairs $a,b$ with $a+b-2\le N$, the unique unordered minimizing pair for $I_\alpha(a,b)$ is $\{M,M+1\}$. \end{theorem} \begin{proof} For $a0 \end{equation} by strict convexity of $\delta$. If $f_m=I_\alpha(m,m+1)$, direct subtraction gives \[ f_{m+1}-f_m =2(\delta_{2m+1}+\delta_{2m+2}) -4(\delta_m+\delta_{m+1})<0. \] It remains to exclude diagonal pairs. For $m\ge2$, \[ I_\alpha(m,m)-I_\alpha(m-1,m) =2(c_0+\delta_{2m-1}+d_1-2\delta_{m-1})>0, \] because $\delta_{m-1}0. \] The latter comparison is feasible since $N\ge1$. Thus the largest admissible adjacent pair is the unique unordered minimizer. \end{proof} \begin{theorem}[The finite-degree third interval] \label{thm:third-degree} For $0<\alpha<1$ and $N\ge5$, the formula and complete minimizer classification in \cref{eq:main-degree-third} hold. With $M=\floor{(N+1)/2}$, \begin{equation} \label{eq:third-error} F_N(2+\alpha)-F(2+\alpha) =2\pi(4d_M+4d_{M+1}-2d_{2M+1}). \end{equation} Writing $p=2\alpha+1$ and $A_\alpha=\Gamma(2\alpha+1)\sin(\pi\alpha)/\pi$, the error is asymptotic to \begin{equation} \label{eq:third-error-asymptotic} 2\pi A_\alpha(8-2^{-2\alpha})M^{-2\alpha-1}. \end{equation} \end{theorem} \begin{proof} By \cref{thm:two-products}, the best mass-two candidate is $D_{M-1}D_M$. Since $M\ge3$ and the adjacent-pair energy decreases, \[ I_\alpha(M,M+1)-T_\alpha \le4d_3+4d_4-2d_7<4d_3+4d_4 =\frac{28}{\alpha+4}d_3. \] The gap from \cref{eq:T-gap} satisfies $g_\alpha=6d_3/\alpha$. Moreover \[ \frac{28d_3/(\alpha+4)}{g_\alpha} =\frac{14\alpha}{3(\alpha+4)}<1 \quad(0<\alpha<1). \] Thus every polynomial with positive mass at least three in its second difference has strictly larger energy, by \cref{lem:two-moments}. All minimizers have mass two and are classified by \cref{thm:two-products}. Their monomial shifts are exactly those allowed by the degree bound. Formula \eqref{eq:third-error} is immediate; the gamma-ratio asymptotic $d_n\sim A_\alpha n^{-p}$ gives \cref{eq:third-error-asymptotic}. \end{proof} \subsection{Separated clusters of exponents} \begin{proposition}[A product limit] \label{prop:product-limit} Fix $0<\alpha<1$, positive integers $u_1,\ldots,u_k$ and integers $v_1,\ldots,v_k$. Put $L_i(t)=u_it+v_i$ and, for $S\subseteq\{1,\ldots,k\}$, write \[ U_S=\sum_{i\in S}u_i,\quad V_S=\sum_{i\in S}v_i, \quad\varepsilon_S=(-1)^{|S|}. \] As $t\to\infty$ through integers for which all lengths are positive, \begin{equation} \label{eq:product-limit} \cE_{k+\alpha}(P_{L_1(t),\ldots,L_k(t)}) \longrightarrow \sum_{\substack{S,T\subseteq\{1,\ldots,k\}\\U_S=U_T}} \varepsilon_S\varepsilon_Tc_{V_S-V_T}(\alpha). \end{equation} \end{proposition} \begin{proof} Expand the binomial product as $\sum_S\varepsilon_Sz^{U_St+V_S}$ and insert it into the Fourier quadratic form. Terms with $U_S\ne U_T$ tend to zero, while those with $U_S=U_T$ remain constant. \end{proof} For example, the lengths $(t,t+1,2t+1)$ give \[ (1-z^t)(1-z^{t+1})(1-z^{2t+1}) =1-z^t-z^{t+1}+z^{3t+1}+z^{3t+2}-z^{4t+2}. \] There are two isolated coefficients and two adjacent two-term clusters. Consequently \begin{equation} \label{eq:fourth-upper} F(3+\alpha)\le2\pi(6c_0+4c_1) =4\pi\frac{\alpha+3}{\alpha+1}\cC(\alpha). \end{equation} In particular $F(7/2)\le112/3<12\pi=F(3)$. \subsection{A Pell construction on the fifth interval} \begin{theorem}[An eight-term Pell construction] \label{thm:pell-upper} For every $0<\alpha<1$, \begin{equation} \label{eq:pell-upper} F(4+\alpha)\le 2\pi\bigl(8c_0(\alpha)+6c_1(\alpha)\bigr) =4\pi\frac{\alpha+4}{\alpha+1}\cC(\alpha). \end{equation} More precisely, let \[ u_n+w_n\sqrt2=(3+2\sqrt2)^n,\qquad n\ge2, \] and put \begin{align*} A_n&=\left\{0,\frac{u_n-1}{2},\frac{u_n+1}{2},u_n\right\},\\ B_n&=\left\{\frac{u_n-w_n-1}{2},\frac{u_n-w_n+1}{2}, \frac{u_n+w_n-1}{2},\frac{u_n+w_n+1}{2}\right\}. \end{align*} The polynomials \[ Q_n(z)=\sum_{a\in A_n}z^a-\sum_{b\in B_n}z^b, \qquad P_n(z)=\frac{Q_n(z)}{(1-z)^4} \] are well defined, $P_n$ is monic with integer coefficients and degree $u_n-4$, and \begin{equation} \label{eq:pell-limit} \lim_{n\to\infty}\cE_{4+\alpha}(P_n) =8c_0(\alpha)+6c_1(\alpha). \end{equation} \end{theorem} \begin{proof} The Pell recurrence \[ u_{n+1}=3u_n+4w_n,\qquad w_{n+1}=2u_n+3w_n \] shows that $u_n,w_n$ are positive integers, $u_n$ is odd, $w_n$ is even, and $u_n^2-2w_n^2=1$. For $n\ge2$ the displayed sets are nonnegative, disjoint, and each has four elements. Under the affine change $x\mapsto2x-u_n$, they become \[ \{-u_n,-1,1,u_n\},\qquad \{-w_n-1,-w_n+1,w_n-1,w_n+1\}. \] Their odd power sums agree by symmetry, and their second power sums agree because \[ u_n^2+1=(w_n+1)^2+(w_n-1)^2. \] Thus the original sets have equal moments of orders zero through three. Their fourth-moment difference is \[ \sum_{a\in A_n}a^4-\sum_{b\in B_n}b^4 =\frac{w_n^2(w_n^2-4)}4>0. \] Consequently $Q_n$ has a zero of exactly order four at one. Division by $(1-z)^4$ gives an integer polynomial, and the leading coefficient of $Q_n$ is one, so $P_n$ is monic of degree $u_n-4$. Write $a_n=(u_n-1)/2$ and $b_n=(u_n-w_n-1)/2$. The eight terms of $Q_n$ form the five clusters \[ \{0\},\quad\{b_n,b_n+1\},\quad\{a_n,a_n+1\}, \quad\{u_n-b_n-1,u_n-b_n\},\quad\{u_n\}. \] Each of the three two-point clusters has coefficients of the same sign. Since $u_n/w_n\to\sqrt2$ and $w_n\to\infty$, the distances between distinct clusters tend to infinity. In the Fourier expression for $\cE_\alpha(Q_n)$, all interactions between different clusters therefore vanish in the limit. The eight diagonal terms contribute $8c_0(\alpha)$, and the three adjacent same-sign pairs contribute $6c_1(\alpha)$. This proves \cref{eq:pell-limit}. Finally, $\cE_{4+\alpha}(P_n)=\cE_\alpha(Q_n)$ and $c_1(\alpha)=-\alpha c_0(\alpha)/(\alpha+1)$, which gives \cref{eq:pell-upper}. \end{proof} The first two pairs are \[ (A_2,B_2)=(\{0,8,9,17\},\{2,3,14,15\}),\qquad (A_3,B_3)=(\{0,49,50,99\},\{14,15,84,85\}). \] Their quotient degrees are $13$ and $95$. The first quotient is $D_1D_2D_4D_6$; none of the subsequent quotients is a product of four Dirichlet polynomials. \begin{corollary}[Nonproduct members of the Pell family] \label{cor:pell-nonproduct} For $n\ge3$, the polynomial $P_n$ in \cref{thm:pell-upper} is not a product of four Dirichlet polynomials. \end{corollary} \begin{proof} Put $u=u_n$, $w=w_n$ and $a=(u-w-1)/2$. The first negative terms of $Q_n$ occur at $a,a+1$. If $Q_n$ were a product of four binomials, these would force its first lengths to be $a,a+1$. The vanishing coefficient at $2a+1$ then forces the third length to be $2a+1$. The fourth length is determined by the total degree as $d=2w-u$. For $u>17$, the Pell identity gives \[ 2a+12u$ follows from $9w^2-4u^2=(u^2-9)/2>0$, and the middle inequality follows from \[ (5u-1)^2-49w^2=(u-17)(u-3)/2>0. \] The first three factors have their next nonzero term at $3a+1$. Multiplication by $1-z^d$ therefore produces coefficient $-1$ at exponent $d$, where $Q_n$ has coefficient zero. This is impossible. \end{proof} The Pell construction gives the explicit bounds \begin{equation} \label{eq:pell-special} F(9/2)\le48,\qquad \frac{F(17/4)}{2\pi}\le\frac{34}{5}\cC(1/4). \end{equation} It also yields additional information about the shape of $F$. \begin{corollary}[Three strict local maxima] \label{cor:three-maxima} The parameters $2$, $3$ and $4$ are strict local maxima of $F$. \end{corollary} \begin{proof} For $1\le l\le4$ put \[ H_l(\alpha)=4\pi\frac{\alpha+l}{\alpha+1}\cC(\alpha). \] On $0<\alpha<1$, the exact interval formulas and \cref{eq:fourth-upper,eq:pell-upper} give $F(l+\alpha)\le H_l(\alpha)$. For $k=2,3,4$, the integer values imply \[ H_k(0)=H_{k-1}(1)=4\pi k=F(k). \] Since $\cC'(0)=0$ and $\cC'(1)=2$, \[ H_k'(0)=4\pi(1-k)<0,\qquad H_{k-1}'(1)=2\pi(k+2)>0. \] The respective upper bounds are strictly below $F(k)$ in right and left punctured neighborhoods of $k$, which proves the assertion. \end{proof} \subsection{Persistence of arithmetic obstructions} \label{subsec:product-obstructions} Let \[ \mathcal B_m=\left\{\prod_{i=1}^m(1-z^{L_i}): L_1,\ldots,L_m\ge1\right\}. \] Repeated lengths are allowed and the degrees are unrestricted. The coefficient-length results of Maltby \cite{Maltby1997}, recorded explicitly in \cite[p.~39]{Cipu2004}, give \begin{equation} \label{eq:pure-lower} \min_{Q\in\mathcal B_7}\norm{Q}_1=16,\qquad \min_{Q\in\mathcal B_9}\norm{Q}_1=20,\qquad \min_{Q\in\mathcal B_{10}}\norm{Q}_1=24. \end{equation} These are also the minima of $\norm{Q}_2^2$. Integrality gives the lower bounds, and the respective length vectors \[ (1,2,3,4,5,7,11),\quad (1,2,3,4,5,7,9,11,13),\quad (1,2,3,4,5,7,9,11,13,17) \] in \cite[Table~1]{Cipu2004} attain them by direct expansion. Thus pure products have strict squared-norm gaps above $\cM_7=14$, $\cM_9=18$ and $\cM_{10}=20$. \begin{lemma}[Comparison with the coefficient norm] \label{lem:norm-comparison} For $-1/2<\beta<1$ and every real-coefficient polynomial $Q$, \begin{equation} \label{eq:norm-comparison} \left|\frac{\cE_\beta(Q)}{\cC(\beta)}-\norm{Q}_2^2\right| \le\frac{|\beta|}{1+\beta} \bigl(\norm{Q}_1^2-\norm{Q}_2^2\bigr). \end{equation} \end{lemma} \begin{proof} The Fourier recurrence in \cref{lem:fourier} shows that \[ \frac{|c_j(\beta)|}{\cC(\beta)}\le \frac{|\beta|}{1+\beta}\qquad(j\ge1), \] since the successive absolute ratios are $(j-\beta)/(j+\beta+1)<1$ when $\beta\ne0$. At zero all off-diagonal coefficients vanish. Sum the absolute off-diagonal terms of the quadratic form, using $\sum_{i\ne j}|b_i b_j|=\norm{Q}_1^2-\norm{Q}_2^2$. \end{proof} \begin{theorem}[Persistence of a pure-product gap] \label{thm:gap-persistence} Suppose $Q_*=(1-z)^mP_*$ with $P_*\in\cP$, and put $h=\norm{Q_*}_2^2$, $L=\norm{Q_*}_1$. Assume $\norm{Q}_2^2\ge G>h$ for every $Q\in\mathcal B_m$. If \begin{equation} \label{eq:elementary-gap-radius} 0<\delta<\min\left\{\frac12, \frac{G-h}{4^m+L^2-2h}\right\}, \end{equation} then, for $|\lambda-m|\le\delta$ and $k=\floor\lambda$, \begin{equation} \label{eq:strict-separation} \cE_\lambda(P_*)< \inf_{\substack{s\ge0,\ L_1,\ldots,L_k\ge1}} \cE_\lambda\left(z^s\prod_{i=1}^kD_{L_i-1}\right). \end{equation} \end{theorem} \begin{proof} Write $\beta=\lambda-m$ and $r=|\beta|/(1+\beta)$. Every $Q\in\mathcal B_m$ has $\norm{Q}_1\le2^m$. Hence \cref{lem:norm-comparison} gives \[ \frac{\cE_\beta(Q)}{\cC(\beta)}\ge G-r(4^m-G), \qquad \frac{\cE_\beta(Q_*)}{\cC(\beta)}\le h+r(L^2-h). \] The difference is at least $(G-h)-r(4^m+L^2-G-h)$, uniformly in $Q$. It is strictly positive by \cref{eq:elementary-gap-radius} and $r\le\delta/(1-\delta)$. If $\lambda\ge m$, then $k=m$ and \[ (1-z)^m\prod_{i=1}^kD_{L_i-1}=\prod_{i=1}^m(1-z^{L_i}) \in\mathcal B_m. \] If $\lambda0$, the off-diagonal Fourier coefficients are negative. Discard the nonnegative interactions between opposite signs and apply the row-sum bound to each same-sign block. If $\beta<0$, the off-diagonal Fourier coefficients are positive. Discard the nonnegative same-sign interactions. The remaining rectangular matrix has row sums and column sums at most $H_\delta(p)$, and therefore has operator norm at most $H_\delta(p)$. Apply $2\norm{u}_2\norm{v}_2\le\norm{u}_2^2+\norm{v}_2^2$ to the two coefficient vectors. At $\beta=0$ the assertion is immediate. \end{proof} Suppose $G\le2^m$ is an even lower bound for the squared norm on $\mathcal B_m$, and $H_\delta(M)<1$. Define the finite minimum \begin{equation} \label{eq:B-bound} B_{m,G}(\delta)= \min_{\substack{G/2\le s\le M\\s\in\Z}} 2s\bigl(1-H_\delta(s)\bigr). \end{equation} For $s\ge M$, the right side of \cref{eq:sparse-support} increases linearly in $s$. Thus \cref{eq:B-bound} is a uniform lower bound for $\cE_\beta(Q)/\cC(\beta)$ over all $Q\in\mathcal B_m$ and all $|\beta|\le\delta$. For a height-one polynomial $Q_* = \sum_{a\in A}z^a-\sum_{b\in B}z^b$ with disjoint sets $A,B$ satisfying $|A|=|B|=m$, put \begin{equation} \label{eq:U-bound} U_\delta(A,B)=2\max\left\{ \sum_{a\in A,\,b\in B}\rho_\delta(|a-b|),\quad \sum_{\{a,a'\}\subset A}\rho_\delta(|a-a'|) +\sum_{\{b,b'\}\subset B}\rho_\delta(|b-b'|) \right\}. \end{equation} The within-set sums are over unordered two-element subsets. Keeping only the interactions that can increase the energy gives \begin{equation} \label{eq:witness-upper} \frac{\cE_\beta(Q_*)}{\cC(\beta)}\le2m+U_\delta(A,B) \qquad(|\beta|\le\delta). \end{equation} \begin{theorem}[Explicit product exclusion intervals] \label{thm:product-intervals} For $(m,\delta,\eta)$ equal to \[ (7,1/40,1/50),\qquad(9,1/60,1/4),\qquad(10,1/35,1/10), \] let $P_m$ be the monic quotient of the corresponding pair in \cref{eq:large-pte}. For $|\lambda-m|\le\delta$ and $k=\floor\lambda$, \begin{equation} \label{eq:quantitative-product-exclusion} \inf_{\substack{s\ge0,\ L_1,\ldots,L_k\ge1}} \cE_\lambda\left(z^s\prod_{i=1}^kD_{L_i-1}\right) \ge\cE_\lambda(P_m)+\eta\cC(\lambda-m). \end{equation} In particular, this product family cannot attain or approach the full infimum anywhere in the indicated closed intervals. \end{theorem} \begin{proof} Use the squared-norm bounds in \cref{eq:pure-lower}. At the indicated rational values of $\delta$, every term in \cref{eq:rho,eq:B-bound,eq:U-bound} is rational. Direct evaluation gives the strict bounds in \cref{tab:product-intervals}; each row also satisfies $H_\delta(2^{m-1})<1$. \begin{table}[htbp] \centering \caption{Rational bounds used in the product exclusion intervals.} \label{tab:product-intervals} \small \begin{tabular}{@{}ccccl@{}} \toprule $m$&$G$&$\delta$&$B_{m,G}(\delta)>$&$2m+U_\delta(A_m,B_m)<$\\ \midrule 7&16&$1/40$&$14243/1000$&$14222/1000$\\ 9&20&$1/60$&$18418/1000$&$18146/1000$\\ 10&24&$1/35$&$20392/1000$&$20281/1000$\\ \bottomrule \end{tabular} \end{table} The differences between the last two columns exceed $1/50$, $1/4$ and $1/10$, respectively. The finite minima in \cref{eq:B-bound} require $57$, $247$ and $501$ rational comparisons; the program \path{anc/verify_pte.py} performs them using integers and fractions. Combining \cref{eq:sparse-support,eq:witness-upper} gives the uniform gaps for pure $m$-factor products. The transformation in the proof of \cref{thm:gap-persistence} covers both sides of each integer and yields \cref{eq:quantitative-product-exclusion}. \end{proof} \section{Finite-degree structure and exact computation} \label{sec:computation} \subsection{Rational energy functions and algebraic transitions} Dividing the energy by $\cC(\lambda)$ makes every Fourier coefficient rational in $\lambda$. At a fixed degree, positive definiteness bounds the coefficients of all possible minimizers locally uniformly in the parameter. These two facts reduce changes of minimizer to crossings of finitely many rational functions. Define \begin{equation} \label{eq:rational-fourier} r_0(\lambda)=1,\qquad r_j(\lambda)=\frac{c_j(\lambda)}{\cC(\lambda)} =(-1)^j\prod_{h=0}^{j-1}\frac{\lambda-h}{\lambda+h+1} \quad(j\ge1). \end{equation} For $P(z)=\sum_{i=0}^Na_iz^i$, write \begin{equation} \label{eq:rational-energy} R_P(\lambda)=\frac{\cE_\lambda(P)}{\cC(\lambda)} =\sum_{i=0}^Na_i^2+ 2\sum_{j=1}^Nr_j(\lambda)\sum_{i=0}^{N-j}a_i a_{i+j}. \end{equation} This is a rational function with no pole on $(0,\infty)$. Multiplication by a monomial leaves $R_P$ unchanged. We regard two polynomials as equivalent if removing their maximal monomial factors gives the same polynomial. Each such class with a representative of degree at most $N$ has a unique representative of exact degree $N$. \begin{theorem}[Local finiteness and algebraic transitions] \label{thm:rational-phases} Fix $N\ge0$ and a compact interval $K\subset(0,\infty)$. There is a finite collection of monic integer polynomials of exact degree $N$ such that \begin{equation} \label{eq:finite-envelope} \frac{F_N(\lambda)}{2\pi\cC(\lambda)} =\min_P R_P(\lambda)\qquad(\lambda\in K). \end{equation} Two energy functions in this collection are identical if and only if the corresponding coefficient autocorrelations are identical. After identical functions are merged, every crossing of two branches is an algebraic number of degree at most $N$, and there are only finitely many such crossings in $K$. Between them, $F_N$ is real analytic and the set of minimizing polynomials, modulo monomial factors, is constant. \end{theorem} \begin{proof} Let $T_N(\lambda)=(r_{|i-j|}(\lambda))_{0\le i,j\le N}$. Its quadratic form at a nonzero real vector is the positive integral in \cref{eq:energy-definition}, divided by $\cC(\lambda)$. The weight is positive almost everywhere, so $T_N(\lambda)$ is positive definite. Continuity and compactness give a uniform lower bound $\nu_K>0$ for its smallest eigenvalue. The monomial $z^N$ has normalized energy one. Thus every minimizing integer vector satisfies \[ \nu_K\sum_{i=0}^Na_i^2\le1. \] There are only finitely many such vectors with $a_N=1$. This proves existence of the finite-degree minimum and \cref{eq:finite-envelope}. The common denominator \[ \Delta_N(\lambda)=\prod_{h=1}^N(\lambda+h) \] is positive on $K$, and $\Delta_NR_P$ is an integer polynomial of degree at most $N$. Differences between distinct branches therefore have at most $N$ zeros, each algebraic of degree at most $N$. For $q$ distinct branches, their total number of crossings is at most $N\binom q2$. The endpoints of $K$ need not be algebraic. Finally, $r_0,\ldots,r_N$ are linearly independent as rational functions. For $N=0$ this is immediate. For $N\ge1$, only $r_N$ has a pole at $-N$, with nonzero residue. Successively examining $-N,-N+1,\ldots,-1$ proves independence. The formula in \cref{eq:rational-energy} gives the autocorrelation assertion. On an interval without crossings, the ordering of all distinct branches is constant, proving the final statements. \end{proof} \subsection{Exact enumeration of the full coefficient lattice} For rational $\lambda>0$ and $N\ge1$, the normalized energy is a positive definite rational quadratic form in the free integer coefficients. Completing the square and using a triangular decomposition gives finite bounds at each stage of the enumeration. Write $P=z^N+\sum_{i=0}^{N-1}a_i z^i$, and set \[ T=(r_{|i-j|})_{0\le i,ji}L_{ji}(a_j-c_j)\right)^2. \end{equation} \begin{proposition}[Completeness of the enumeration] \label{prop:enumeration} Start with any admissible polynomial and its exact rational energy $B$. At each step of \cref{eq:triangular-search}, enumerate all integers whose contribution does not exceed the remaining bound, and replace $B$ whenever a smaller value is reached. If all branches are exhausted, the final value equals $F_N(\lambda)/(2\pi\cC(\lambda))$. Retaining equality branches also determines all minimizers modulo monomial factors. \end{proposition} \begin{proof} Every diagonal contribution is nonnegative. A branch whose partial sum exceeds the current bound cannot contain an improving point or a tie. Conversely, every vector with value at most the current bound satisfies all partial inequalities and is visited. Positive definiteness makes the initial ellipsoid bounded, so it contains finitely many integer vectors. Bound updates only remove points whose values exceed an already attained value. Complete exhaustion therefore leaves the minimum and all ties. The monomial representation at exact degree $N$ covers every polynomial of degree at most $N$. \end{proof} All branch endpoints are computed without rounding. If a conditional center is $p/q$ in lowest terms and the squared radius is $R\ge0$, then admissible integers satisfy \[ |qa_i-p|\le\operatorname{isqrt}\bigl(\floor{q^2R}\bigr), \] where $\operatorname{isqrt}$ denotes the integer square root. The implementation clears denominators in \cref{eq:triangular-search}, so each pruning decision uses only integer arithmetic. At every leaf it independently reevaluates \cref{eq:rational-energy}. The search imposes no symmetry or sign restriction on the coefficients. The program \path{anc/exact_toeplitz.py} uses the Python standard library. Its output records the rational objective, all minimizing representatives, the recursive node count, the completion status and a hash of the program. A node limit yields only an admissible upper bound. The separate product enumeration visits all nondecreasing tuples $L_1\le\cdots\le L_k$ with $\sum_i(L_i-1)\le N$ and evaluates their sparse binomial products exactly. \subsection{Finite-degree values} \Cref{tab:degree34} gives the quarter-step grid through six at degree bound $34$. The first three intervals follow from the analytic theorems, and the integer rows follow from the displayed ideal pairs. All nine noninteger rows above three were obtained by complete enumeration of the full integer lattice. In those rows a product representative attains the finite-degree minimum. The listed lengths mean $\prod_iD_{L_i-1}$; factors of length one are omitted. The displayed decimals approximate exact values of the form $r\cC(\lambda)$, with $r\in\Q$ recorded in the accompanying data. \begin{table}[htbp] \centering \caption{Exact minima $F_{34}(\lambda)/(2\pi)$ on the quarter-step grid.} \label{tab:degree34} \small \begin{tabular}{@{}crrll@{}} \toprule $\lambda$&$F_{34}(\lambda)/(2\pi)$°ree&lengths&basis\\ \midrule $1/4$&1.0787052024&0&$\varnothing$&Interval theorem\\ $1/2$&1.2732395447&0&$\varnothing$&Interval theorem\\ $3/4$&1.5737874654&0&$\varnothing$&Interval theorem\\ $1$&2.0000000000&0&$\varnothing$&Integer theorem\\ $5/4$&2.1593372006&34&$(35)$&Interval theorem\\ $3/2$&2.5469988852&34&$(35)$&Interval theorem\\ $7/4$&3.1476575393&34&$(35)$&Interval theorem\\ $2$&4.0000000000&1&$(2)$&Integer theorem\\ $9/4$&3.9032487457&33&$(17,18)$&Interval theorem\\ $5/2$&4.2519542939&33&$(17,18)$&Interval theorem\\ $11/4$&4.9479850131&33&$(17,18)$&Interval theorem\\ $3$&6.0000000000&3&$(2,3)$&Integer theorem\\ $13/4$&5.6866081883&31&$(8,9,17)$&Enumeration\\ $7/2$&5.9830183496&31&$(8,9,17)$&Enumeration\\ $15/4$&6.7578167415&31&$(8,9,17)$&Enumeration\\ $4$&8.0000000000&7&$(2,3,5)$&Integer theorem\\ $17/4$&7.8857045648&34&$(5,6,11,16)$&Enumeration\\ $9/2$&8.2295955039&13&$(2,3,5,7)$&Enumeration\\ $19/4$&8.9152179618&13&$(2,3,5,7)$&Enumeration\\ $5$&10.0000000000&13&$(2,3,5,7)$&Integer theorem\\ $21/4$&10.5879471968&30&$(3,4,7,10,11)$&Enumeration\\ $11/2$&10.2373568706&30&$(3,4,7,10,11)$&Enumeration\\ $23/4$&10.7270972270&30&$(3,4,7,10,11)$&Enumeration\\ $6$&12.0000000000&16&$(2,3,4,5,7)$&Integer theorem\\ \bottomrule \end{tabular} \end{table} The larger searches are summarized in \cref{tab:larger-searches}. At $7/2$ and $N=80$, complete enumeration gives a unique minimizer modulo monomial factors, represented by $D_{19}D_{20}D_{40}$ of degree $79$, and \begin{equation} \label{eq:seven-halves-80} \frac{F_{80}(7/2)}{2\pi\cC(7/2)} =\frac{397421143002128}{2488650531906825}. \end{equation} At $9/2$ and $11/2$, the degree-$80$ runs stopped at $10^7$ nodes; their values in the table are upper bounds. These computations do not determine the unrestricted infimum for $\lambda>3$. \begin{table}[htbp] \centering \caption{Larger exact lattice searches. Completed searches determine the minimum; searches stopped at the node limit give upper bounds.} \label{tab:larger-searches} \small \begin{tabular}{@{}ccrrl@{}} \toprule $\lambda$&$N$&energy $\cE_\lambda$&nodes&status\\ \midrule $17/4$&61&7.7625934099&3\,894\,122&Complete\\ $3/2$&80&2.5465761241&1\,398&Complete\\ $5/2$&80&4.2455882227&149\,987&Complete\\ $7/2$&80&5.9487961058&2\,226\,136&Complete\\ $17/4$&80&7.7625934099&34\,239\,539&Complete\\ $9/2$&80&8.2295955039&10\,000\,000&Stopped: upper bound\\ $11/2$&80&9.9081780706&10\,000\,000&Stopped: upper bound\\ \bottomrule \end{tabular} \end{table} \FloatBarrier \subsection{A nonproduct interval at degree bounds 61 through 80} Set \begin{equation} \label{eq:new-nonproduct} \begin{split} Q_*(z)&=1-z^9-z^{10}+z^{31}+z^{34}-z^{55}-z^{56}+z^{65},\\ P_*(z)&=Q_*(z)/(1-z)^4. \end{split} \end{equation} Its positive and negative supports $\{0,31,34,65\}$ and $\{9,10,55,56\}$ are symmetric about $65/2$. The identity $65^2+3^2=47^2+45^2$ gives equal second centered moments; symmetry gives the odd moments. Thus $Q_*$ is divisible by $(1-z)^4$ and $P_*\in\cP$ has degree $61$. The fourth power-sum difference is $1\,108\,800$, so the order of vanishing is exactly four. \begin{theorem}[A nonproduct interval from exact enumeration] \label{thm:nonproduct-region} For every $61\le N\le80$, \begin{equation} \label{eq:nonproduct-minimum} \frac{F_N(17/4)}{2\pi\cC(17/4)} =\frac{3278824635541247815125114181047484955184155424876416} {40639891549624156371693126611609552341028229338209875}. \end{equation} All minimizing polynomials are $z^sP_*$ with $0\le s\le N-61$. There is an $\varepsilon>0$ such that the same complete minimizer classification holds for every $|\lambda-17/4|<\varepsilon$ and every $61\le N\le80$. None of these minimizers is a monomial times a product of four Dirichlet polynomials. \end{theorem} \begin{proof} Complete enumeration using \cref{prop:enumeration} at $N=61$ visits $3\,894\,122$ nodes and returns $P_*$ as its sole representative. At $N=80$ it visits $34\,239\,539$ nodes and returns the same sole representative. The rational value is \cref{eq:nonproduct-minimum}; the records for these runs include the full coefficient vector and report no other minimizer modulo monomial factors. The degree-$80$ computation also proves the assertion at every intermediate degree, since $P_*$ is admissible from degree $61$ onwards. For the open interval assertion, apply \cref{thm:rational-phases} on a compact neighborhood of $17/4$. Only finitely many coefficient vectors can minimize. At $17/4$ the exact-degree-$80$ representative of $P_*$ has strictly smaller energy than every other vector in this finite collection. Continuity preserves these inequalities in a sufficiently small neighborhood. The argument again covers all intermediate degree bounds. Finally, if $Q_*$ were a product of four binomials, its lowest nonzero coefficients would force the first two lengths to be $9,10$. The zero coefficient at exponent $19$ forces the third length to be $19$. The total degree then forces the fourth length to be $27$. That product has coefficient $-1$ at exponent $27$, contrary to $Q_*$. The nonzero constant term also excludes a nontrivial monomial factor in $Q_*$ itself. \end{proof} Numerically, $\cE_{17/4}(P_*)=7.7625934099\ldots$. Enumeration of all $92\,582$ four-factor length tuples with degree bound $80$ gives its minimum at $(11,12,23,34)$, with energy $7.8094477516\ldots$. The difference is $0.0468543417\ldots$; its exact value after division by $\cC(17/4)$ is recorded in \path{anc/results/nonproduct_17_4.json}. The Pell construction in \cref{thm:pell-upper} gives the smaller unrestricted upper bound $F(17/4)/(2\pi)\le7.3351953762\ldots$. Exact rational comparison gives $\cE_{17/4}(P_*)/\cC(1/4)>34/5$. Thus the finite-degree minimum in \cref{thm:nonproduct-region} lies strictly above the unrestricted infimum. \section{Further questions} \label{sec:questions} The next analytic problem is to determine whether the explicit constructions on $(3,4)$ and $(4,5)$ are sharp. Combining \cref{thm:interpolation} with their upper bounds gives \begin{align} 8\pi\cC(\alpha) &\le F(3+\alpha) \le4\pi\frac{\alpha+3}{\alpha+1}\cC(\alpha),\label{eq:next-window-3}\\ 10\pi\cC(\alpha) &\le F(4+\alpha) \le4\pi\frac{\alpha+4}{\alpha+1}\cC(\alpha), \qquad0<\alpha<1.\label{eq:next-window-4} \end{align} The third-interval proof suggests seeking energy gaps under three or four vanishing moment constraints. Such gaps would have to distinguish the possible equal-moment supports, whose arithmetic is richer than the two-binomial factorization in \cref{lem:two-moments}. \begin{question} Does either upper bound in \cref{eq:next-window-3,eq:next-window-4} hold with equality throughout the corresponding parameter interval? What are the possible shapes of minimizing sequences there? \end{question} The Pell family shows that separated clusters of exponents can occur along a sparse arithmetic sequence of degrees. A classification of such limiting configurations may therefore require Diophantine parametrizations in addition to inequalities for the Fourier kernel. At $17/4$, the degree-$61$ example establishes nonproduct minimizers for every degree bound from $61$ to $80$, whereas the Pell family gives a strictly smaller unrestricted upper bound. \begin{question} For which $\lambda>0$ is $F(\lambda)$ attained, and is $F$ continuous on all of $(0,\infty)$? \end{question} The function $F$ is upper semicontinuous, being an infimum of continuous energy functions. The interval formulas settle continuity and characterize attainment on $(0,3]$, while the integer reduction gives attainment at every positive integer. Lower semicontinuity outside the determined region requires control as the polynomial degree grows. \begin{question} Does a multiplicity-free ideal PTE pair of size eleven exist? Equivalently, is $F(11)=44\pi$ or $48\pi$? \end{question} More generally, the arithmetic sequence $\cM_m$ is even, nondecreasing and submultiplicative, with $2m\le\cM_m\le m(m-1)+2$. Determining its growth would sharpen the interpolation bounds for the full real-parameter problem. For each fixed degree, \cref{thm:rational-phases} gives finitely many algebraic transitions on compact parameter intervals. Determining their exact locations is a separate finite problem: once all relevant energy functions are known, their crossings reduce to polynomial root isolation. Quantitative bounds on the size of the relevant coefficient set would make this description effective over whole parameter intervals, and would clarify how the possible minimizers change as the degree bound grows. \subsection*{Computational material} The accompanying \path{anc} directory contains the complete programs and exact data used in this paper. Its \path{README.yaml} specifies the normalization, commands and completion states. The PTE and product-exclusion identities are checked by \path{verify_pte.py}; the Pell identities by \path{verify_pell.py}; the full coefficient-lattice and restricted-product enumerations by \path{exact_toeplitz.py} and \path{exact_products.py}; and the table consistency by \path{verify_results.py}. The critical-point decimals are calculated separately by \path{critical_point.py}. 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