\documentclass[11pt,reqno]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,microtype,booktabs,needspace} \usepackage[margin=1.03in]{geometry} \usepackage[hidelinks]{hyperref} \hypersetup{pdftitle={Davenport constants with growing binary rank}, pdfauthor={Chao Liu}} \allowdisplaybreaks[2] \emergencystretch=1.5em \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\F}{\mathbb F} \newcommand{\Z}{\mathbb Z} \newcommand{\D}{\mathsf D} \newcommand{\Ds}{\mathsf D^*} \newcommand{\dd}{\mathsf d} \newcommand{\supp}{\operatorname{supp}} \newcommand{\Sp}{\operatorname{sp}} \newcommand{\wt}{\operatorname{wt}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\kap}{\kappa} \newcommand{\rk}{\operatorname{rank}} \newcommand{\Id}{\mathbf 1} \newcommand{\Vb}[2]{V_{#1}(#2)} \newcommand{\ceil}[1]{\left\lceil#1\right\rceil} \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} \newcommand{\ind}{\mathbf 1} \title[Davenport constants with growing binary rank] {Davenport constants with growing binary rank} \author{Chao Liu} \address{Department of Mathematics, University of Mississippi} \email{chaoliuac@gmail.com} \date{Working draft, revised October 28, 2026} \subjclass[2020]{11B75, 20K01, 94B05, 68Q25} \keywords{Davenport constant, zero-sum sequence, binary linear code, Fourier sparsity, modular constraint, group program} \begin{document} \begin{abstract} We study the correction to the standard lower bound $\Ds$ for Davenport constants of finite abelian groups with a growing elementary two-component. For every fixed nontrivial abelian $p$-group $H$ of odd order, we prove $\D(H\oplus C_2^r)=\Ds(H\oplus C_2^r) +(\D(H)-1)\log_2r+O_H(1)$, with a uniform version for varying $H$. For $C_3\oplus C_2^r$, we determine exact values on explicit infinite intervals of ranks. More generally, the correction is logarithmic or smaller precisely when the fixed group has at most one odd prime divisor. We show that the two-primary part affects the logarithmic coefficient: for $C_{24}\oplus C_2^r$, the ratio $(\D-\Ds)/\log_2r$ has limsup at least $2.4933231\ldots>2$, whereas for $C_3\oplus C_2^r$ the limit is $2$. These upper limiting coefficients are unbounded even with the odd prime fixed. The proofs use binary codes and finite-field Fourier analysis. \end{abstract} \maketitle \section{Introduction}\label{sec:introduction} For a finite abelian group $G$, the Davenport constant $\D(G)$ is the least positive integer $N$ such that every sequence of $N$ elements of $G$ contains a nonempty zero-sum subsequence. Repeated elements are allowed. If \[ G\cong C_{n_1}\oplus\cdots\oplus C_{n_s}, \qquad 10$ such that, for all sufficiently large $r$, \begin{equation}\label{eq:main-superlog} \D(H\oplus C_2^r)\ge\Ds(H\oplus C_2^r) +c_H\left(\frac{\log r}{\log\log r}\right)^t. \end{equation} \end{enumerate} \end{theorem} The multi-prime lower mechanism is the Barrington--Beigel--Rudich construction and its coding interpretation~\cite{BBR,BGG}. The positive qualitative bound also follows from existing group-program results through the explicit reduction in Section~\ref{sec:programs}. Our direct approach supplies constants and a common framework for the arithmetic and fiber statements. \subsection{Higher two-primary exponents} For comparing logarithmic coefficients, it is convenient to use \[ E_H(r)=\D(H\oplus C_2^r)-\Ds(H\oplus C_2^r), \qquad \underline\lambda(H)=\liminf_{r\to\infty}\frac{E_H(r)}{\log_2r}, \qquad \overline\lambda(H)=\limsup_{r\to\infty}\frac{E_H(r)}{\log_2r}. \] Theorem~\ref{thm:dichotomy} determines both coefficients for odd $p$-groups. Higher two-primary exponents can change them. \begin{theorem}[Two-primary dependence]\label{thm:main-coefficients} For $r\ge512$, \[ \D(C_{24}\oplus C_2^r) \le\Ds(C_{24}\oplus C_2^r)+32\log_2r+3. \] Moreover, \begin{equation}\label{eq:main-C24} \frac{9}{5H_2(1/5)} \le\overline\lambda(C_{24})\le32, \qquad \frac{9}{5H_2(1/5)}=2.4933231062\ldots>2, \end{equation} where $H_2(u)=-u\log_2u-(1-u)\log_2(1-u)$ is binary entropy. For every fixed odd prime $p$ and integer $a\ge1$, \begin{equation}\label{eq:main-unbounded} \overline\lambda(C_{p2^a})\ge\frac1{H_2(2^{-a})} \sim\frac{2^a}{a+\log_2e}\qquad(a\to\infty). \end{equation} Thus these lower-bound coefficients are unbounded with $p$ fixed. \end{theorem} The lower bound in \eqref{eq:main-C24} holds along an explicit sequence of ranks. It does not determine the exact Davenport constant, the optimal coefficient, or the existence of a coefficient limit. The underlying transfer, Theorem~\ref{thm:box}, converts finite zero-sum-free configurations in an abelian two-group into entropy lower bounds with a fixed elementary odd-primary target. Appendix~\ref{sec:nonabelian} gives corresponding logarithmic-order conclusions for both small and large nonabelian Davenport constants. \subsection{Connections with earlier work} Logarithmic lower bounds for large elementary two-components have a history in zero-sum theory. For cyclic odd $H$, the lower coefficient $\D(H)-1$ is already contained in Liu~\cite[Theorem~4.3 and Corollary~4.4]{Liu}. The simplex construction in Section~\ref{sec:uniform} gives a direct form of this non-dispersive block method for arbitrary odd-order abelian label groups. Coding-theoretic approaches also appear in Plagne--Schmid~\cite{PS}. Brakensiek, Gopi, and Guruswami~\cite{BGG} relate modular constraints on binary codes to sparse polynomial and weak-OR representations. We make the exact passage to Davenport constants explicit in Section~\ref{sec:transfer}. The primary algebraic input is Olson's group-ring method. The finite sphere-covering inequality in Corollary~\ref{cor:sphere} also follows directly from Han--Zhang~\cite[Proposition~2.4]{HZ}, by taking the Hall subgroup to be $C_2^r$. We prove the stronger code-level covering-radius constraint and combine it with matching simplex constructions, uniform estimates, and signed Fourier identities for the exact $C_3$ intervals. Section~\ref{subsec:covering} compares direct substitution into the same criterion for $C_{24}\oplus C_2^r$. Idziak, Kawa{\l}ek, Krzaczkowski, and Wei{\ss}~\cite{IKKW} establish Boolean-conjunction lower bounds for programs over certain fixed groups and deduce polynomial lower bounds on nonempty solution densities. Section~\ref{sec:programs} constructs an explicit fixed wreath-product simulation of local maps on affine codes. It places our fixed nilpotent targets in their setting and recovers the qualitative fiber estimates and logarithmic Davenport upper bounds. The quantitative estimates here have independent proofs by Fourier support and primary group algebras. \subsection{Organization} Section~\ref{sec:preliminaries} develops the selection-code and primary representations. Section~\ref{sec:uniform} proves the sharp odd-primary and uniform estimates. Sections~\ref{sec:transfer} and~\ref{sec:C3} give the exact Fourier transfer and the $C_3$ intervals. Sections~\ref{sec:intersection}--\ref{sec:programs} treat mixed-characteristic intersections, local fibers, and group programs. Section~\ref{sec:Davenport} proves the general threshold and the $C_{24}$ upper bound. Section~\ref{sec:entropy} gives the entropy construction, and Section~\ref{sec:questions} discusses its limitations and further questions. The appendices contain the nonabelian transfer and the large-rank $C_3$ localization. AI tools developed by OpenAI were used in this work. \section{Binary codes and primary representations}\label{sec:preliminaries} Unless specified otherwise, logarithms have base two. Write \[ \Vb{b}{L}=\sum_{j=0}^b\binom Lj, \] with $\binom Lj=0$ for $j>L$. We set $\D(1)=1$, so that $\D(1)-1=0$. The algebraic degree of a binary function is the degree of its unique multilinear polynomial over $\F_2$. The zero function may be assigned degree $-\infty$. \subsection{The code correspondence} \begin{lemma}\label{lem:constant-offset} For every fixed finite abelian group $H$, there is an integer $c_H$ such that \[ \Ds(H\oplus C_2^r)=r+c_H \] for all sufficiently large $r$. \end{lemma} \begin{proof} For $r$ sufficiently large, the rank of the two-primary component of $H\oplus C_2^r$ is at least the rank of every odd-primary component. Every nontrivial invariant factor is then even. Adding a further $C_2$ inserts one initial invariant factor equal to two, leaving all the others fixed. It increases $\Ds$ by one. \end{proof} If $h_1,\ldots,h_L\in H$, a weighted sum on a binary code is the map \[ \sigma_h(x)=\sum_{i=1}^L x_i h_i. \] The bits are interpreted as the integers zero and one; this map is usually not a homomorphism from the binary additive group to $H$. \begin{proposition}[Selection kernels and realizations]\label{prop:code} Let $H$ be a finite abelian group. \begin{enumerate} \item If $((h_i,v_i))_{i=1}^L$ is zero-sum-free over $H\oplus C_2^r$, then the binary code \[ V=\{x\in\F_2^L:\textstyle\sum_i x_iv_i=0\} \] has dimension at least $L-r$, and $\sigma_h^{-1}(0)\cap V=\{0\}$. \item Conversely, suppose $V\le\F_2^L$ has dimension $d$ and $\sigma_h^{-1}(0)\cap V=\{0\}$. A full-rank parity-check matrix $B$ of $V$ gives a zero-sum-free sequence $((h_i,B_i))_{i=1}^L$ over $H\oplus C_2^{L-d}$. For every $r\ge L-d$, padding by independent binary basis elements gives a zero-sum-free sequence of length $r+d$ over $H\oplus C_2^r$. \end{enumerate} \end{proposition} \begin{proof} A binary selection vector describes a subsequence, and its two components sum to zero exactly when it belongs to $V$ and its $H$-sum vanishes. This proves both correspondences. In the second part, the appended basis elements use new binary coordinates, so a zero-sum selection cannot contain any of them. \end{proof} For later use, if $T:\F_2^d\to\F_2^R$ is linear, the graph code $\{(x,Tx):x\in\F_2^d\}$ has parity-check matrix $[\,T\mid I_R\,]$. It has length $d+R$, dimension $d$, and parity-check rank exactly $R$. \begin{lemma}[Binary minimum weight]\label{lem:binary-weight} A nonzero binary polynomial of degree at most $e$ in $d$ variables has at least $2^{d-e}$ nonzero values. \end{lemma} \begin{proof} The assertion is immediate for a nonzero constant. Otherwise write $f(x',x_d)=g(x')+x_dh(x')$ with $h\ne0$, after relabeling a variable on which $f$ depends. The degree of $h$ is at most $e-1$. For every $x'$ with $h(x')=1$, the two values of $f$ differ, so at least one is nonzero. Induction gives at least $2^{(d-1)-(e-1)}=2^{d-e}$ such values. \end{proof} \subsection{Fourier support in characteristic different from two} For a field $\mathbb K$ with $\operatorname{char}\mathbb K\ne2$, put $\chi_u(x)=(-1)^{u\cdot x}$ and \[ f(x)=\sum_{u\in\F_2^d}\widehat f(u)\chi_u(x), \qquad \widehat f(u)=2^{-d}\sum_{x\in\F_2^d}f(x)\chi_u(x), \qquad \Sp(f)=|\supp\widehat f|. \] The characters form a basis over $\mathbb K$. Translation preserves sparsity. If $x=Az+c$ is an affine substitution, each character restricts to a scalar multiple of one character of $z$, so sparsity does not increase. Also \begin{equation}\label{eq:sparsity-product} \Sp(fg)\le\Sp(f)\Sp(g), \end{equation} because Fourier support of a product is contained in the sumset of the two supports. \begin{lemma}[Finite-field uncertainty]\label{lem:uncertainty} For every nonzero $f:\F_2^d\to\mathbb K$, \[ |\supp f|\,\Sp(f)\ge2^d. \] \end{lemma} \begin{proof} Let $\mathcal T_f$ be the linear span of all translates of $f$. Fourier diagonalization shows that its dimension is $\Sp(f)$: character averaging extracts each frequency with nonzero coefficient. Choose a basis of translates. Their supports cover $\F_2^d$, since otherwise every translate of $f$ would vanish at a common point, forcing $f=0$. Each basis function has support size $|\supp f|$. The union bound proves the claim. No absolute values or positivity are used. \end{proof} A multilinear polynomial of degree at most $b$ in $L$ binary coordinates has Fourier support at most $\Vb{b}{L}$. Indeed $x_i=(1-\chi_{e_i})/2$, so only characters supported on at most $b$ of the original coordinates occur. The same bound holds after affine restriction. \subsection{Primary group algebras} For a finite $p$-group $R$, write $I_R$ for the augmentation ideal of $\F_p[R]$ and put \[ \kap(R)=\max\{j:I_R^j\ne0\},\qquad \kap(1)=0. \] The ideal is nilpotent. One elementary proof is induction on $|R|$: choose a central element $z$ of order $p$. The kernel of $\F_p[R]\to\F_p[R/\langle z\rangle]$ is the central ideal generated by $z-1$, whose $p$th power vanishes. Nilpotence in the quotient therefore gives nilpotence in $\F_p[R]$. Jennings' theorem provides the exact general index~\cite{Jennings}. If $R=\bigoplus_i C_{p^{a_i}}$ is abelian, the truncated-polynomial description gives \begin{equation}\label{eq:abelian-kappa} \F_p[R]\cong \F_p[T_1,\ldots,T_s]/(T_1^{p^{a_1}},\ldots,T_s^{p^{a_s}}), \qquad \kap(R)=\sum_i(p^{a_i}-1)=\D(R)-1. \end{equation} Let $g_1,\ldots,g_T$ be group-valued functions. In the relevant primary group algebra, \begin{equation}\label{eq:augmentation} \prod_{j=1}^T[g_j(x)] =\prod_{j=1}^T\bigl(1+([g_j(x)]-1)\bigr). \end{equation} Every term with more than $\kap(R)$ augmentation factors is zero. The expansion preserves the original order, including for nonabelian groups. Taking the coefficient of $[h]$ gives the exact indicator that the group product equals $h$. \section{Sharp estimates for odd-primary label groups}\label{sec:uniform} \subsection{A covering-radius constraint} For a binary linear code $C\subseteq\F_2^L$, its covering radius is \[ \rho(C)=\max_{w\in\F_2^L}\min_{c\in C}d_H(w,c), \] where $d_H$ is Hamming distance. We use the augmentation-ideal identity \eqref{eq:abelian-kappa}. In group-algebra formulas below, $X^h$ denotes the basis element $[h]$. \begin{theorem}[Covering transfer]\label{thm:cover} Let $H$ be a finite abelian $p$-group with $p$ odd, and put $b=\D(H)-1$. For a sequence \[ S=((h_i,v_i))_{i=1}^L\quad\text{over }H\oplus\F_2^r, \] define its binary character code by \[ C(S)=\{(\langle x,v_1\rangle,\ldots,\langle x,v_L\rangle): x\in\F_2^r\}\subseteq\F_2^L. \] If $S$ is zero-sum free, then \begin{equation}\label{eq:radius} \rho(C(S))\le b. \end{equation} \end{theorem} \begin{proof} Let $I$ be the augmentation ideal of $K[H]$, where $K=\F_p$. Fourier decomposition on the elementary two-component gives an injective algebra map \[ K[H\oplus\F_2^r]\longrightarrow\prod_{x\in\F_2^r}K[H],\qquad X^{(h,v)}\longmapsto\bigl((-1)^{\langle x,v\rangle}X^h\bigr)_x. \] Indeed, the character matrix on $\F_2^r$ is invertible because $2$ is invertible in $K$. Suppose that a vector $w\in\F_2^L$ has distance at least $b+1$ from every codeword. Set $\varepsilon_i=(-1)^{w_i+1}$ and consider \[ F=\prod_{i=1}^L(1-\varepsilon_iX^{(h_i,v_i)}). \] In the component indexed by $x$, the factor at position $i$ belongs to $I$ whenever $w_i\ne\langle x,v_i\rangle$, since that factor is then $1-X^{h_i}$. At least $b+1$ factors belong to $I$, so every component vanishes by \eqref{eq:abelian-kappa}. Hence $F=0$. On the other hand, if $S$ is zero-sum free, only the empty selection contributes to the coefficient of $X^0$ in $F$. That coefficient is $1$, which is impossible. \end{proof} \begin{corollary}[Finite sphere-covering bound]\label{cor:sphere} Every zero-sum-free sequence of length $L$ over $H\oplus C_2^r$ satisfies \begin{equation}\label{eq:sphere} 2^{L-r}\le \Vb{b}{L}. \end{equation} Consequently, \begin{equation}\label{eq:finite-upper} \D(H\oplus C_2^r) \le \min\{L\ge1: \Vb{b}{L}<2^{L-r}\}. \end{equation} \end{corollary} \begin{proof} Theorem~\ref{thm:cover} proves \eqref{eq:sphere} by covering $\F_2^L$ with at most $2^r$ balls of size $\Vb{b}{L}$. Here is a direct algebraic verification as well. Let $A:\F_2^L\to\F_2^r$ have columns $v_i$ and put $d=\dim\ker A\ge L-r$. Parameterize its kernel linearly by $x\in\F_2^d$, with coordinates $\ell_i(x)=\langle t_i,x\rangle\in\{0,1\}$. For $\chi_t(x)=(-1)^{\langle t,x\rangle}$, one has $\ell_i=(1-\chi_{t_i})/2$ in $\F_p$. Write $[X^0]$ for the coefficient at $X^0$ in $\F_p[H]$. Zero-sum freeness gives the identity \[ \delta_0(x)=[X^0]\prod_{i=1}^L \bigl(1+\ell_i(x)(X^{h_i}-1)\bigr). \] Indeed, the product is $X^{\sum_i\ell_i(x)h_i}$, and its coefficient at $X^0$ is one precisely when $x=0$. Expand the product. Every term involving more than $b$ factors $X^{h_i}-1$ vanishes by $I^{b+1}=0$. Since $\ell_i=(1-\chi_{t_i})/2$, the Fourier support of the remaining function is contained in \[ \left\{\sum_{i\in J}t_i:J\subseteq\{1,\ldots,L\},\ |J|\le b\right\}, \] which has at most $\Vb{b}{L}$ elements. On the other hand, every Fourier coefficient of $\delta_0$ equals $2^{-d}\ne0$ in $\F_p$. Consequently $2^{L-r}\le2^d\le \Vb{b}{L}$, as required. For fixed $r,b$, the exponential term eventually exceeds the polynomial $\Vb{b}{L}$, so the minimum in \eqref{eq:finite-upper} exists. \end{proof} More precise lower bounds on the size of a covering code can be inserted in the same way. The transfer uses the covering radius of the particular code $C(S)$, and it allows zero columns and repeated columns in the projection matrix. Standard covering-code results, including those of Graham and Sloane~\cite{GS}, are therefore available after their individual hypotheses have been checked. For later comparison with the standard lower bound, write an odd-order group as $H\cong\bigoplus_{j=1}^s C_{m_j}$ in invariant-factor form. Pairing its cyclic factors with $s$ binary factors gives, for $r\ge s$, \begin{equation}\label{eq:Dstar-general} \Ds(H\oplus C_2^r)=r+1+2\sum_{j=1}^s(m_j-1). \end{equation} \subsection{Simplex constructions} The next construction uses only that multiplication by a power of two is an automorphism of $H$. \begin{proposition}[Simplex blocks]\label{prop:simplex} Let $H$ be a nontrivial finite abelian group of odd order, and put $b=\D(H)-1$. For any positive integers $m_1,\ldots,m_b$, set \[ r_0=\sum_{j=1}^b(2^{m_j}-1-m_j). \] For every $r\ge r_0$, \begin{equation}\label{eq:simplex-general} \D(H\oplus C_2^r)\ge r+1+\sum_{j=1}^b m_j. \end{equation} In particular, if $r\ge b(2^m-1-m)$, then \begin{equation}\label{eq:simplex-blocks} \D(H\oplus C_2^r)\ge r+bm+1. \end{equation} \end{proposition} \begin{proof} Choose a zero-sum-free sequence $h_1,\ldots,h_b$ in $H$. Let $\mathcal S_m$ be the binary simplex code whose generator matrix has as columns all nonzero vectors of $\F_2^m$. Its length is $2^m-1$, its dimension is $m$, and each nonzero word has weight $2^{m-1}$. Take the direct sum $\mathcal K=\bigoplus_{j=1}^b\mathcal S_{m_j}$, of length $N=\sum_j(2^{m_j}-1)$ and dimension $\sum_j m_j$. Choose a quotient map $A:\F_2^N\to\F_2^{r_0}$ with kernel $\mathcal K$. For every position in block $j$, use the $H$-label $2^{-(m_j-1)}h_j$, where the inverse power denotes the inverse automorphism of $H$. Pair these labels with the columns of $A$. A selection whose binary sum is zero has its indicator vector in $\mathcal K$. Each nonzero block contributes exactly $h_j$ to its $H$-sum. If the selection is nonempty, this is a nonempty subsequence sum of $h_1,\ldots,h_b$, hence is nonzero. Thus the resulting sequence is zero-sum free. Appending $r-r_0$ new independent binary basis vectors, with zero $H$-labels, preserves this property. Its total length is $N+r-r_0=r+\sum_j m_j$. \end{proof} For $r\ge b$, the choice $m=\lfloor\log_2(1+r/b)\rfloor$ in \eqref{eq:simplex-blocks} gives \begin{equation}\label{eq:simplex-rounded} \D(H\oplus C_2^r) \ge r+b\floor{\log_2(1+r/b)}+1 \ge r+b\log_2(r/b)-b+1. \end{equation} \subsection{Uniform estimates} \begin{theorem}[Uniform second-order estimate]\label{thm:uniform} Let $H$ be any nontrivial finite abelian $p$-group, where $p$ is odd, and put $b=\D(H)-1$ and $G=H\oplus C_2^r$. If $r\ge16b$, then \begin{equation}\label{eq:uniform} \D(G)=\Ds(G)+b\log_2(r/b)+\rho_H(r),\qquad -3b\le \rho_H(r)<0. \end{equation} More precisely, \begin{equation}\label{eq:uniform-rounded} \D(G)\le\Ds(G)+\ceil{b\log_2(r/b)}-1. \end{equation} Consequently, for any sequence of such groups $H_r$, allowing their primes, ranks, and cyclic orders to vary, if $b_r=\D(H_r)-1=o(r)$, then \begin{equation}\label{eq:movingH} \D(H_r\oplus C_2^r) =\Ds(H_r\oplus C_2^r)+(1+o(1))b_r\log_2(r/b_r). \end{equation} Moreover, $\D(H_r\oplus C_2^r)\sim\Ds(H_r\oplus C_2^r)\sim r$. \end{theorem} \begin{proof}[Proof of Theorem~\ref{thm:uniform}] Write $H=\bigoplus_{j=1}^s C_{p^{e_j}}$. By \eqref{eq:abelian-kappa}, $b=\sum_j(p^{e_j}-1)\ge2$. Since $s\le b/2\le r$, \eqref{eq:Dstar-general} becomes \begin{equation}\label{eq:Dstar-p} \Ds(H\oplus C_2^r)=r+2b+1. \end{equation} Subtracting this from \eqref{eq:simplex-rounded} proves the lower bound in \eqref{eq:uniform}. For the upper bound, put $u=r/b\ge16$ and \[ L=\ceil{r+b\log_2u+2b}. \] The inequality $b\ge2$ gives \[ \frac Lr\le1+\frac{\log_2u+5/2}{u}\le\frac{45}{32}. \] For the last inequality, $(\log_2u+5/2)/u$ is decreasing for $u\ge16$ and equals $13/32$ at $u=16$. For $1\le b\le L$, setting $z=b/L\le1$ yields \[ z^b\Vb{b}{L}\le(1+z)^L\le e^b, \qquad \Vb{b}{L}\le(eL/b)^b. \] Using $e<11/4$, we obtain \[ \Vb{b}{L} <\left(\frac{495}{128}\,\frac rb\right)^b <(4r/b)^b\le2^{L-r}. \] Corollary~\ref{cor:sphere} gives $\D(H\oplus C_2^r)\le L$. The rounding is exact: \[ L=r+2b+\ceil{b\log_2(r/b)} =\Ds(G)+\ceil{b\log_2(r/b)}-1. \] This proves \eqref{eq:uniform-rounded}. Since $\lceil y\rceil-12(q-1)$ be the unique solution of \begin{equation}\label{eq:entropy-root} \lambda_q\left[1-H_2\!\left(\frac{q-1}{\lambda_q}\right)\right]=1, \end{equation} where $H_2(u)=-u\log_2u-(1-u)\log_2(1-u)$. For every integer $r\ge1$, \begin{equation}\label{eq:uniform-ratio} \D(C_{2q}^r)\le\floor{\lambda_q r}+1 <1.4162\,\Ds(C_{2q}^r). \end{equation} \end{corollary} \begin{proof} Put $b=r(q-1)$ and $\Phi_q(x)=x[1-H_2((q-1)/x)]$. On $x>2(q-1)$, the derivative $\Phi_q'(x)=\log_2(2(1-(q-1)/x))$ is positive, so $\Phi_q$ increases from zero to infinity. This proves existence and uniqueness in \eqref{eq:entropy-root}. Take $L=\floor{\lambda_qr}+1>2b$ and $z=b/(L-b)$. Then \[ \Vb{b}{L}\le z^{-b}(1+z)^L=2^{L H_2(b/L)}<2^{L-r}, \] where the strict inequality follows from $L[1-H_2(b/L)]=r\Phi_q(L/r)>r$. Corollary~\ref{cor:sphere}, with $H=C_q^r$, proves the first bound in \eqref{eq:uniform-ratio}. For the numerical constant, put $c=7081/5000=1.4162$ and $a=(q-1)/(2q-1)\in[2/5,1/2)$. Define $F_c(a)=c[1-H_2(a/c)]-(1-2a)$. Its derivative is $F_c'(a)=2-\log_2((c-a)/a)>0$, because $c<5a$. The endpoint inequality $F_c(2/5)>0$ is equivalent to the exact integer comparison \[ 2^{14081}5^{6000}5081^{5081}>7081^{7081}. \] Thus $\Phi_q(c(2q-1))>1$ and $\lambda_q1$, we obtain $\floor{\lambda_qr}+1\le\lambda_qr+1\frac4{\sqrt3}\,2^s. \end{equation} \end{corollary} \begin{proof} Write $n=\nu(2s+1)$. Equation~\eqref{eq:pair-even} gives $n\ge5$. If $n\le(4/\sqrt3)2^s$, then \eqref{eq:pair-odd} would imply \[ 2^{2s+1}\le1+\frac{3n^2-2n}{8} \le2^{2s+1}+1-\frac n4<2^{2s+1}, \] a contradiction. \end{proof} If a zero-sum-free sequence over $C_3\oplus C_2^r$ has length $L$ and $L-r\ge d\ge1$, choose a $d$-dimensional subspace of the binary projection kernel. Its label-sum function is admissible, as in Theorem~\ref{thm:exact-transfer}, and its full Fourier support has size at most $L+1$. We will use this even when the full kernel is larger than $d$. \subsection{Proof of the explicit intervals} \begin{proof}[Proof of Theorem~\ref{thm:windows}] For the lower bound, take two simplex blocks of dimension $m$ in Proposition~\ref{prop:simplex}. Their binary quotient dimension is $a_m=2(2^m-1-m)$. Hence for every $r\ge a_m$, \[ \D(C_3\oplus C_2^r)\ge r+2m+1. \] Suppose $m\ge2$ and a zero-sum-free sequence of length $L=r+2m+1$ exists with $r\le z_m$. Put $T=2^m$. Restrict its label-sum function to a $(2m+1)$-dimensional kernel subspace. It has Fourier support size $n\le L+1\le4T/\sqrt3$, contradicting Corollary~\ref{cor:odd-support}. This proves the upper bound. The case $m=1$ is $r=0$ and follows from $\D(C_3)=3$. \end{proof} The isolated values \eqref{eq:special-ranks} also have a short proof using only the sphere volume. At $L=2^{m+1}-1$, \[ \Vb{2}{L}=2^{2m+1}-2^m+1<2^{2m+1}=2^{L-a_m}. \] This gives the upper bound directly from Corollary~\ref{cor:sphere}. \subsection{An integer refinement} For odd $d\ge1$, let $\lambda_d$ be the least positive integer $n$ satisfying \begin{equation}\label{eq:lambda} n\equiv2\pmod3,\qquad 3n^2-2n-9\ind_{\{n\text{ odd}\}}\ge8(2^d-1). \end{equation} This is an explicit arithmetic bound, not an unknown Fourier extremum. For instance, set \[ R_e(d)=\frac{1+\sqrt{24\cdot2^d-23}}3,\qquad R_o(d)=\frac{1+\sqrt{24\cdot2^d+4}}3. \] Then \begin{equation}\label{eq:lambda-closed} \lambda_d=\min\left\{ 2+6\ceil{\frac{R_e(d)-2}{6}},\, 5+6\ceil{\frac{R_o(d)-5}{6}} \right\}. \end{equation} By \eqref{eq:residue}, when $n$ is odd the integer $P-Q$ is odd and divisible by three, so $(P-Q)^2\ge9$. Lemma~\ref{lem:signed-pairs} therefore proves $\nu(d)\ge\lambda_d$. \begin{proposition}[Refined exact intervals]\label{prop:refined} Put \[ \ell_m=2^{m+1}-2m-3+\ind_{\{m\text{ odd}\}},\qquad u_m=\lambda_{2m+1}-2m-3. \] For every $m\ge1$ and every integer $\ell_m\le r\le u_m$, \begin{equation}\label{eq:refined-window} \D(C_3\oplus C_2^r)=r+2m+1. \end{equation} \end{proposition} \begin{proof} On $\F_2^m\oplus\F_2^m$, consider \[ f(x,y)=(1-\delta_0(x))+(1-\delta_0(y))\quad\text{in }\F_3. \] Its only zero is $(0,0)$: elsewhere its value is one or two. There are $2(2^m-1)$ nonconstant Fourier coefficients. The constant coefficient is $2-2^{1-m}$, which is zero for even $m$ and nonzero for odd $m$. Thus the full support has size \[ B_{2m}=2^{m+1}-2+\ind_{\{m\text{ odd}\}}. \] Translate this support by a character so it contains a nonzero constant coefficient. Equations~\eqref{eq:mu-nu} and \eqref{eq:C3transfer} then give the lower bound for $r\ge B_{2m}-1-2m=\ell_m$. For the upper bound, a zero-sum-free sequence of length $r+2m+1$ would yield, on a $(2m+1)$-dimensional kernel subspace, an admissible function of support size at most \[ r+2m+2\le\lambda_{2m+1}-1. \] This contradicts $\nu(2m+1)\ge\lambda_{2m+1}$. \end{proof} \section{A mixed-characteristic intersection theorem}\label{sec:intersection} We first extract a linear space from the support of a Fourier-sparse function. The dimension required for this extraction will depend only on the degree of the binary condition. \begin{lemma}[A flat in a sparse support]\label{lem:flat} Suppose $b:\F_2^d\to\mathbb K$ satisfies $b(0)\ne0$ and $\Sp(b)\le S$. If an integer $t\ge0$ satisfies \begin{equation}\label{eq:flat-condition} d>t+2^t\log S, \end{equation} then $\supp b$ contains a linear subspace of dimension $t+1$. \end{lemma} \begin{proof} Start from $U=\{0\}$. Suppose a $j$-dimensional subspace $U\subseteq\supp b$ has been constructed. Define \[ H_U(x)=\prod_{u\in U}b(x+u). \] It is nonzero at zero, and \eqref{eq:sparsity-product} gives $\Sp(H_U)\le S^{2^j}$. Hence \[ |\supp H_U|\ge 2^dS^{-2^j} \] by Lemma~\ref{lem:uncertainty}. If $d>j+2^j\log S$, this support is larger than $|U|$. Choose $v\in\supp H_U\setminus U$. Then $v+U\subseteq\supp b$, and $U\cup(v+U)$ is a $(j+1)$-dimensional linear subspace in $\supp b$. The function $j+2^j\log S$ is nondecreasing, so \eqref{eq:flat-condition} permits all the required extensions. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:main-intersection}] First suppose $Z$ is a singleton. Translate its point to zero. If \eqref{eq:main-isolation} failed, Lemma~\ref{lem:flat} with $t=D$ would give a $(D+1)$-dimensional subspace $U\subseteq\supp b$. On $U$, the binary function $a$ is one at zero and zero elsewhere. In basis coordinates it is $\prod_{i=1}^{D+1}(1+x_i)$, whose degree is $D+1$. Affine substitution cannot increase degree, a contradiction. For general nonempty $Z$, translate a point of $Z$ to zero and choose an inclusion-maximal linear subspace $W$ with $W\cap Z=\{0\}$. Put $k=\dim W$. The singleton bound on $W$ gives $k\le D+2^D\log S$. Every coset of $W$ meets $Z$: a missing coset $v+W$ would make $W\cup(v+W)$ a larger isolating subspace. Therefore \[ |Z|\ge2^{d-k}\ge2^{d-D}S^{-2^D}. \] All ingredients are stable under affine restriction, as required. \end{proof} \begin{remark}\label{rem:intersection} The theorem is not obtained by multiplying separate lower bounds for $\{a=1\}$ and $\{b\ne0\}$; two sets of large density can have a very small intersection. The argument instead uses the algebraic degree of one condition and the translation structure of the other. Section~\ref{subsec:obstruction} shows that the coefficient of $\log S$ in the singleton bound cannot universally be replaced by one, even when $D=3$ and $b$ takes only the values zero and one. \end{remark} \section{Fibers of local group-valued products}\label{sec:fibers} Let $V\subseteq\F_2^L$ be an affine code of dimension $d$. Fix scopes $S_j\subseteq[L]$, with $|S_j|\le\beta$, and maps $g_j:\F_2^{S_j}\to K$. We consider the ordered product \begin{equation}\label{eq:local-word} F(x)=g_1(x|_{S_1})\cdots g_T(x|_{S_T}),\qquad x\in V. \end{equation} The arity $\beta\ge1$ is fixed; the scopes may overlap arbitrarily. The order in \eqref{eq:local-word} is part of the data. \begin{theorem}[Explicit local fibers]\label{thm:local} Let $Q$ be a finite two-group and $P$ a finite odd $p$-group, and take $K=Q\times P$. Define \[ D_0=\beta\kap(Q),\qquad W=\sum_{j=0}^{\kap(P)}2^{\beta j}\binom Tj. \] Every attainable $h\in K$ satisfies \begin{equation}\label{eq:local-density} \frac{|F^{-1}(h)\cap V|}{|V|}\ge2^{-D_0}W^{-2^{D_0}}. \end{equation} If a fiber is a singleton, then \begin{equation}\label{eq:local-isolation} d\le D_0+2^{D_0}\log W. \end{equation} \end{theorem} \begin{proof} Write $h=(h_Q,h_P)$ and parametrize $V$ by $\F_2^d$. First work in $\F_2[Q]$. Before restriction to $V$, each local augmentation-valued function $[g_j^Q(x)]-1$ has a multilinear representation of degree at most $\beta$ with coefficients in $I_Q$. By \eqref{eq:augmentation}, terms with more than $\kap(Q)$ such factors vanish. Thus the coefficient of $[h_Q]$ in the ordered product is a binary polynomial $a$ of degree at most $D_0$. It is the exact $Q$-target indicator. The degree bound persists after the affine parametrization over $\F_2$. Now work in $\F_p[P]$. A surviving term in the augmentation expansion has its factors indexed by $J\subseteq[T]$ with $|J|\le\kap(P)$. It depends on at most $\beta|J|$ original binary coordinates. On $V$, those coordinates are affine linear forms, so any scalar coefficient of this term has Fourier support in a space of at most $2^{\beta|J|}$ characters. Summing over $J$ shows that the exact $P$-target indicator $b$ has $\Sp(b)\le W$. The desired fiber is precisely $\{a=1,b\ne0\}$. Apply Theorem~\ref{thm:main-intersection}. \end{proof} \begin{corollary}[Search consequences]\label{cor:algorithms} For fixed $Q,P,\beta$, existence of a target preimage in \eqref{eq:local-word} can be tested in randomized polynomial time with one-sided error. A preimage can be found with the same error guarantee. Deterministic search and emptiness certification take time $\exp(O_{Q,P,\beta}((\log(T+2))^2))$, in addition to polynomial input-reading and linear-algebra costs. \end{corollary} \begin{proof} Project $V$ onto the union of the scopes. Its dimension is at most $\beta T$, and projection has equal-size fibers, preserving the density of every target fiber. Uniform sampling succeeds with inverse polynomial probability by \eqref{eq:local-density}. Every sampled candidate is checked directly; in an empty fiber the algorithm never outputs a witness. For deterministic search, identify the projected code with $\F_2^{d_0}$. Choose a solution $z$ of minimum Hamming weight $k$. The coordinate subspace on its support contains no other solution: every other point has fewer than $k$ nonzero coordinates. Applying \eqref{eq:local-isolation} to that subspace gives $k=O_{Q,P,\beta}(\log(T+2))$. Enumerating all vectors up to this weight gives the asserted running time, including when no solution exists. A projected witness can be lifted to $V$ by linear algebra. \end{proof} \subsection{Sharper estimates for weighted abelian sums} Let $Q$ be abelian of two-power order and $P$ abelian of odd prime-power order, allowing either group to be trivial. Set $b_Q=\D(Q)-1$ and $b_P=\D(P)-1$. \begin{corollary}[Weighted fibers]\label{cor:weighted} For labels $h_i\in Q\oplus P$ on an affine binary code $V$ of dimension $d$ and length $L$, every attainable weighted sum has at least \begin{equation}\label{eq:weighted-density} 2^{d-b_Q}\Vb{b_P}{L}^{-2^{b_Q}} \end{equation} preimages. In particular, a singleton fiber implies \begin{equation}\label{eq:weighted-isolation} d\le b_Q+2^{b_Q}\log\Vb{b_P}{L}. \end{equation} \end{corollary} \begin{proof} The $Q$ indicator has binary degree at most $b_Q$. For the odd part, expand \[ \prod_{i=1}^L\bigl(1+x_i([h_i^P]-1)\bigr) \quad\text{in }\F_p[P]. \] Its target coefficient is a multilinear polynomial of degree at most $b_P$ in the original bits. It therefore has Fourier support at most $\Vb{b_P}{L}$, also after affine restriction. Theorem~\ref{thm:main-intersection} gives the claim. \end{proof} When $Q=1$, the singleton bound is just $d\le\log\Vb{b_P}{L}$, and the density bound also follows directly from Lemma~\ref{lem:uncertainty}. The mixed intersection argument is what controls a simultaneous nontrivial two-primary condition. \section{A connection with programs over fixed groups}\label{sec:programs} This section gives a second, qualitative route to the preceding fiber bounds. A program over a fixed finite group $G$ is an ordered list of instructions, each of which chooses one of two group elements according to one input bit. An accepting subset of $G$ specifies the Boolean output. Constants can be inserted as instructions with equal choices. Let $\gamma_G(n)$ denote the minimum program length representing an $n$-variable singleton indicator, with value infinity if no such program exists. We use the following consequence of existing results. By \cite[Theorem 17]{IKKW}, if \begin{equation}\label{eq:IKKW-class} G\le N\times(R\rtimes A), \end{equation} where $N$ is finite nilpotent, $R$ is a finite $p$-group, and $A$ is finite abelian, then $\gamma_G(n)\ge2^{c_G n}$ for all sufficiently large $n$, with $c_G>0$. Their Proposition~9 converts this into an inverse-polynomial lower bound for every nonempty accepting set of a program of length $m$. \begin{proposition}[Fixed wreath-product simulation]\label{prop:wreath} Fix $K=Q\times P$ and $\beta$ as in Theorem~\ref{thm:local}. There is a fixed finite group $W$ satisfying \eqref{eq:IKKW-class} such that, after parametrizing the relevant projection of $V$, each target condition $F(x)=h$ in \eqref{eq:local-word} is recognized by a $W$-program of length $O_\beta(T^2)$. The simulation preserves the fraction of satisfying inputs. \end{proposition} \begin{proof} Let $A=\F_2^\beta$ and use the regular translation action on $K^A$. Write \[ W=K^A\rtimes A,\qquad (f,u)(g,v)=(f\,\tau_u g,u+v),\qquad (\tau_u g)(a)=g(a+u). \] Multiplication in $K^A$ is pointwise and ordered. Project the code onto the coordinates that occur in a scope, and parametrize it by $z\in\F_2^{d_0}$, where $d_0\le\beta T$. Each local scope is an affine map $y_j(z)\in A$; append unused coordinates when its size is smaller than $\beta$. Regard the local table as $f_j\in K^A$. The cursor $(1,y_j(z))$ can be computed with at most $d_0$ bit-dependent translation instructions and one constant instruction. Consequently \begin{equation}\label{eq:conjugate} (1,y_j(z))(f_j,0)(1,y_j(z))^{-1} =(\tau_{y_j(z)}f_j,0) \end{equation} is a program of length at most $2d_0+3$. At coordinate $a=0$, the right-hand side has value $g_j(y_j(z))$. The product of these conjugates has cursor zero, and its base coordinate at zero is exactly $F(x)$. Take the accepting subset \[ \{(f,0)\in W:f(0)=h\}. \] This gives length at most $T(2d_0+3)$ and the desired accepting condition. Finally the diagonal cursor map embeds $W$ into \begin{equation}\label{eq:wreath-embedding} (Q^A\rtimes A)\times(P^A\rtimes A). \end{equation} The first factor is a two-group and hence nilpotent; the second is an odd $p$-group extended by the abelian group $A$. Thus \eqref{eq:IKKW-class} applies. Uniform binary parameters give uniform points of the projected affine code, and projection has equal-size fibers, so all accepting fractions are preserved. \end{proof} \begin{corollary}[Qualitative density from group programs]\label{cor:IKKW-density} For every fixed $Q,P,\beta$ there is $C=C(Q,P,\beta)$ such that every nonempty fiber in Theorem~\ref{thm:local} has density at least $(T+2)^{-C}$. \end{corollary} \begin{proof} Apply the program density consequence above to Proposition~\ref{prop:wreath}. Its length is polynomial in $T$, and its group is fixed. The resulting density is the original fiber density. \end{proof} The explicit estimate \eqref{eq:local-density} and this reduction serve different quantitative purposes. The former gives an exponent in terms of primary augmentation indices and applies a field-uniform intersection theorem. The latter explains why the qualitative phenomenon is compatible with fixed-group satisfiability theory. Thus the polynomial-density and search consequences can be obtained through this fixed-group connection, while the direct Fourier argument also provides the explicit estimates needed below. \section{Logarithmic Davenport corrections}\label{sec:Davenport} \subsection{One odd prime: explicit upper bounds} Let $H=Q\oplus P$, where $Q$ is a fixed abelian two-group and $P$ a fixed abelian odd $p$-group. Put \[ b_Q=\D(Q)-1,\qquad b_P=\D(P)-1, \qquad A_H(L)=b_Q+2^{b_Q}\log\Vb{b_P}{L}. \] \begin{proposition}[A finite-rank bound]\label{prop:D-upper} If $r\ge1$ and $\lfloor A_H(2r)\rfloor\le r-1$, then \begin{equation}\label{eq:D-upper-finite} \D(H\oplus C_2^r)\le r+1+\lfloor A_H(2r)\rfloor. \end{equation} Consequently \begin{equation}\label{eq:D-upper-asymptotic} \D(H\oplus C_2^r) \le r+2^{b_Q}b_P\log r+O_H(1). \end{equation} \end{proposition} \begin{proof} If a zero-sum-free sequence has length $L$, its selection kernel in Proposition~\ref{prop:code} has dimension at least $L-r$. The zero fiber of its $H$-sum is a singleton. Thus \begin{equation}\label{eq:length-obstruction} L-r\le A_H(L) \end{equation} by Corollary~\ref{cor:weighted}. Take $L=r+1+\lfloor A_H(2r)\rfloor\le2r$. Then $L-r=1+\lfloor A_H(2r)\rfloor>A_H(2r)\ge A_H(L)$, contradicting \eqref{eq:length-obstruction}. This proves \eqref{eq:D-upper-finite}. For fixed $b_P>0$, $\log\Vb{b_P}{2r}=b_P\log r+O_H(1)$, so the finite condition holds for all sufficiently large $r$ and yields \eqref{eq:D-upper-asymptotic}. If $b_P=0$, the same conclusion holds with a bounded correction, and Olson's formula gives the exact value $\D(H\oplus C_2^r)=r+\D(H)$. \end{proof} \begin{remark} Corollary~\ref{cor:IKKW-density} gives the qualitative conclusion of Proposition~\ref{prop:D-upper} by the same selection-kernel argument. If its unique fiber has dimension $d$, its density $2^{-d}$ must be at least $L^{-O_H(1)}$. Hence $L-r\le d=O_H(\log L)$, which implies $L=r+O_H(\log r)$. The elementary proof above identifies an explicit coefficient. \end{remark} \subsection{Lower bounds and the odd-prime threshold} We first record the standard simplex-block argument in the form needed for the logarithmic lower bound. \begin{corollary}[Logarithmic simplex lower bound]\label{lem:simplex} For every nontrivial fixed finite abelian group $H$ of odd order, \[ \D(H\oplus C_2^r)\ge r+(\D(H)-1)\log r-O_H(1). \] In particular, for every odd prime $p$, \begin{equation}\label{eq:simplex} \D(C_p\oplus C_2^r)\ge r+(p-1)\log r-O_p(1). \end{equation} \end{corollary} \begin{proof} Apply \eqref{eq:simplex-rounded} with $b=\D(H)-1$ fixed. \end{proof} If an odd prime $p$ divides $|H|$, then $C_p\le H$, so \eqref{eq:simplex} and Lemma~\ref{lem:constant-offset} give $E_H(r)\gg_H\log r$. \begin{lemma}[Composite-modulus construction]\label{lem:composite} Let $H$ be a fixed finite abelian group with $t=t(H)\ge2$. Then \eqref{eq:main-superlog} holds for all sufficiently large $r$. \end{lemma} \begin{proof} Choose the distinct odd primes $p_1,\ldots,p_t$ dividing $|H|$ and set $M=\prod_i p_i$. For a large integer $B$, let $Q_i$ be the largest power of $p_i$ not exceeding $B+1$, and set $d=\prod_iQ_i-1$. Thus $d\gg_H B^t$. For $q=p^k$, the polynomial \begin{equation}\label{eq:digit-polynomial} u_q(x)=\sum_{j=0}^{q-1}(-1)^j e_j(x)\quad\text{over }\F_p \end{equation} is the indicator of $q\mid\wt(x)$ on any binary cube; $e_j$ denotes the elementary symmetric polynomial and $e_0=1$. To verify this, at weight $w\ge1$ use \[ \sum_{j=0}^{q-1}(-1)^j\binom wj =(-1)^{q-1}\binom{w-1}{q-1}. \] Since $q$ is odd, the sign is positive, and Lucas's theorem makes the last binomial coefficient equal to one modulo $p$ precisely when $q\mid w$, and zero otherwise. At $w=0$, the sum is one. By the Chinese remainder theorem, combine the polynomials $1-u_{Q_i}$ into one polynomial $f$ over $\Z/M\Z$ of degree at most $B$. Its zero set on $\F_2^d$ is exactly $\{0\}$: the simultaneous conditions $Q_i\mid\wt(x)$ force $\prod_iQ_i\mid\wt(x)$, while $0\le\wt(x)\le d$. This is the symmetric composite-modulus mechanism of~\cite{BBR}, used in the sparse-polynomial/code setting in~\cite{BGG}. Because two is invertible modulo $M$, expand $f$ in parity characters. At most $\Vb{B}{d}$ frequencies occur. The identity $f(0)=0$ gives \[ f(x)=\sum_{u\ne0}c_u(\chi_u(x)-1) =\sum_{u\ne0}(-2c_u)\ell_u(x), \] where $\ell_u(x)=u\cdot x\pmod2$ is regarded as a bit. Use one coordinate for each nonzero coefficient $c_u$. The resulting linear map to a binary code is injective: a vector in its kernel would also be a zero of $f$. The code has dimension $d$ and length $L\le\Vb{B}{d}-1$, with a unique zero for its weighted $C_M$-sum. Now $\log\Vb{B}{d}=O_H(B\log B)$. For each sufficiently large $r$, choose $B=\lfloor\varepsilon_H\log r/\log\log r\rfloor$, where $\varepsilon_H>0$ is a sufficiently small fixed constant. Then $L\le r$, and Proposition~\ref{prop:code} gives a zero-sum-free sequence of length $r+d$ in $C_M\oplus C_2^r$. The subgroup $C_M\le H$ and $\Ds(H\oplus C_2^r)=r+O_H(1)$ finish the proof. \end{proof} The same construction has a finite form, useful without an unspecified asymptotic constant. \begin{proposition}[Finite composite-modulus bound]\label{prop:CRT-finite} Let $M=p_1\cdots p_t$ be a product of distinct odd primes, with $t\ge1$. Choose $Q_i=p_i^{a_i}$, with $a_i\in\Z_{\ge0}$, and put $Q=\prod_iQ_i$ and $d_0=\max_i(Q_i-1)$. Then \begin{equation}\label{eq:CRT-finite} \D(C_M\oplus C_2^r)\ge r+Q \quad\text{whenever}\quad r\ge\Vb{d_0}{Q-1}-Q. \end{equation} \end{proposition} \begin{proof} Combine the polynomials $1-u_{Q_i}$ in \eqref{eq:digit-polynomial} by CRT on the cube of dimension $Q-1$. For $Q_i=1$, set $u_1=1$. The resulting polynomial has a unique zero and degree at most $d_0$, so $\mu_{C_M}(Q-1)\le\Vb{d_0}{Q-1}-1$. Theorem~\ref{thm:exact-transfer} now applies. If $Q=1$, the same conclusion follows from $\mu_{C_M}(0)=0$. \end{proof} \begin{example} For $M=15$, take $Q_1=3^a$ and $Q_2=5^b$. On the cube of dimension $3^a5^b-1$, the polynomial is explicitly \[ f=10\sum_{j=1}^{3^a-1}(-1)^{j+1}e_j +6\sum_{j=1}^{5^b-1}(-1)^{j+1}e_j\pmod {15}. \] Its only zero is the origin, and its degree is at most $\max(3^a,5^b)-1$. Both prime components test the same Boolean weight, over a range of length $3^a5^b$. \end{example} \begin{proposition}[An explicit bound at every large rank]\label{prop:mixed-growth} Let $H$ be fixed and abelian, with distinct odd prime divisors $p_1,\ldots,p_t$, where $t\ge2$, and put $M=\prod_i p_i$. For $r\ge4$, set \[ R=\log_2r,\qquad d=\floor{\frac{R}{(t-1)\log_2R+t+2}},\qquad Q_i=p_i^{\lfloor\log_{p_i}(d+1)\rfloor}. \] Whenever $d\ge1$, \begin{equation}\label{eq:mixed-explicit} \D(H\oplus C_2^r)\ge r+\prod_iQ_i >r+\frac{(d+1)^t}{M}. \end{equation} In particular, \begin{equation}\label{eq:mixed-order} \D(H\oplus C_2^r)\ge\Ds(H\oplus C_2^r) +\left(\frac1{M(t-1)^t}+o_H(1)\right) \left(\frac{\log_2r}{\log_2\log_2r}\right)^t. \end{equation} \end{proposition} \begin{proof} The group $H$ contains $C_M$. The largest allowed powers satisfy $(d+1)/p_i(d+1)^t/M$. Put $k=Q-1$ and $d_0=\max_i(Q_i-1)\le d$. If $k\ge d\ge1$, the binomial estimate used in the proof of Theorem~\ref{thm:uniform} gives \begin{align*} \Vb{d_0}{k}&\le\Vb{d}{k}\le(ek/d)^d\\ &\le(e2^td^{t-1})^d <2^{d((t-1)\log_2R+t+2)}\le2^R=r. \end{align*} Here $k<(d+1)^t\le2^td^t$, $d\le R$, and $e<4$. If $k3+2j$, this is larger than the old span, so an extending vector exists. The identity \[ q(x\oplus y)=q(x)+q(y)-2\sum_iq_ix_iy_i\pmod4 \] shows that the enlarged span is again zero. Iteration gives the stated dimension. \end{proof} \begin{lemma}[The residual cubic]\label{lem:residual-cubic} Let $q(x)=\sum_iq_ix_i\pmod8$ on a binary linear code $U$. If every value is divisible by four, then $q/4:U\to\F_2$ has algebraic degree at most three. \end{lemma} \begin{proof} In basis coordinates $z$ on $U$, every original bit is a parity. Its integer multilinear coefficient on a monomial of size $j$ is divisible by $2^{j-1}$. Consequently $q$ has a multilinear representation modulo eight with no terms of degree at least four. Since all its values are divisible by four, integer M\"obius inversion shows that all its coefficients are divisible by four modulo eight. Divide by four and reduce modulo two. \end{proof} \begin{theorem}[Explicit $C_{24}$ bounds]\label{thm:C24-upper} For any weighted $C_{24}$-sum on a binary affine code of length $L$ and dimension $d$, every attainable fiber has at least \begin{equation}\label{eq:C24-density} \frac{2^d}{512\,\Vb{2}{L}^{16}} \end{equation} points. A singleton fiber implies \begin{equation}\label{eq:C24-isolation} d\le9+16\log\Vb{2}{L}. \end{equation} For every $r\ge512$, \begin{align} \D(C_{24}\oplus C_2^r) &\le r+10+\lfloor16\log(2r^2+r+1)\rfloor, \label{eq:C24-upper-finite}\\ E_{C_{24}}(r)&\le32\log r+3.\label{eq:C24-upper-log} \end{align} \end{theorem} \begin{proof} First suppose the zero fiber on a linear code is $\{0\}$. Apply Lemma~\ref{lem:mod4} to the reduction modulo four of its $C_8$-sum. On a subcode $U$ of dimension $k\ge\max\{0,\lfloor(d-2)/2\rfloor\}$, the remaining $C_8$ zero indicator is a binary polynomial of degree at most three, by Lemma~\ref{lem:residual-cubic}. The $C_3$ zero indicator has Fourier support at most $\Vb{2}{L}$. Theorem~\ref{thm:main-intersection} gives $k\le3+8\log\Vb{2}{L}$. Since $d\le2k+3$, apart from the harmless case $d\le3$, this proves \eqref{eq:C24-isolation}. Translate an arbitrary attained affine target to zero. Toggling a bit $x_i$ about a fixed solution replaces its weight by $(-1)^{(x_0)_i}h_i$ and subtracts the target. Thus the singleton bound is stable under every affine restriction. The maximal isolating-subspace argument in the proof of Theorem~\ref{thm:main-intersection} gives \eqref{eq:C24-density}. Use the proof of Proposition~\ref{prop:D-upper} with $A(L)=9+16\log\Vb{2}{L}$. Now $\Vb{2}{2r}=2r^2+r+1$, and the necessary finite cutoff holds for $r\ge512$. Indeed $A(2r)\le41+32\log r$, and $r-42-32\log r$ is positive at $512$ and increasing thereafter. This proves \eqref{eq:C24-upper-finite}. Finally \[ 16\log(2r^2+r+1) \le32\log r+16+16\log(1+1/r) <32\log r+17 \] in this range. Since $\Ds(C_{24}\oplus C_2^r)=r+24$, \eqref{eq:C24-upper-log} follows. \end{proof} \subsection{Direct comparison with a covering criterion} \label{subsec:covering} Han--Zhang's covering criterion~\cite[Proposition 2.4]{HZ}, applied with odd primary part $C_3$ and complementary group $C_8\oplus C_2^r$, requires \begin{equation}\label{eq:HZ-C24} 8\,2^r\sum_{j=0}^2\binom Nj7^{N-j}<8^N. \end{equation} At $N=r+O(\log r)$ even the $j=0$ term prevents this inequality: its ratio to $8^N$ has binary logarithm $3+r-N\log(8/7)$, which tends to infinity. Thus this direct substitution does not give \eqref{eq:C24-upper-log}. This observation concerns that substitution, not a non-implication claim about every possible use of the criterion. The alternative qualitative derivation through fixed-group programs was established in Section~\ref{sec:programs}. \section{Entropy lower bounds from finite configurations}\label{sec:entropy} We now turn from fiber lower bounds to constructions of singleton fibers. The following exact identity permits the internal prime-power scale to grow while the odd target group remains fixed. \subsection{An exact prime-power detector} For $U\subseteq[n]$, write $\ell_U(x)=\bigoplus_{i\in U}x_i\in\{0,1\}$ and $\chi_U(x)=(-1)^{\ell_U(x)}$. \begin{lemma}[Truncated Fourier detector]\label{lem:detector} Let $p$ be an odd prime, $k\in\Z_{\ge1}$, and $A\in\Z_{\ge2}$. Set $t=p^k$ and $n=At-1$. Define $b(x)=\Id_{t\mid\wt(x)}$ on $\F_2^n$, with values in $\F_p$. Then \begin{equation}\label{eq:detector-fourier} b(x)=\sum_{\substack{U\subseteq[n]\\|U|2$, bound the sum between its largest term and $At$ times that term and apply Stirling's formula. For $A=2$, the sum equals $2^{2t-2}-1$, giving the same estimate. Since the number of blocks is fixed, \[ d_k=t\sum_i A_i+O(1),\qquad \log R_k=t\max_i\{A_iH_2(1/A_i)\}+o(t). \] The sequence length gives $\D(Q\oplus P\oplus C_2^{R_k})\ge R_k+d_k+1$. Subtract $\Ds=R_k+O_{Q,P}(1)$ and divide by $\log R_k$. This proves \eqref{eq:box-entropy}. \end{proof} \begin{definition}[Finite entropy bound]\label{def:box-invariant} For a positive integer $b$, define \[ \mathcal B_b(Q)=\max\left( \{0\}\cup \left\{ \frac{\sum_{i=1}^s(a_i+1)} {\max_i\{(a_i+1)H_2(1/(a_i+1))\}}: \begin{array}{l} 1\le s\le b,\\[-2pt] (q_i,a_i)_{i=1}^s\text{ admissible in }Q \end{array} \right\}\right). \] \end{definition} This is a finite maximum: every admissible list satisfies $\sum_i a_i\le\D(Q)-1$, and the labels are chosen in a fixed finite group. Theorem~\ref{thm:box} gives \begin{equation}\label{eq:finite-entropy-bound} \overline\lambda(Q\oplus C_p^v)\ge\mathcal B_{v(p-1)}(Q). \end{equation} This parameter packages a family of lower bounds; it is not asserted to equal the Davenport coefficient. \begin{example}[Elementary two-groups] For $Q=C_2^u$, every admissible list has $a_i=1$ and linearly independent labels. Thus $\mathcal B_b(C_2^u)=\min\{u,b\}$. On the other hand, the weighted odd-primary bound and $v$ independent simplex constructions give \[ E_{C_2^u\oplus C_p^v}(r)=v(p-1)\log r+O_{p,u,v}(1). \] For the lower bound, use $p-1$ simplex blocks in each independent $C_p$ coordinate; their total dimension is $v(p-1)m$, and $m=\log r+O_{p,v}(1)$. For the upper bound absorb $C_2^u$ into the binary rank and use $b_P=v(p-1)$ in Corollary~\ref{cor:weighted}. In particular \eqref{eq:finite-entropy-bound} need not be sharp. \end{example} \subsection{The \texorpdfstring{$C_{24}$}{C24} family and unbounded coefficient growth} \begin{corollary}[An explicit $C_{24}$ lower family]\label{cor:C24-lower} For each integer $k\ge1$, put $t=3^k$ and \[ n_1=5t-1,\qquad n_2=4t-1,\qquad R_k=\sum_{j=1}^{t-1} \left(\binom{n_1}{j}+\binom{n_2}{j}\right). \] Then \begin{equation}\label{eq:C24-lower-finite} E_{C_{24}}(R_k)\ge9\cdot3^k-25, \end{equation} and \begin{equation}\label{eq:C24-lower-coefficient} \overline\lambda(C_{24})\ge\frac{9}{5H_2(1/5)}. \end{equation} \end{corollary} \begin{proof} In $Q=C_8$, let $q$ be a generator and take two blocks with $(q_1,q_2)=(q,q)$ and $(a_1,a_2)=(4,3)$. If $0\le z_1\le4$ and $0\le z_2\le3$, then $0\le z_1+z_2\le7$. Hence the configuration is admissible. Take $P=C_3$ and $h_1=h_2$ a generator. Theorem~\ref{thm:box} gives a zero-sum-free sequence of length $R_k+9t-2$. Since $\Ds(C_{24}\oplus C_2^{R_k})=R_k+24$, this proves \eqref{eq:C24-lower-finite}. The larger of the two entropy exponents is $5H_2(1/5)$, which proves \eqref{eq:C24-lower-coefficient}. \end{proof} Here is the sequence without matrix notation. Let $g$ have order 24, and let $e_{i,U}$, with $i=1,2$ and $\varnothing\ne U\subseteq[n_i]$, $|U|\sqrt2\,2^m$, so $A_m+2\log_2A_m\ge(15/8)2^m$. On the other side, $A_m-1<2^{m+1}$ implies \[ A_m-1+2\log_2(A_m-1)<(15/8)2^m. \] These two inequalities identify every integer boundary exactly. Theorem~\ref{thm:even-construction} makes dimension $2m$ feasible in \eqref{eq:C3transfer}. Dimension $2m+3$ is impossible: Corollary~\ref{cor:odd-support}, with $s=m+1$, gives \[ \nu(2m+3)>\frac8{\sqrt3}2^m, \] whereas integer $r