\documentclass[11pt]{amsart} \usepackage[margin=1.1in,footskip=30pt]{geometry} \usepackage{amsmath,amssymb,amsthm,booktabs,mathtools} \usepackage{graphicx,array,needspace} \usepackage[hidelinks]{hyperref} \frenchspacing \setlength{\emergencystretch}{1.5em} \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \theoremstyle{plain} \newtheorem{theoremA}{Theorem} \renewcommand{\thetheoremA}{A} \renewcommand{\thepart}{\Roman{part}} \makeatletter \renewcommand\part{\@startsection{part}{0}{\z@}{2\linespacing\@plus\linespacing}{\linespacing}{\normalfont\large\bfseries\centering}} \makeatother \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\e}{\mathbf{e}} \newcommand{\coef}[1]{[t^{#1}]\,} \newcommand{\Res}{\operatorname{Res}} \newcommand{\Q}{\mathcal{Q}} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\coord}{coord} \DeclareMathOperator{\dig}{digits} \providecommand{\R}{\mathbb{R}} \providecommand{\N}{\mathbb{N}} % arc figures (produced separately); guarded so that the build never breaks \IfFileExists{figs/arc_captions.tex}{\input{figs/arc_captions}}{} \providecommand{\arcAnatomyCaption}{The mechanism of a graph-form net (figure pending).} \providecommand{\arcSmallCaption}{Small-entry regime (figure pending).} \providecommand{\arcSkeletonCaption}{Skeleton regime (figure pending).} \providecommand{\arcLargeCaption}{Large-entry regime (figure pending).} \providecommand{\arcSplitCaption}{Split frame (figure pending).} \providecommand{\arcTwoCosetCaption}{A two-coset layer at $m=3$ (figure pending).} \providecommand{\arcWhyLatCaption}{What linearity forces (figure pending).} \providecommand{\arcGKCaption}{The two-dimensional Gr\"unschlo{\ss}--Keller nets as lattice image plus jitter (figure pending).} \IfFileExists{figs/part1_captions.tex}{\input{figs/part1_captions}}{} \providecommand{\thmADesignCaption}{Theorem~A design (figure pending).} \providecommand{\baseTwoThreeCaption}{The deep-shift lift (figure pending).} \providecommand{\projLinesCaption}{Lines in a two-dimensional projection (figure pending).} \IfFileExists{figs/col_captions.tex}{\input{figs/col_captions}}{} \providecommand{\colNetCaption}{A column net (figure pending).} \providecommand{\colDriftCaption}{Drift and jitter (figure pending).} \providecommand{\colHalfCaption}{Dimension five, one resolution (figure pending).} \IfFileExists{figs/w_captions.tex}{\input{figs/w_captions}}{} \providecommand{\wNetCaption}{A rank-one layer at $m=3$ (figure pending).} \newcommand{\arcfig}[1]{\IfFileExists{figs/#1.pdf}{\includegraphics[width=\linewidth,height=0.62\textheight,keepaspectratio]{figs/#1.pdf}}{\fbox{\parbox{0.9\textwidth}{\centering figure \texttt{\detokenize{#1}} pending}}}} \title[Quasi-uniform digital nets in every dimension]{Low-discrepancy quasi-uniform digital nets in every dimension} \author{Leonhard Gruenschloss} \thanks{ORCID iD: \url{https://orcid.org/0009-0004-6476-2191}} \hypersetup{pdfauthor={Leonhard Gruenschloss}} % Version line. PLACEHOLDER: \VersionDate = the release date of version 1 (YYYY-MM-DD); it is set in this one line % when the author approves the first push of the deposit repository (DEPOSIT_CHECKLIST step A2). The paper carries no % DOI: the Zenodo DOI goes only into the metadata of the deposit (README, CITATION.cff, Palomar yaml, Hexagon). \newcommand{\VersionDate}{2026-10-13} % Interactive companions (GitHub Pages of the deposit repository; live once Pages is enabled). \newcommand{\interactive}[1]{ Interactive version: \protect\href{https://lgruen.github.io/separated-nets-lean/explainers/#1}{\protect\texttt{explainers/#1}} (Section~\protect\ref{sec:intro}).} \date{Version~1, \VersionDate. The Lean formalization of this version is release v1.0.0 of \url{https://github.com/lgruen/separated-nets-lean}.} \begin{document} \begin{abstract} Dick, Goda and Suzuki asked for low-discrepancy digital nets that are quasi-uniform, with minimum distance $\delta$ and covering radius both of the optimal order $N^{-1/s}$, and stated the problem as open in dimension $s\ge3$. We construct such nets in every dimension. For every $s\ge3$, every base $q$ whose prime factors are at least $s-1$, and every $k\ge1$, we give explicit digital $(0,sk,s)$-nets over $\Z/q\Z$ with $N=q^{sk}$ points, covering radius at most $\sqrt s\,N^{-1/s}$, and $\delta N^{1/s}\ge1-a^*_s/q$ once $q\ge5a^*_s+4$. The constant $a^*_s\le e^{s\log s+O(s)}$ is computed from $s$ by a deterministic polynomial-time lattice algorithm. In dimension three every $N=p^m$, $m\ge3$, is covered for each prime $p\ge37$, and, with computer assistance, every $N=2^m$ and $N=3^m$. In prime bases these families are well separated in the sense of Dick, Goda and Suzuki, and they stay so under every digital shift. One mechanism underlies all constructions. For $N=q^{sk}$ it reads as follows. The first $k$ base-$q$ digits of a point's coordinates select a cube of side $q^{-k}$, and a $(0,sk,s)$-net has exactly one point in each such cube. Digit $k+1$ of each coordinate is a linear function of the first $k$ digits of all coordinates, because the net is digital. We choose this linear function so that, between the planes where this digit wraps around modulo $q$ or where a carry occurs among the first $k$ digits, the points lie on a translate of one lattice; for the main theorem this lattice is a slightly distorted copy of $\Z^s$. All later digits move a point by less than $1/q$ of the cube side. Part~II studies the constant $K$ in $\delta N^{1/s}=1-K/q+O(q^{-2})$. We prove lower bounds on $K$ when the linear maps have bounded coefficients. At $N=q^3$ we find the optimal $K$ among nets whose points lie on lines parallel to the main diagonal. With the cube-tiling theorems of Haj\'os and Keller we show that idealized lattices with one point per cube, whose points sit independently inside their cubes, cannot exceed $\delta N^{1/s}=1$. When the positions inside the cubes are coupled, the value $1$ can be exceeded: $\delta N^{1/3}\to\sqrt{281}/16$ at $N=p^3$, $\delta N^{1/5}>1$ for $(0,5,5)$-nets from $p=37$ on, and $\delta N^{1/s}\to\min(\sqrt s/2,\sqrt2)$ at $N=p^{sk}$, which is the $E_8$ value at $s=8$ (limits as $p\to\infty$). The results are formalized in Lean~4, apart from some finite computations, which are checked by scripts. \end{abstract} \maketitle \enlargethispage{6pt} \section{Introduction}\label{sec:intro} A point set $P\subset[0,1)^s$ of $N$ points is \emph{quasi-uniform} if its covering radius $h_P$ and its minimum distance $\delta_P$ are both of the optimal order $N^{-1/s}$. Dick, Goda and Suzuki \cite{DGS2} asked for digital nets that are both low-discrepancy and quasi-uniform. Their example is the two-dimensional Larcher--Pillichshammer net \cite{LP01}; for $s\ge3$ they state the problem as open, and Suzuki \cite{Suz26} recently stated that no such family is known. Without the word \emph{digital} there are answers in every dimension: Frolov point sets, Kronecker sequences, and rank-$1$ lattices with prime $N$ \cite[Thms.~3.1, 3.9, 3.7]{DGLPS1}. A rank-$1$ lattice with prime $N$ is formally a digital $(0,1,s)$-net, but over $\F_N$, in a base that grows with $N$. We work in the setting of \cite{DGS2,Suz26}: digital nets in a fixed base $q$, along $N=q^m$ with $m\to\infty$, with their exact stratification by elementary boxes. In \cite{DGS2} the base is a prime $b$ and a digital net is defined over $\F_b$ \cite[Def.~2.1]{DGS2}. Our nets are digital over $\F_p$ for prime bases $q=p$, which is the case that answers their question, and over the ring $\Z/q\Z$ for composite $q$, in the sense of Niederreiter \cite{Nie87} (see also \cite{LNS96}). All of them are $(0,m,s)$-nets, so they have low discrepancy (Remark~\ref{rem:lowdisc}), and the question is only about separation. We answer the question for nets, that is, for families of nets indexed by $m$, in every dimension; the corresponding question for sequences, whose initial segments would form such a family, remains open (Remark~\ref{rem:open}(10)). Our nets start from a classical construction of Niederreiter \cite[Sec.~6]{Nie87}, in which coordinate $c$ reads the base-$q$ Taylor coefficients of one polynomial at a point $a_c$ of the projective line. We call this a \emph{frame} and the $a_c$ its \emph{places}. For the places $\infty,0$ the frame is the Hammersley net \cite{Ham60}, and for $\infty,0,1$ it is, up to lower-triangular scrambles, the Hammersley net of the two-dimensional Faure sequence \cite{Fau82} (Proposition~\ref{prop:faure}); we call the frame of the places $\infty,0,1,\dots,s-2$ the \emph{Faure--Hammersley frame} (Section~\ref{sec:frames}). We scramble each coordinate by a lower-triangular matrix, chosen so that digit $k+1$ of each coordinate depends on the first $k$ digits of all coordinates through small integer matrices. These matrices form the \emph{design} $U^*_s$, which a lattice algorithm computes from $s$ alone (Definition~\ref{def:Ustar}). \begin{theoremA}[Every dimension]\label{thm:A} Let $s\ge3$, let $q\ge2$ be an integer all of whose prime factors are at least $s-1$, and let $k\ge1$. Let $\mathcal P^{U^*_s}_{k,q}$ be the digital net over $\Z/q\Z$ of Definition~\ref{def:Ns} for the places $\infty,0,1,\dots,s-2$ and the design $U^*_s$ of Definition~\ref{def:Ustar}, with $N=q^{sk}$ points. Then: \begin{enumerate} \item $\mathcal P^{U^*_s}_{k,q}$ is a digital $(0,sk,s)$-net over $\Z/q\Z$, every cell $\prod_i[n_iq^{-k},(n_i+1)q^{-k})$ holds exactly one point, and $h\le\sqrt s\,N^{-1/s}$; \item if $q\ge5a^*_s+4$, its toroidal (hence also Euclidean) minimum distance satisfies $\delta\ge(1-a^*_s/q)\,N^{-1/s}$, for every $k\ge1$; for $k\ge2$ the constant $a^*_s$ is first-order sharp for this net: $\delta N^{1/s}=1-a^*_s/q+O(q^{-2})$ as $q\to\infty$. \end{enumerate} Here $a^*_3=2$, and $a^*_s\le2(s-1)^{3/2}(3/2)^{s-2}(s-2)!=e^{s\log s+O(s)}$ for $s\ge4$. \end{theoremA} The range of $q$ in (1) is the largest possible for digital nets over $\Z/q\Z$ \cite[Cor.~2]{LNS96}, and for prime $q$ the nets are digital in the sense of \cite{DGS2}. The exact values are $a^*_s=2,5,16,43,170,543,2842,13640$ for $s=3,\dots,10$ (Table~\ref{tab:astar}), so for $s=3,4,5$ the separation bound holds from $q=14$, $29$ and $84$ on (for $s=4,5$ with the computed values $a^*_4=5$, $a^*_5=16$; $a^*_3=2$ is proved in Section~\ref{sec:Ustar}). The design $U^*_s$ is computed from $s$ by a deterministic algorithm in time polynomial in $s$ (Section~\ref{sec:Ustar}). A simpler, monomial design $U_s$ (Remark~\ref{rem:Us}) gives the same theorem, but with the constant $a_s=e^{\Theta(s^2)}$ ($a_5=17407$). Theorem~A uses the same frame and the same diagonal-dominance argument (Proposition~\ref{prop:D}) as $U_s$; it only replaces the entries of the design by short vectors of a lattice, which brings the constant down to $e^{s\log s+O(s)}$ (Section~\ref{sec:dims}). The separation bound is proved by a lower bound that uses only digit $k+1$ and treats all deeper digits as worst case (Lemma~\ref{lem:Ss}); we call it the \emph{one-digit relaxation}. It holds for every $k\ge1$, including $k=1$, i.e.\ $N=q^s$, which is the size at which the nets are practical. Whether the base threshold can be polynomial in $s$ is open (Part~\ref{part:constants}). Write $\delta N^{1/s}=1-K/q+O(q^{-2})$ and call $K$ the \emph{first-order deficit}. Smaller bases are certified for particular designs: for a given design the certified deficit is a finite maximum that an exact computation determines (Proposition~\ref{prop:Dprime}), searched designs have the certified deficits $2,5,10$ for $s=4,5,6$, against $a^*_s=5,16,43$ (Table~\ref{tab:Kdesigns}), and at $k=1$, where the matrix of digit $k+1$ need not come from a frame, searched integer matrices give, for instance, $\delta N^{1/5}\ge0.8$ from $N=13^5\approx3.7\cdot10^5$ points and $\delta N^{1/6}\ge0.8$ from $N=19^6\approx4.7\cdot10^7$ (Proposition~\ref{prop:k1}, Table~\ref{tab:practical}, computer-assisted). \emph{Natural candidates.} The base-$2$ $(0,m,3)$-net of Gr\"unschlo{\ss} and Keller \cite[Sec.~2]{GK09} was designed for large minimum distance at fixed $m$; in our computation its normalized separation $\delta N^{1/3}$ falls, not monotonically, from $0.75$ ($m=3$) to $0.18$ ($m=13$) (Appendix~\ref{app:verify}). The $p$-dimensional Faure sequence is not quasi-uniform \cite[Thm.~4.3, Cor.~4.4]{DGS2}, \cite[Thm.~2(3)]{GHS26}, and Goda, Hofer and Suzuki \cite[Remark~3]{GHS26} extend this to the $(0,p)$-sequences $(I,L_2P,\dots,L_pP^{p-1})$ with nonsingular lower-triangular $L_i$, along $N=p^m$, $m=(p-1)p^w$; for $p=3$ these are three-dimensional. This does not conflict with our results: those nets are prefixes of one sequence whose first coordinate is unscrambled, whereas ours are Hammersley-type nets that depend on $m$ (so they answer the question for nets, not for sequences) and scramble every coordinate. \emph{Dimension three: every resolution.} In dimension three the nets cover every $N$, not only $N=q^{3k}$: \begin{itemize} \item $m\equiv0\pmod3$, $m\ge6$, every base $q\ge4$: one integer lower-triangular scramble, the same for all three coordinates, of the frame of the places $\infty,0,1$ in its M\"obius form (Section~\ref{sec:frames}), with $\delta N^{1/3}\ge1-1/q$ for $q\ge8$ (Theorem~\ref{thm:main}); \item $m=3$, every base $q\ge5$ coprime to $6$: column nets (points on lines parallel to $(1,1,1)$), whose minimum distance is known exactly, $\delta N^{1/3}=1-\frac1{3q}+O(q^{-2})$ or $1-\frac2{3q}+O(q^{-2})$ according to $q\bmod3$, and $\delta N^{1/3}>0.9059$ (Theorem~\ref{thm:col}); \item every $m\ge3$ and every prime $p\ge37$: $\delta N^{1/3}>0.2715$, and $>0.4250$ for $p\ge83$, uniformly in $m$, with a bounded mesh ratio for each $p$ (Corollary~\ref{cor:allm}); \item every $m\ge1$ in the bases $2$ and $3$: $\delta N^{1/3}\ge0.4129$ resp.\ $0.3538$ with an explicit digital shift, and $\ge0.1718$ resp.\ $0.0770$ without a shift and under every digital shift (Theorem~\ref{thm:base23}, computer-assisted); \item every prime $5\le p\le31$: every $m\ge3$ except $m\in\{4,7\}$, and except $m=5$ for $p\le7$ (Theorems~\ref{thm:col} and~\ref{thm:main}, and nets designed separately for each prime, Appendix~\ref{app:perprime}). \end{itemize} So in every prime base there are well-separated families of digital $(0,m,3)$-nets in the sense of \cite[Def.~3.8]{DGS2} (no two points of $\mathcal P_m$ lie in a common shifted elementary box with $m/3+O(1)$ digits in every coordinate), on all of $\N$ (Corollary~\ref{cor:dgs}, Remark~\ref{rem:dgsfinite}). The prime-base families of Theorems~A, \ref{thm:main}, \ref{thm:S1} and Corollary~\ref{cor:allm} stay well separated under every digital shift (Proposition~\ref{prop:shift}, Corollaries~\ref{cor:allmshift} and~\ref{cor:dgs}), and so do the per-prime nets of Appendix~\ref{app:perprime} (Remark~\ref{rem:ppshift}) and, in the bases two and three, the nets of Theorem~\ref{thm:base23} (part~(3)). So in every prime base there are families that are well separated under every digital shift, on all of $\N$ (Remark~\ref{rem:dgsfinite}). When $m$ is not a multiple of $s$, the boxes that hold one point each cannot all be cubes. We then give some coordinates, the \emph{skeleton coordinates}, one cell digit fewer than the others. For such a coordinate the first digit after its cell digits, the \emph{skeleton digit}, places the point in one of $p$ thinner slices of its cell, so that every point lies in a cube of side $p^{-k-1}$, a \emph{fine cell}. Inside each cube of side $p^{-k}$ (a \emph{coarse block}) the occupied fine cells form a lattice defined by congruences modulo $p$, for instance a Korobov lattice (Section~\ref{sec:skel}). In every dimension this covers $m\equiv-1$ and $m\equiv-2\pmod s$ for all large primes (Theorems~\ref{thm:S1} and~\ref{thm:S2}). \emph{One mechanism.} All nets in this paper work the same way (Figure~\ref{fig:arc}). We explain it for Theorem~A, with $m=sk$, in the order in which it suggests itself: a frame, the trouble with its closest pair, and two fixes. Fixing the first $k$ base-$q$ digits of every coordinate selects a cube of side $q^{-k}$; we call these $N$ cubes the \emph{cells}. In units of the cell side, cell $n=(n_1,\dots,n_s)$ is the unit cube $n+[0,1)^s$, and $n_c$, the integer with the first $k$ digits of coordinate $c$, is its \emph{cell index} in coordinate $c$. A $(0,sk,s)$-net has exactly one point in each cell. So the cells can serve as the index of the points, and the covering radius is at most the cell diameter $\sqrt s\,q^{-k}$. A net written in this way, as the graph of the map from the cell digits to the remaining (\emph{offset}) digits, is a \emph{graph form} (Section~\ref{sec:prelim}). \emph{The frame and its closest pair.} The Faure--Hammersley frame is a $(0,sk,s)$-net for free: each of its box minors (the determinant that decides $t=0$ for one shape of box, Section~\ref{sec:prelim}) is a product of differences $a_c-a_d$ of its places, hence $\pm1$ for the places $\infty,0,1$ and a unit modulo $q$ whenever the prime factors of $q$ are at least $s-1$ (Lemma~\ref{lem:Ts}). Its separation is another matter. Inside its cell a point is placed, up to $1/q$ of the cell side, by digit $k+1$ of each coordinate, the first offset digit; we call it the \emph{geometric digit} $r_c$. In a digital net the vector $r=(r_1,\dots,r_s)$ is a linear function, modulo $q$, of the cell digits. Suppose we move from a cell $n$ to a neighbouring cell $n+e$, $e\in\{-1,0,1\}^s$, and only the last digit of each $n_d$ changes, that is, no carry occurs. Then $r$ changes by $\tilde Me\bmod q$ for one fixed integer $s\times s$ matrix $\tilde M$, the \emph{geometric layer}; a step whose carry reaches further changes $r$ by another matrix, which depends on how far the carry reaches. As long as no carry occurs and no coordinate of $r$ leaves the range $0,\dots,q-1$ (no \emph{wrap}), the point of cell $n$ therefore sits at $n+\bigl(r^{(0)}+\tilde M(n-n^{(0)})\bigr)/q$, up to the deeper digits, where $r^{(0)}$ is the geometric digit of a fixed cell $n^{(0)}$. On such a patch of cells the points lie on a translate of the lattice $(I+\tilde M/q)\Z^s$. So the closest pair is decided by $\tilde M$ and by the matrices of the steps with carries, and the frame does not control them: its geometric digit depends on the cell digits through coefficients that depend on $k$ and on how far a carry reaches (Lemma~\ref{lem:Rs}), its geometric layer is, off the diagonal, a Cauchy matrix in the places up to row and column scalars (Proposition~\ref{prop:Ns}), and nothing in the frame keeps these residues modulo $q$ small. The frame comes with no separation bound, and $t=0$ does not supply one: the coding data of a $(0,m,s)$-net do not determine its separation (Section~\ref{sec:frames}), and families of $(0,m,s)$-nets need not be quasi-uniform (the natural candidates above). \emph{First fix: scrambles that keep $t=0$.} A lower-triangular scramble of a coordinate adds to each of its digits a combination of the earlier digits of the same coordinate. It maps every elementary box onto a box of the same shape, so every box still holds exactly one point. This is classical \cite{Mat98,Tez95,DP10}, and it holds as well for Owen's nested scrambles \cite{Owe95}, which permute each digit depending on the earlier digits of its coordinate. In the frame these scrambles, like the gauge moves of Lemma~\ref{lem:box}, multiply every box minor by a unit \eqref{eq:scr}, so $t=0$ holds in the whole frame orbit (Section~\ref{sec:frames}). They change the geometric digits, and that is all we use them for. The digits after the geometric digit are not designed: they are what the frame, after the scramble, supplies. We call this bounded, deterministic displacement the \emph{jitter} (Figure~\ref{fig:arc}(b)); it is not the random jitter of stochastic sampling in computer graphics \cite{Coo86}. The jitter does necessary work: a box that fixes more than $k+1$ digits of some coordinate has a box minor that involves the deeper offset rows (Lemma~\ref{lem:box}), so the jitter is what keeps these thin boxes filled. And it is cheap: it moves each point by less than $1/q$ of a cell in each coordinate, which costs one unit of $1/q$ per coordinate in the separation bound (Lemma~\ref{lem:Ss}), the summand $1$ of the first-order deficit $K$ (Remark~\ref{rem:K}). \emph{Second fix: move the lattice into each cell.} Within this freedom we re-choose only the geometric digit. A design prescribes it on the last few levels of the cell indices, by small integer matrices: the geometric layer and the matrices of the steps whose carries reach these levels; the scramble removes its dependence on all more significant levels, for every $k$ (Proposition~\ref{prop:Gs}). On each patch the points then lie on a translate of $(I+\tilde M/q)\Z^s$, and when the entries of $\tilde M$ are small compared with $q$, this lattice is a slightly rotated and contracted copy of $\Z^s$. Where some $r_c$ wraps, the patch ends and the next patch starts about one cell further on. We call these boundaries \emph{slip planes} (Figure~\ref{fig:arc}(a)); they are what keeps one point in every cell. So the separation is at least the minimum of the local lattice minus a jitter cost of order $1/q$, and every separation proof in the paper is a version of this sentence (Lemma~\ref{lem:Ss}). This is the division of labour: the jitter can be borrowed from the frame, since whatever the frame supplies costs less than $1/q$ of a cell, whereas the lattice decides the separation and has to be moved into each cell, through the choice of its geometric digit. \emph{The result.} Theorem~A carries this out in every dimension: its design $U^*_s$ takes the entries of these matrices as short vectors of a lattice, small and independent of $k$ (Section~\ref{sec:dims}). In dimension three one scramble serves all three coordinates (Theorem~\ref{thm:main}). Faure--Hammersley frames also underlie the nets with one skeleton coordinate (Section~\ref{sec:skel}) and, in dimension three, the nets for $m\equiv0,1,2\pmod 3$, $m\ge4$ (Section~\ref{sec:dim3}); frames with other places underlie the remaining skeletons; and split frames (Section~\ref{sec:large}), which read each coordinate at two places, have unit box minors by the same confluent Vandermonde determinant (Lemma~\ref{lem:Tsplit}). \emph{Other nets.} Not every net here is a frame scramble. The column nets at $m=3$ (Section~\ref{sec:m3}), the nets $\mathcal T_p$ and $\mathcal W_p$ (Section~\ref{sec:large}), the nets $\mathcal Q^*_p$ (Proposition~\ref{prop:half5}) and the nets at $k=1$ whose geometric layer is chosen directly (Proposition~\ref{prop:k1designs}) choose all their offset digits directly, and their $t=0$ is proved by checking that their box minors are units (Lemmas~\ref{lem:nie} and~\ref{lem:box}). The nets in the bases two and three are built by a recursion $m\to m+3$ (Section~\ref{sec:base23}), in which copies of the old net play the role of the lattice and shifted lower digits are the jitter. The same pattern is visible already in dimension two (Remark~\ref{rem:gk2d}, Figure~\ref{fig:arcgk}): the points of the even-$m$ permutation nets of Gr\"unschlo{\ss} and Keller \cite[Sec.~3]{GK09} lie on the lines of a lattice, and van der Corput slides along these lines are the jitter; since these nets are a nested scrambling of the Hammersley net, the frame of the places $\infty,0$, the slides carry $t=0$. \begin{figure}[tbp] \centering \arcfig{arc_anatomy} \caption{\arcAnatomyCaption\interactive{anatomy.html}} \label{fig:arc} \end{figure} \emph{Separation constants.} Part~II asks how large the constant can be, that is, how small the first-order deficit $K$ in $\delta N^{1/s}=1-K/q+O(q^{-2})$ can be. If the geometric layer $\tilde M$ is \emph{bounded}, i.e.\ its entries are bounded independently of $q$, and has no zero on the diagonal, then $K$ cannot be small: $K\ge1$ in dimension three (Corollary~\ref{cor:ceil3}), and $K\ge\sqrt{(s-3)/8}$ in general (Theorem~\ref{thm:L}). Geometric layers whose entries grow with $q$ do better. The column nets at $m=3$ (Section~\ref{sec:m3}), whose points lie on lines parallel to the main diagonal, have $K=\frac13$ and $\frac23$, according to $q\bmod3$, and these values are optimal among all column nets (Theorem~\ref{thm:O}). Part~II explains them by a bookkeeping with two terms (Section~\ref{sec:drift}). The first term, the \emph{drift}, is forced by the torus: the column nets imitate the reflected cube $H\Z^3$ ($\Z^3$ reflected in the plane perpendicular to $(1,1,1)$), and when $3\nmid q$, $q$ steps of the reflected cube along an axis do not close up around the torus. The mismatch has to be spread over these $q$ steps, and its share per step gives the deficits $\frac13$ and $\frac23$. The second term is the cost of the \emph{jitter}, the displacement by the deeper digits. In dimension five the same bookkeeping gives digital $(0,5,5)$-nets with $\delta N^{1/5}>1$ for every prime $p\ge37$ (Proposition~\ref{prop:half5}). At the resolutions $m=sk$ the cells are cubes. By the theorems of Haj\'os and Keller, an idealized lattice with one point per cube cannot exceed the value $1$ of $\Z^s$ (the \emph{cubic value}) if the positions of its points inside their cubes vary independently (Proposition~\ref{prop:cube}). Nets whose positions inside the cubes are coupled go beyond $1$: \begin{itemize} \item At $m=3$, a geometric layer of rank one makes all three coordinates wrap together. It gives $\delta N^{1/3}>1$ for every prime $p\ge17$ and $\delta N^{1/3}\to\sqrt{281}/16=1.0476\ldots$ (Proposition~\ref{prop:wnet}). The optimum of this family of layers is $\mu^*=\sqrt{(72-8\sqrt{31})/25}=1.0480\ldots$, reached as $p\to\infty$ (Proposition~\ref{prop:wfamily}), and a union of two lattice cosets gives $\sqrt{66}/8$ (Proposition~\ref{prop:twocoset}). \item A variant of the frame in which each coordinate reads its cell digits and its offset digits at two different places (a \emph{split frame}) gives the packing $D_s^+$ locally, with $\delta N^{1/s}\to\min(\sqrt s/2,\sqrt2)$ as $p\to\infty$ at every resolution $m=sk$; this is the $E_8$ value at $s=8$ (Theorem~\ref{thm:HL}). \end{itemize} \begin{table}[tbp] \centering\footnotesize\setlength{\tabcolsep}{1.75pt} \begin{tabular}{@{}llllll@{}} \toprule $s$ & resolution $N$ & base & bound on $\delta N^{1/s}$ & status & result\\ \midrule \multicolumn{6}{@{}l}{\emph{Part I. Existence}}\\ $s$ & $q^{sk}$, $k\ge1$ & $\operatorname{lpf}(q)\ge s-1$, $q\ge5a^*_s+4$ & $1-a^*_s/q$ & Lean & Thm.~\ref{thm:A}\\ 3 & $q^{3k}$, $k\ge2$ & every $q\ge4$ & $1-1/q$ ($q\ge8$) & Lean & Thm.~\ref{thm:main}\\ $s$ & $p^{sk+s-1}$, $k\ge0$ & prime $p>2^s$ & $(\rho-1)p^{-1/s}>\frac13$, $\to1$ & Lean & Thm.~\ref{thm:S1}\\ $s\ge4$ & $p^{sk+s-2}$, $k\ge1$ & prime $p\ge p_0(s)$ & $(\rho-4)p^{-1/2}$; $(B-F)p^{-2/s}$ & Lean & Thm.~\ref{thm:S2}\\ 3 & $p^{3k+1}$, $k\ge1$ & prime $p\ge37$ & $(\rho^2-4)p^{-2/3}\to1$ & Lean & Thm.~\ref{thm:allm}\\ 3 & $q^{3}$ & $\gcd(q,6)=1$, $q\ge5$ & exact; $>.9059$, $\to1$ & Lean & Thm.~\ref{thm:col}\\ 3 & $p^m$, every $m\ge3$ & prime $p\ge37$ & $>.2715$; $>.4250$ ($p\ge83$) & Lean & Cor.~\ref{cor:allm}\\ 3 & $p^{3k+r}$, $r\in\{1,2\}$ & primes $5\le p\le79^{\ddagger}$ & $.50$--$1.03$ per prime & Lean & App.~\ref{app:perprime}\\ 3 & $2^m$, $3^m$, every $m\ge1$ & $b=2,3$ & $.4129$, $.3538$ (explicit shift) & comp.-ass. & Thm.~\ref{thm:base23}\\ & & & $.1718$, $.0770$ (every shift) & Lean &\\ \multicolumn{6}{@{}l}{\emph{Part II. Separation constants}}\\ 4 & $q^{4k}$, $k\ge1$ & odd $q\ge17$ & $1-2/q$, sharp for $k\ge2$ & Lean & Prop.~\ref{prop:4D2}\\ 5, 6 & $q^s$ & Tab.~\ref{tab:practical} & $\ge.8$ ($q\ge13$ resp.\ $19$) & comp.-ass. & Prop.~\ref{prop:k1}\\ $s\ge4$ & $q^{sk}$, $k\ge3$ & bounded $\tilde M$ & $<1-\sqrt{(s-3)/8}/q+O(q^{-2})$ & Lean & Thm.~\ref{thm:L}\\ $s$ & $p^{sk+s-1}$, $k\ge0$ & prime $p\ge s-1$ & up to $1.238$ ($s=7$) & Lean & Prop.~\ref{prop:S1E}\\ 3 & $q^{3}$ & $3\nmid q$, column nets & $\le1-\frac1{3q}$ resp.\ $1-\frac2{3q}$, $+O(q^{-2})$ & Lean & Thm.~\ref{thm:O}\\ 3 & $p^{3}$ & prime $p\ge41$ & $\sqrt{\frac{33}{32}-\frac{41}{16p}}-\frac{\sqrt3}p\to\frac{\sqrt{66}}8=1.0155$ & Lean & Prop.~\ref{prop:twocoset}\\ 3 & $p^{3}$ & prime $p\ge17$ & $>1$, $\to\frac{\sqrt{281}}{16}=1.0476$ & Lean & Prop.~\ref{prop:wnet}\\ 3 & $p^{3}$ & rank-one family & $<\mu^*+\frac{\sqrt3}p$; $\mathcal W^*_p\to\mu^*=1.0480$ & Lean & Prop.~\ref{prop:wfamily}\\ $s\ge4$ & $p^{sk}$, $k\ge1$ & prime $p\ge\max(2s,2B_s+2)$ & $\to\min(\frac{\sqrt s}2,\sqrt2)$ & Lean & Thm.~\ref{thm:HL}\\ 5 & $p^{5}$ & prime $p\ge37$ & $>1$, $\to\frac{\sqrt5}2=1.118$ & Lean & Prop.~\ref{prop:half5}\\ 5 & $p^{5k}$, $k\ge1$ & prime $p\ge1511$ & $>1.0265$, $\to\frac{\sqrt5}2=1.118$ & Lean & Prop.~\ref{prop:HL5}\\ \bottomrule \end{tabular} \medskip \caption{Overview of the separation results: Part~I by the resolutions covered, Part~II by the constant. Here $q$ is an integer base, $p$ a prime, $b$ a prime base and $\operatorname{lpf}(q)$ the least prime factor of $q$. All nets are explicit digital $(0,m,s)$-nets (digitally shifted where stated) with one point in every cell (for $m=sk+r$, $0k_c$ and \emph{under} if $d_c1$. Otherwise $E=e\in\{-1,0,1\}^s\setminus\{0\}$ and $\Delta=e+\theta$ with $|\Delta_c|<2$; if $|\Delta_c|\le G/2$, then $\operatorname{tor}_c=|\Delta_c|\ge b_c/q$, and otherwise $\operatorname{tor}_c\ge G-|\Delta_c|>G-2\ge1$. So either some coordinate has toroidal distance at least one cell, or the distance is at least $(\sum_ib_i^2)^{1/2}/q$ cells; as a cell has side $q^{-k}$, this is the claim. \end{proof} Like Theorem~\ref{thm:sep}, Lemma~\ref{lem:Ss} uses only that the cells determine the point and the shape of the first offset rows, not $t=0$. The lemma is formally verified for every $k\ge1$. We call a step $e$ together with carry depths $j=(j_d)$ on its moving axes a \emph{configuration} $(e,j)$. At $k=1$ there are no carries, and the configurations are those with all $j_d=0$, a subset of the configurations for $k\ge2$; so for a given design the value of $\mathrm{LB}$ at $k=1$ is at least its value with more levels, and every lower bound proved below through Lemma~\ref{lem:Ss} for $k\ge2$ holds at $k=1$ as well. \begin{remark}[First-order expansion]\label{rem:K} Call a moving coordinate \emph{backward} if $e_i\sigma_i\le0$ and \emph{forward} otherwise. For a fixed design and $q\to\infty$: if some configuration with at least two moving axes has no backward coordinate, or some diagonal partial sum $M^{(j)}[a][a]$ is positive, then $\mathrm{LB}=O(1/q)$; otherwise $\mathrm{LB}=1-K/q+O(q^{-2})$ with $K$ the maximum of $|M^{(j)}[a][a]|$ and of $|\sigma_b|+1$ over the configurations with at least two moving axes and a single backward coordinate $b$. (Two backward coordinates cost nothing at first order, and the jitter costs the $+1$.) In dimension three, $M=-I+S$ with $S$ skew with entries $\pm1$ gives $K=1$. \end{remark} \subsection{The lattice picture} The mechanism of the introduction underlies every net in this paper. Write the point of a graph form with depths $(k,\dots,k)$ in cell units as $n+(r+f)/q$, with $n$ its cell, $r$ its geometric digits and $f\in[0,1)^s$ the contribution of the deeper digits (the \emph{jitter}); we call $n+r/q$ its \emph{lattice image}. For a step $n\to n+e$ that changes only the last cell digits, the geometric digits change by $\tilde Me$ modulo $q$. So on a patch of cells where no geometric digit wraps, the lattice images form a piece of a translate of the lattice $(I+\tilde M/q)\Z^s$, and the points differ from them by $f/q$. Where a geometric digit wraps, the patch ends at a \emph{slip plane}, and every cell keeps exactly one point. This picture is behind every separation proof here: the separation is the minimum of the lattice, or union of lattice cosets, that the geometric digit realizes locally, minus a jitter cost of less than $1/q$ of a cell per coordinate and, at unequal depths, a bounded cost from shifts of the lattice between neighbouring blocks of cells (\emph{faults}, Section~\ref{sec:skel}). Lemma~\ref{lem:Ss} makes this exact for $m=sk$, and Lemma~\ref{lem:skelred} for $m=sk+r$. Three regimes occur (Figure~\ref{fig:arc}): (I) bounded centred entries give a slightly rotated and contracted copy of $\Z^s$, of minimum $1-O(1/q)$ (Sections~\ref{sec:dims} and~\ref{sec:dim3}); (II) when $m\not\equiv0\pmod s$, some coordinates have one cell digit fewer, their first offset digit selects a thinner box (a \emph{fine cell}, Section~\ref{sec:fine}), and the occupied fine cells form lattices of index $p^\ell$, with minimum at the scale of $p^{\ell/s}$ fine cells (Section~\ref{sec:skel}); and (III) entries of size $\Theta(q)$, $\tilde M\approx qA$ with a rational $A$, put the point of cell $n$ near $n+\{An\}$, a lattice or packing other than $\Z^s$ (the column nets of Section~\ref{sec:m3}, and Section~\ref{sec:large} in Part~II). \begin{remark}[A two-dimensional example]\label{rem:gk2d} The permutation $(0,m,2)$-nets of Gr\"unschlo{\ss} and Keller \cite[Sec.~3]{GK09} for even $m$ show this mechanism in the smallest case, base~$2$ and dimension two (Figure~\ref{fig:arcgk}). For even $m=2r\ge4$, $Q=2^r$, $N=Q^2$ and $0\le i,j{\raggedright\arraybackslash}p{11.5cm}@{}} \toprule $s$, $q$, $p$ & dimension; base (any integer $q\ge2$, the ring $\Z/q\Z$); a prime base; $b\in\{2,3\}$ in Section~\ref{sec:base23}\\ $m$, $N$, $k$, $\mathbf k$ & number of digits, $N=q^m$ points; number of cell levels ($m=sk$, or $m=sk+r$); depths $\mathbf k=(k_1,\dots,k_s)$ of a graph form\\ $n_c$, $r_c$, $f_c$ & cell index of coordinate $c$ (levels $0,\dots,k-1$, most significant first); geometric digit; jitter $\in[0,1)$\\ $\varpi$, $P_c$, $a$ & in Section~\ref{sec:skel}: fine unit $\varpi=p^{m-k-1}$; fine cell of coordinate $c$; centred fine difference (Section~\ref{sec:fine})\\ $\Phi_{cd}$, $\tilde M$ & blocks of a graph form; geometric layer (Definition~\ref{def:geom})\\ $F_n$, $M^{(j)}$ & design (target at carry depth $n$, i.e.\ level $k-1-n$); its partial sums\\ $e$, $j_d$, $\sigma$, $b_i$ & cell step; carry depths; change of the geometric digits; per-coordinate bounds of the relaxation\\ $\omega$ & wraps of a step: $\omega_c=1$ if the digit in question of coordinate $c$ wraps around modulo $q$ on the step, else $0$ (geometric digits in Lemma~\ref{lem:phase}; last cell digits in the proof of Theorem~\ref{thm:sep}; third digits in Lemma~\ref{lem:drift})\\ $E$, $\Delta$ & difference of two cells, or a step; difference of two points in cell units\\ $K$, $K(F)$, $K_{\mathrm{pair}}$ & first-order deficit of a family of nets, $c=1-K/q+O(q^{-2})$; certified deficit of a design (Definition~\ref{def:K}); lower bound from two-axis steps (Definition~\ref{def:Ktrue})\\ $\sigma_{\max}(F)$, $q_{\mathrm{tail}}$ & largest digit change of a design; base from which Proposition~\ref{prop:Dprime} certifies $1-K(F)/q$\\ $\Sigma n$ & coordinate sum $\sum_cn_c$ of a cell or a step\\ $c$ & normalized separation $c=\delta N^{1/s}$; also a coordinate index (in Proposition~\ref{prop:ceil} the coordinates are $a,b,b'$)\\ $\mathcal K$ & lower-triangular Toeplitz matrix of $(1-t)^{-k}$ (used in Definition~\ref{def:net})\\ $\Psi$, $T$ & M\"obius map of the frame of the places $0,1,\infty$, and $T=s_k\Psi$, $s_k=(-1)^{k+1}$, with $T^3=1$ (Section~\ref{sec:frames})\\ $\mathcal P_{k,q}$, $\mathcal B^1$, $\mathcal B^2$ & the nets of Theorem~\ref{thm:main} (Definition~\ref{def:net}); the nets for $m\equiv1,2$ in dimension three (Definition~\ref{def:B})\\ $\delta$, $h$ & toroidal minimum distance (at most the Euclidean one, so lower bounds carry over); covering radius\\ $a_c$; $u_c$, $v_c$ & places of a frame (numeric indices $a_1,a_2,\dots$ in Section~\ref{sec:skel}); offset and cell points of a split frame (in Section~\ref{sec:sep}: the last two cell digits)\\ $\alpha_c$, $a^*_s$ & diagonal magnitudes of a dominant design (Proposition~\ref{prop:D}; in Lemma~\ref{lem:CH} $[\alpha_c:\beta_c]$ are homogeneous coordinates of the places); $a^*_s=\max_c\alpha_c$ for $U^*_s$ (Definition~\ref{def:Ustar})\\ $\Lambda_j$, $\operatorname{lc}_j$, $\gamma_j$, $\eta_j$ & lattice of admissible design columns at carry depth $j$; leading coefficient; generator of its values; projection length (Section~\ref{sec:lattice})\\ $\lambda_c$, $\mu_d$, $D_{cd}$ & row and column scalars and place differences of a Cauchy layer $\lambda_c\mu_d/D_{cd}$\\ $\ell$, $\rho$, $\Lambda_A$ & number of skeleton coordinates; Korobov or congruence parameter; skeleton lattice\\ $S_z$, $\kappa$, $\Lambda_\kappa$ & skeleton digit; its coefficients on the full coordinates; the skeleton lattice for $\ell=1$\\ $\lambda_\infty$, $\lambda_2$, $\lambda_{\mathrm{jit}}$ & lattice minimum in the maximum norm; in the Euclidean norm; Euclidean minimum after one fine cell of jitter per coordinate (Proposition~\ref{prop:S1E})\\ $H$, $D_s^+$, $\gamma_s$ & reflection in $\mathbf 1^\perp$ (reflected cube $H\Z^3$); the packing $D_s^+$; Hermite constant\\ $\mathbb T$, $\varphi$, $\mathcal C_q$ & columns $(\Z/q\Z)^3/\langle\mathbf 1\rangle$; heights of a column point set; column net (Section~\ref{sec:m3})\\ $a$, $b$, $w$, $\xi$, $\varepsilon_i$, $\eta$ & shared row $a=\frac13(q\mathbf 1+w)$ of a column net and its third-digit row $b$ (Definition~\ref{def:col}); $w$ gives its drift $\frac13qw_i$ per step; excess of a step; excess of a step along $\e_i$ without a wrap; largest excess (Section~\ref{sec:drift})\\ $b=\beta/p$ & glue of a step: change of the phase (in-cell position) in cell units, modulo $1$ (Lemma~\ref{lem:phase})\\ $\mathrm{dr}$, $\mathrm{dj}$ & drift and drift plus jitter of a half-integer layer at $s=5$ (Section~\ref{sec:halfms})\\ $Y$, $\psi$ & phase group of a lattice (Section~\ref{sec:whylat}); phase subtorus of a layer at $m=3$, and a point $\psi$ of it (Section~\ref{sec:wnet})\\ $\mathcal T_p$, $\mathcal W_p$, $\zeta$, $\varepsilon$ & two-coset and rank-one nets; the digit read by $z$ in $\mathcal W_p$; $\varepsilon=\rho/(2p)$ with $\rho=2\lfloor p/48\rfloor+1$ (Sections~\ref{sec:twocoset}, \ref{sec:wnet})\\ $B_s$, $\tau$ & constant of Theorem~\ref{thm:HL}; a digital shift (Definition~\ref{def:dshift})\\ \bottomrule \end{tabular} \caption{Notation.} \label{tab:notation} \end{table} \emph{Digital shifts.} Dick, Goda and Suzuki also consider digitally shifted nets, the digital analogue of the random shifts of Cranley and Patterson \cite{CP76}, and so do we (Sections~\ref{sec:base23} and~\ref{sec:dgs}). \begin{definition}[Digital shift {\cite[Def.~2.2]{DGS2}}]\label{def:dshift} For a point set given by digit vectors $x_c\in(\Z/q\Z)^m$ and $\tau=(\tau_1,\dots,\tau_s)\in(\Z/q\Z)^{s\times m}$, the \emph{digitally shifted} point set $P\oplus\tau$ has the digit vectors $x_c+\tau_c$, added digit by digit modulo $q$ without carries. For prime $q$ this is \cite[Def.~2.2]{DGS2}. A \emph{digital net} is unshifted ($\tau=0$) unless stated otherwise. \end{definition} A digital shift maps every elementary box of shape $d$ bijectively onto one of the same shape, so it preserves the $(0,m,s)$-property; in graph form it only adds a constant to the offsets (Lemma~\ref{lem:shift}). \section{Frames: \texorpdfstring{$t=0$}{t=0} for free}\label{sec:frames} A \emph{frame} is a digital point set whose coordinate $c$ reads the base-$q$ Taylor coefficients (the \emph{jets}) of one polynomial $V$ at a point $a_c$ of the projective line, its \emph{place}; these are the points $\beta_i$ of Niederreiter's construction \cite[Sec.~6]{Nie87}. For the places $\infty,0$ the frame is the Hammersley net \cite{Ham60}. For the places $\infty,0,1,\dots,s-2$ it is the Hammersley net of the $(s-1)$-dimensional Faure sequence \cite{Fau82}, whose generator matrices, the powers of the Pascal matrix, read the jets at $0,1,\dots,s-2$; we call it the \emph{Faure--Hammersley frame}. A frame gives one point per cell and $t=0$ for free (Lemma~\ref{lem:hermite}), and $t=0$ survives every scramble and gauge move. The nets obtained from a frame by lower-triangular scrambles and the gauge moves of Lemma~\ref{lem:box} form its \emph{frame orbit}. In the order of the introduction: the frame supplies $t=0$ and the deeper digits (the jitter), and the moves are used to re-choose the geometric digit, which decides the separation (Section~\ref{sec:prelim}). What these moves cannot change is the shape of the geometric layer: it is a Cauchy matrix in the places, up to row and column scalars and a free diagonal (Proposition~\ref{prop:Ns}), and its Pl\"ucker relations (Lemma~\ref{lem:CH}) are the obstructions specific to frames; the obstructions that hold for every digital net, or for every net, are those of Section~\ref{sec:whylat}. For a commutative ring $R$ and $V\in R[t]_{0$; by (i), $\gamma_j\in R_S$. The vector $(w_cy_c^j)_c$ of $g=X^j$ has $S$-unit or zero entries, so some product $P$ of primes in $S$ makes $P(w_cy_c^j)_c$ integral, with $\operatorname{lc}_j=P$; hence $P\in\gamma_j\Z$, and $\gamma_j$ has no prime factor outside $S$ in its numerator either. (iii) is (i) together with the definition. (iv) If $v\in\Lambda_j$ has $\operatorname{lc}_j(v)=t\gamma_j$, then $v-tE_j\in\Lambda_j\cap\ker\operatorname{lc}_j=\Lambda_{j-1}$; induction on $j$. \end{proof} So the admissible design columns at carry depth $j$ are the lattice vectors on the affine hyperplanes $\operatorname{lc}_j=\pm\gamma_j\cdot(S\text{-unit})$ of $\Lambda_j$, and the smallest ones are governed by the distance between consecutive hyperplanes. That distance is a product of local Chebyshev constants, one for each place of $\mathbb Q$ that matters (in the sense of number theory: the archimedean absolute value and the $p$-adic ones; not the places of a frame). For the archimedean absolute value and each $p\in S$ let \[ A_{j,\R}=\min\bigl\{\|E\|_2:E\in V_j\otimes\R,\ \operatorname{lc}_j(E)=1\bigr\},\qquad A_{j,p}=\min\bigl\{\max_c|E_c|_p:E\in V_j,\ \operatorname{lc}_j(E)=1\bigr\}, \] the least size, at that place, of the vectors $(w_cg(y_c))_c$ of monic polynomials $g$ of degree $j$ (the minimum at $p$ exists because these sizes are powers of $p$ and are bounded below by Lagrange interpolation). \begin{lemma}[Adelic Chebyshev identity]\label{lem:adelic} Let $\pi_j$ be the orthogonal projection of $\R^{s-1}$ onto the line $V_j\cap V_{j-1}^\perp$. Then $\pi_j(\Lambda_j)$ is generated by a vector of length \[ \eta_j=A_{j,\R}\prod_{p\in S}A_{j,p},\qquad\text{and}\qquad \gamma_j=\prod_{p\in S}A_{j,p}. \] Every $E\in\Lambda_j\setminus V_{j-1}$ has $\|E\|_2\ge \eta_j$. \end{lemma} \begin{proof} \emph{The $p$-adic places, $p\in S$.} If $E\in V_j$ has $\operatorname{lc}_j(E)=1$ and $\max_c|E_c|_p=p^e$, then $p^eE$ is $p$-integral, and $D'p^eE\in\Lambda_j$ for an integer $D'$ prime to $p$; so $D'p^e\in\gamma_j\Z$ and $v_p(\gamma_j)\le e$. Conversely, for $v\in\Lambda_j$ with $\operatorname{lc}_j(v)=\gamma_j$ the vector $v/\gamma_j$ has $\operatorname{lc}_j=1$ and $\max_c|v_c/\gamma_j|_p\le p^{v_p(\gamma_j)}$. So $A_{j,p}=p^{v_p(\gamma_j)}$, and as $\gamma_j>0$ is an $S$-unit, $\gamma_j=\prod_{p\in S}A_{j,p}$. \emph{The archimedean place.} On $V_j$, $\operatorname{lc}_j(E)=\langle\theta,E\rangle$ for a unique $\theta\in V_j$, and $\theta\perp V_{j-1}$. Hence $\|\pi_j(E)\|=|\operatorname{lc}_j(E)|/\|\theta\|$, and the least norm with $\operatorname{lc}_j=1$ is $A_{j,\R}=1/\|\theta\|$, attained at $\theta/\|\theta\|^2$. So $\pi_j(\Lambda_j)=\gamma_j\Z\,\theta/\|\theta\|^2$, generated by a vector of length $\gamma_jA_{j,\R}=\eta_j$, and $\|E\|\ge\|\pi_j(E)\|\ge \eta_j$ if $\operatorname{lc}_j(E)\ne0$. \end{proof} This is the product formula at work: an $S$-unit scalar can move the $p$-adic cost of a collision of nodes (Proposition~\ref{prop:Ns}) to the archimedean place, but never remove it. For the upper bounds below it suffices to exhibit, \emph{independently at each place}, one monic polynomial that is small there. At the place $p$ a monic polynomial is small on the nodes if its roots sit $p$-adically near the nodes that collide modulo $p$ (two finite rows $c,c'$ collide if $p\mid c-c'$, and then $y_c$ is $p$-adically large); different primes want the roots in different places, so no single closed-form basis serves all primes, and the lattice reduction chooses the roots for each prime automatically. \begin{lemma}[Babai rounding along the flag]\label{lem:babai} Let $E_i\in\Lambda_i$ with $\operatorname{lc}_i(E_i)=\gamma_i$ for $i0$, and for $1\le j\le \nu$ let $E_{d,j}$ be the unique $E\in\Lambda_j$ with $\operatorname{lc}_j(E)=+\gamma_j$ whose orthogonal projection onto $V_{j-1}$ has all its coefficients in the basis $E_{d,0},\dots,E_{d,j-1}$ in $[-\frac12,\frac12)$. The design $U^*_s=(F_0,\dots,F_\nu)$ has the partial sums $M^{(j)}[c][d]=E_{d,j}[c]$ ($c\ne d$, $j\le \nu$), the diagonal $F_0[c][c]=-\alpha_c$ at carry depth $0$ with $\alpha_c=\sum_{d\ne c}\max_j|E_{d,j}[c]|$, and zero diagonal at the carry depths $n\ge1$. Let $a^*_s=\max_c\alpha_c$. \end{definition} By Lemma~\ref{lem:sat}(iv) the set $\{E\in\Lambda_j:\operatorname{lc}_j(E)=\gamma_j\}$ is a coset of $\Lambda_{j-1}=\bigoplus_{i\nu$, so $\gcd(q,\nu!)=1$, and Corollary~\ref{cor:tsk} gives a digital $(0,sk,s)$-net with one point in every cell, for every $k\ge1$. A cell has diameter $\sqrt s\,q^{-k}=\sqrt s\,N^{-1/s}$, which bounds $h$. (2) \emph{Lower bound.} (H1) and (H2) of Lemma~\ref{lem:Ss} hold by Proposition~\ref{prop:Gs} and Definition~\ref{def:Ns}, with $L=\min(k,s-1)$. $U^*_s$ is diagonally dominant by construction, with the maximum over all carry depths, so Proposition~\ref{prop:D} gives $\delta q^k\ge1-a^*_s/q$ for $q\ge5a^*_s+4$ and every $k\ge1$. \emph{Sharpness} ($k\ge2$). Let $i$ attain $a^*_s=\alpha_i$. For $q>2a^*_s$ the entries of $F_0$ are centred, so $\tilde M=F_0$. By (H2) the own-axis step $e_i$ from a cell whose last digit of $n_i$ is not $q-1$ changes coordinate $i$ by exactly $1-a^*_s/q$ cells if its first offset digit is at least $a^*_s$ (no wrap, no jitter), and every other coordinate $u$ by less than $(|\tilde M_{ui}|+1)/q$ if $r_u$ avoids the $|\tilde M_{ui}|$ values from which adding $\tilde M_{ui}$ wraps. For each $u\ne i$ the $t=0$ box with depths $k+1$ on $i$ and $u$ and $0$ elsewhere (total $2k+2\le sk$) makes $(n_i,r_i,n_u,r_u)$ jointly uniform (that box fixes exactly these $2k+2$ digits, so every value of them occurs equally often), so among the cells satisfying the conditions on $n_i$ and $r_i$ the fraction with a wrapping $r_u$ is $|\tilde M_{ui}|/q$; for $q>\sum_{u\ne i}|\tilde M_{ui}|$ a union bound leaves a cell. Its pair of points lies inside the unit cube, so \[ \delta N^{1/s}<\Bigl(\bigl(1-\tfrac{a^*_s}q\bigr)^2+\sum_{u\ne i}\frac{(|F_0[u][i]|+1)^2}{q^2}\Bigr)^{1/2}=1-\frac{a^*_s}q+O(q^{-2}). \] \emph{The bound on $a^*_s$.} Let $s\ge4$. By Lemma~\ref{lem:babai} it suffices to show $\eta_{d,j}\le \Xi_s:=2\sqrt{s-1}\,(s-2)!$ for every column $d$ and carry depth $j$; then every entry of $U^*_s$ is at most $\max_{d,j}\|E_{d,j}\|_2\le(3/2)^{s-2}\Xi_s$, and $a^*_s\le(s-1)(3/2)^{s-2}\Xi_s=2(s-1)^{3/2}(3/2)^{s-2}(s-2)!$. By Lemma~\ref{lem:adelic} it suffices to give, at each place of $\mathbb Q$, one monic $g$ of degree $j$. \emph{Finite column $d$.} Archimedean place: all nodes $y_c$ lie in $[-1,1]$ and $|w_c|\le1$, so the monic Chebyshev polynomial $2^{1-j}T_j$ ($j\ge1$; $g=1$ for $j=0$) gives $A_{j,\R}\le2\sqrt{s-1}$. At $p\in S$: let $\mathcal R_p$ be the finite rows $c\ne d$ with $p\mid c-d$, $e_c=v_p(c-d)\ge1$ for $c\in \mathcal R_p$, let $\mathcal R'\subset \mathcal R_p$ consist of $\min(j,|\mathcal R_p|)$ rows with the largest $e_c$, and $g_p=\prod_{r\in \mathcal R'}(X-y_r)\cdot X^{j-|\mathcal R'|}$. For $c\in \mathcal R'$ the entry vanishes. For $c,r\in \mathcal R_p$, $|y_c-y_r|_p=|(r-c)/((c-d)(r-d))|_p\le p^{\max(e_c,e_r)}$; so for $c\in \mathcal R_p\setminus \mathcal R'$ (then $|\mathcal R'|=j$ and $e_c\le e_r$ for $r\in \mathcal R'$) the entry has $p$-adic size at most $p^{e_c+\sum_{r\in \mathcal R'}e_r}\le p^{\sum_{r\in \mathcal R_p}e_r}$. For a finite row $c\notin \mathcal R_p$, and for the row $\infty$ ($y=0$, $w=1$), $|w_c|_p=1$, $|y_c|_p\le1$ and $|y_c-y_r|_p\le p^{e_r}$, so the size is at most $p^{\sum_{r\in \mathcal R'}e_r}$. Hence $A_{j,p}\le p^{\sum_{r\in \mathcal R_p}e_r}=p^{v_p(d!\,(\nu-d)!)}$, and $\prod_{p\in S}A_{j,p}\le d!\,(\nu-d)!\le \nu!$, since every prime factor of $d!(\nu-d)!$ lies in $S$. So $\eta_{d,j}\le2\sqrt{s-1}\,\nu!$. \emph{Column $\infty$.} The nodes are $y_c=c\in\{0,\dots,\nu\}$ with $w_c=1$. At the archimedean place the monic Chebyshev polynomial of $[0,\nu]$ gives $A_{j,\R}\le2\sqrt{s-1}\,(\nu/4)^j$. At $p\in S$ the falling factorial $X(X-1)\cdots(X-j+1)$ takes the values $j!\binom cj$, so $A_{j,p}\le|j!|_p$ and $\prod_pA_{j,p}\le1/j!$. So $\eta_{\infty,j}\le2\sqrt{s-1}\,(\nu/4)^j/j!\le2\sqrt{s-1}\,e^{\nu/4}\le2\sqrt{s-1}\,\nu!$ for $\nu\ge2$. Finally, with $\nu=s-2$ the elementary bounds $\nu^\nu e^{-\nu}\le \nu!\le \nu^\nu$ give $\bigl|\log\bigl(2(s-1)^{3/2}(3/2)^{s-2}(s-2)!\bigr)-s\log s\bigr|\le3s$ for $s\ge4$, so the bound is $e^{s\log s+O(s)}$; and $a^*_3=2$ is the display before Table~\ref{tab:astar}. (At the archimedean place the Chebyshev polynomial even gives $A_{j,\R}\le\sqrt{s-1}$ for a finite column, since $|2^{1-j}T_j|\le1$ on $[-1,1]$ for $j\ge1$; the factor $2$ in $\Xi_s$ is slack.) \end{proof} The bound is far from the computed values: $5.58\cdot10^7$ against $a^*_{10}=13640$ (Table~\ref{tab:astar}). The proof shows where the size comes from: the archimedean Chebyshev constants are harmless, and the cost sits at the primes $p\le s-2$, through the factorials $d!(s-2-d)!$ of the place differences. Whether other row scalars $\lambda'$, other places or another lower-bound argument bring the base threshold down to a polynomial in $s$ is open. \begin{figure}[tbp] \centering \IfFileExists{figs/thmA_design.pdf}{\includegraphics[width=\linewidth,height=0.62\textheight,keepaspectratio]{figs/thmA_design.pdf}}{\fbox{\parbox{0.9\textwidth}{\centering figure \texttt{thmA\_design} pending}}} \caption{\thmADesignCaption} \label{fig:thmA} \end{figure} \begin{remark}[The monomial design $U_s$]\label{rem:Us} The every-dimension theorem also holds with the monomial design $U_s$: put $L_s=\operatorname{lcm}(1,\dots,s-2)^{s-1}$, $\lambda'_c=L_s$ for finite $c$, $\lambda'_\infty=1$, and $F_n[c][d]=L_s/(c-d)^{n+1}$ for finite $c,d$, $F_n[\infty][d]=[n=0]$, $F_n[c][\infty]=L_s\binom cn$, with a diagonal that is dominant in the sense of Proposition~\ref{prop:D}. Its increments are monomials in the nodes, and the proof of Theorem~A applies verbatim with its constant $a_s=\max_c\alpha_c$ in place of $a^*_s$ (for $k\ge2$ the net property, the separation bound for $q\ge5a_s+4$ and the sharpness are formally verified for this design). But $\log a_s=(1+o(1))s^2$: for finite $c$, $\alpha_c\ge|F_0[c][\infty]|=L_s$, and $\log L_s=(s-1)\psi(s-2)$ with Chebyshev's function $\psi$, while $a_s\le L_s(s^2+2^{s-2})$; so $a_s=e^{\Theta(s^2)}$ already by Chebyshev's bounds $\psi(x)\asymp x$, and the constant $1$ uses the prime number theorem \cite{Had96,dlVP96}. In the language above the reason is $p$-adic: a monomial has all its roots at $0$, so at a collision of nodes it is as large as possible, $|y^j|_p=p^{j\,v_p(c-d)}$, at every carry depth and in every column, and the single row scalar $\lambda'$ must clear all of it. Theorem~A puts the roots near the colliding nodes, separately for each prime, and so replaces $e^{\Theta(s^2)}$ by the upper bound $e^{s\log s+O(s)}$ (Table~\ref{tab:astar}: $a_5=17407$ against $a^*_5=16$). \end{remark} \section{Lattice skeletons: \texorpdfstring{$m=sk+r$}{m=sk+r}}\label{sec:skel} For $N=p^m$, $m=sk+r$ with $1\le r\le s-1$, the boxes with one point each cannot all be cubes (regime~II of Section~\ref{sec:prelim}, Figure~\ref{fig:arcskeleton}). We give $r$ coordinates $k+1$ cell digits (the \emph{full} coordinates) and the other $\ell=s-r$ coordinates only $k$ (the \emph{skeleton} coordinates). The first offset digit of a skeleton coordinate, its \emph{skeleton digit}, places the point, along that coordinate, in one of the $p$ sub-intervals of side $p^{-(k+1)}$ of its cell. So we measure positions in \emph{fine cells}, the cubes of side $p^{-(k+1)}$: a cell (with the depths above) consists of $p^\ell$ fine cells, and each point lies in one of them. A \emph{coarse block} is a cube of side $p^{-k}$, fixed by the first $k$ digits of every coordinate; it consists of $p^s$ fine cells and holds $p^r$ points. The skeleton digits build the lattice (the `skeleton') on which the occupied fine cells lie: inside each coarse block they form a coset of a lattice of index $p^\ell$, a Construction~A lattice of an MDS code (Remark~\ref{rem:codes}). From one block to the next this coset is shifted by a small vector, a \emph{fault}; the faults come from the coefficients $g_{ij}$ of Lemma~\ref{lem:skelnec}(N2). So the separation, in fine cells, is at least the lattice minimum minus the largest fault minus one fine cell for the deeper digits (Lemma~\ref{lem:skelred}). One skeleton coordinate needs no faults (Theorem~\ref{thm:S1}); two force them (Theorems~\ref{thm:S2} and~\ref{thm:allm}). For three or more, a Pl\"ucker identity satisfied by every net in the frame orbit (Section~\ref{sec:frames}) rules out the Korobov skeleton digits of the nets $\mathcal B^1$ when the faults are small (Corollary~\ref{cor:frameid}); these residues are open for all large primes. \emph{Conventions.} Here $p$ is a prime, $s\ge3$ and $\ell=s-r$. The $r$ \emph{full} coordinates $x_1,\dots,x_r$ have depth $k+1$, the $\ell$ skeleton coordinates $z_1,\dots,z_\ell$ depth $k$, and $S_z$ denotes the skeleton digit of $z$. Level $l$ of the cells is $x_{d,l}$, $z_{j,l}$ (level $0$ is the most significant; the letter $x$ has nothing to do with the variable $x$ of Lemma~\ref{lem:Rs}). As in Section~\ref{sec:prelim}, the depth of a coordinate is its number of cell digits, and the last level of a coordinate of depth $k_c$ is at carry depth $0$, the level $k_c-1-n$ at carry depth $n$. In dimension three ($m=3k+r$) the coordinates are $(x,y,z)$, with depths $\mathbf k=(k+1,k,k)$ for $r=1$ ($\ell=2$) and $\mathbf k=(k+1,k+1,k)$ for $r=2$ ($\ell=1$), and cell levels $x_l$, $y_l$, $z_l$. \subsection{Fine cells and the separation reduction}\label{sec:fine} Fine cells have side $p^{-(k+1)}$ in every coordinate, and the fine unit is $\varpi=p^{m-k-1}$ integer units. Put $P_c=n_c$ for a full and $P_c=p\,n_c+S_c$ for a skeleton coordinate, where $n_c$ is the cell index as an integer. So $P_c$ is the position of the point along axis $c$ measured in fine cells: for a full coordinate its $k+1$ cell digits, for a skeleton coordinate its $k$ cell digits followed by the skeleton digit. Then $P_c\in[0,p^{k+1})$ is the \emph{fine cell} of coordinate $c$, and the integer coordinate is $X_c=P_c\varpi+\theta_c$ with $0\le\theta_c<\varpi$. Since $n_c=\lfloor P_c/p\rfloor$ for skeleton coordinates, $P$ determines all cell digits, hence the index: distinct points have distinct $P$. For a full coordinate, level $k-j$ is $\lfloor P_c/p^j\rfloor\bmod p$; for a skeleton coordinate, level $k-1-j$ is $\lfloor P_c/p^{j+1}\rfloor\bmod p$, and $P_c\equiv S_c\pmod p$. \begin{lemma}[Separation reduction]\label{lem:skelred}\label{lem:fine} Let $v\ne w$ be two points, and let $a_c$ be the centred residue ($-p^{k+1}/2(p^{k+1}-|a_c|-1)\varpi\ge(|a_c|-1)\varpi$, as $|a_c|\le p^{k+1}/2$. So $\delta N\ge(M-1)\varpi$, and $\varpi N^{1/s}/N=p^{m/s-k-1}=p^{-\ell/s}$. (ii) Write $P_c(w)=P_c(v)+a_c+\varepsilon p^{k+1}$ with $\varepsilon\in\{-1,0,1\}$. The difference of the floors is $\lfloor(P_c(v)+a_c)/p^j\rfloor-\lfloor P_c(v)/p^j\rfloor+\varepsilon p^{k+1-j}$. As $|a_c|\sqrt2\,B$ and $\nu_2,\dots,\nu_{r-1}$ be integers with $\nu_l^2+1>B$. Put $\zeta_l=\nu_l+i$ and $\Omega_l=\zeta_1\cdots\zeta_l\in\Z[i]$, and define $A\in\Z^{2\times r}$ by \[ A_1x+i\,A_2x=\sum_{l=1}^{r-2}x_l\,\Omega_l+(x_{r-1}+ix_r)\,\Omega_{r-1}. \] If $B(1+\|A_j\|_1)(p^{\ell/s}-1)/2$; this says nothing about (N1)--(N3) or reachability. \begin{definition}\label{def:G2rule} Let $s\ge5$, $r=s-2$ and $B\ge1$. Let $A(B)$ be the matrix of Lemma~\ref{lem:skellat}(G) with $\nu_1=\lfloor\sqrt2B\rfloor+1$ and $\nu_l=\lfloor\sqrt B\rfloor+1$ for $2\le l\le r-1$, and, for a prime $p$, let $B(p)$ be the largest integer $\beta\ge1$ with $\beta\,(1+\max_j\|A_j(\beta)\|_1)(2(r-1)/\pi)^2$. \end{enumerate} \end{lemma} \begin{proof} (a) Columns $r-1$ and $r$ of $A$ are $(\operatorname{Re}\Omega_{r-1},\operatorname{Im}\Omega_{r-1})$ and $(-\operatorname{Im}\Omega_{r-1},\operatorname{Re}\Omega_{r-1})$, so $\|A_j\|_1\ge|\Omega_{r-1}|=\prod_j(\nu_j^2+1)^{1/2}\ge\sqrt5\,(\nu_1^2+1)^{1/2}$, as $r\ge3$ and $\nu_l\ge2$. Hence $p>B\sqrt5\,(\nu_1^2+1)^{1/2}$. Since $\nu_1\le\sqrt2B+1$, we have $\nu_1^2+1\le2B^2+2\sqrt2B+2\le5B^2$ for $B\ge2$, and $\nu_1^2+1=5=5B^2$ for $B=1$. So $p>\nu_1^2+1\ge \nu_j^2+1$. (b) The arguments of all $\Omega_l$ lie in $(0,\pi/2)$, so the entries are nonzero integers, of absolute value $\sqrt B$ (Definition~\ref{def:G2rule}), so the sum is $<(r-1)/\sqrt B$. \end{proof} For $s\ge9$ part (c) gives no threshold of the form $B\ge g^*+1$: at $B=g^*+1$ the sum is $1.7194\ldots>\pi/2$ for $s=9$, and larger for $s\ge10$. \begin{remark}[Codes]\label{rem:codes} The lattices of this section are Construction~A lattices \cite[Ch.~5]{CS99} (stated there for binary codes; the $p$-ary version is the same): with the systematic $[s,r]$ code $C_A=\{(x,Ax):x\in\F_p^r\}\subset\F_p^s$ of the skeleton digits, $\Lambda_A=C_A+p\Z^s$. For every prime $p$ the nonzero vectors of $p\Z^s$ have maximum and Euclidean norm at least $p$, and the centred lift of a codeword is a shortest lift in every coordinate. So $\lambda_\infty(\Lambda_A)$ is the least largest centred entry of a nonzero codeword, and $\lambda_2(\Lambda_A)^2=\min(p^2,w_E(C_A))$, with $w_E$ the least Euclidean (squared Lee) weight. We call skeleton digits \emph{pure} if they depend only on the last levels $x_{d,k}$ of the full coordinates (as in Definition~\ref{def:S1}). For pure skeleton digits, $P_{x_d}\equiv x_{d,k}$ and $P_z\equiv S_z\pmod p$ constrain only the lowest fine digit, so the occupied fine cells lie in $\Lambda_A$ (for $\ell=1$ they fill it, Theorem~\ref{thm:S1}(2)); the faults shift $C_A$ to cosets that vary between coarse blocks. In this language: (a) Condition (N1) of Lemma~\ref{lem:skelnec} says that $C_A$ is MDS \cite[Ch.~11, Thm.~8]{MS77}, which says nothing about the Lee or Euclidean minimum: $A=(1,\dots,1)$ is MDS with $\lambda_\infty(\Lambda_A)=1$. (b) Inside the frame orbit the layer at carry depth $0$ is a generalized Cauchy matrix in the places (Proposition~\ref{prop:skelcomp}(i)), so $C_A$ is a generalized Reed--Solomon code \cite{RS85} (doubly extended if $s=p+1$). This restricts the code only if $r\ge3$ and $\ell\ge3$, since every MDS code of dimension or codimension at most two is generalized Reed--Solomon; for $r\le2$ or $\ell\le2$ the restriction comes instead from the faults: the code and the faults must be realized with the same row scalars $\lambda'$ (Proposition~\ref{prop:F}), which ties their values together. Lemma~\ref{lem:frameid} is the Pl\"ucker relation (Lemma~\ref{lem:CH}(X)) applied jointly to the code and the faults; Corollary~\ref{cor:frameid} concerns the Korobov column with bounded faults, not other codes. (c) For $\ell=1$, $\Lambda_A=\{a\in\Z^s:h\cdot a\equiv0\pmod p\}$ with the (parity-)check vector $\mathbf h=(\kappa_1,\dots,\kappa_r,-1)$ is the dual of the rank-$1$ lattice $\Z^s+\Z\mathbf h/p$ \cite{SJ94}, and $\lambda_2(\Lambda_A)$ is its spectral test \cite{CM67}, \cite[Sec.~3.3.4]{Knu97}; for $\kappa_d=\rho^d$ (Definition~\ref{def:S1}) it is Knuth's spectral-test lattice of the multiplier $\rho$ in dimension $s$, after $a_z\mapsto-a_z$. For $\rho=\lfloor p^{1/s}\rfloor$ the normalized minimum $\lambda_2(\Lambda_A)p^{-1/s}$ lies between $\rho p^{-1/s}$ and $\sqrt{\rho^2+1}\,p^{-1/s}$ and tends to $1$, against the upper bound $\sqrt{\gamma_s}$ valid for every $\mathbf h$ (Remark~\ref{rem:S1Elim}). So $\kappa_d=\rho^d$ is optimal in the maximum norm but not for the spectral test, which Proposition~\ref{prop:S1E} (Part~II) exploits. \end{remark} \needspace{8\baselineskip} \subsection{One skeleton coordinate} \begin{theorem}[One skeleton coordinate]\label{thm:S1} Let $s\ge3$, let $p$ be a prime, let $\rho\ge2$ be an integer with $\rho^s2^s$), and let $k\ge0$. The net $\mathcal S^1_{k,p,\rho}$ of Definition~\ref{def:S1}, whose skeleton digit is $S_z=\sum_d\rho^dx_{d,k}$, with $N=p^{sk+s-1}$ points, is a digital $(0,sk+s-1,s)$-net over $\F_p$, and: \begin{enumerate} \item its toroidal (hence also Euclidean) minimum distance satisfies $\delta N^{1/s}\ge(\rho-1)\,p^{-1/s}$; for $\rho=\lfloor p^{1/s}\rfloor$ the bound exceeds $(\rho-1)/(\rho+1)$, so it is $>1/3$ for every prime $p>2^s$, $>1/2$ for $p>3^s$, and tends to $1$; \item its occupied fine cells are exactly the cells of a lattice of index $p$ with minimum $\lambda_\infty=\rho$, which for $\rho=\lfloor p^{1/s}\rfloor$ is the largest minimum of any lattice of index $p$; \item for $\rho=\lfloor p^{1/s}\rfloor$ its covering radius satisfies $hN^{1/s}\le\sqrt s\,(\mu+1)\,p^{-1/s}$ with $\mu=\lceil p/\rho^{s-1}\rceil<\rho(1+1/\rho)^s+1$, so the mesh ratio $h/\delta\le\sqrt s\,(\mu+1)/(\rho-1)$ is bounded uniformly in $p$ and $k$, and is at most $\sqrt s\,(1+o(1))$ as $p\to\infty$. \end{enumerate} \end{theorem} \begin{definition}\label{def:S1} Let $s\ge3$, $\rho\ge2$ with $\rho^s2^s\ge s-1$, and the prescribed coefficients $\rho^d$ are units since $\rho^{s-1}2^s$) and $\ge1/2$ for $\rho\ge3$ ($p>3^s$). (2) There are $N=p^{s(k+1)-1}$ points with distinct $P$, all in $\Lambda_A$, and $\Lambda_A$ has $p^{s(k+1)}/p=N$ classes modulo $p^{k+1}\Z^s$. So the occupied fine cells are exactly the cells of $\Lambda_A$. Optimality is Lemma~\ref{lem:skellat}(L). (3) Let $Q\in\{0,\dots,p^{k+1}-1\}^s$ be the fine cell of a point $y\in[0,1)^s$. We look for a cell $a=Q-v\in\Lambda_A$ in the same range with $|v|_\infty\le\mu:=\lceil p/\rho^{s-1}\rceil$. Prescribe the sign $\epsilon_c$ of $v_c$: $-1$ if $Q_c0$, where $g^*=1+\lceil(s-2)/2\rceil$ and $B(p)\asymp p^{2/s}$ is defined in Definition~\ref{def:G2rule}. \item ($s\ge9$.) The same holds for every prime $p$ with $B(p)>g^*$ and $\sum_{j=1}^{s-3}\arctan(1/\nu_j)<\pi/2$ for the $\nu_j$ of Definition~\ref{def:G2rule} at $B=B(p)$; this holds if $B(p)>\max\{g^*,(2(s-3)/\pi)^2\}$, in particular for all sufficiently large $p$. \end{enumerate} \end{theorem} \begin{definition}\label{def:S2} Let $s\ge4$, $r=s-2$, $k\ge1$, $m=sk+s-2$. For $s=4$ let $A$ be the matrix of Lemma~\ref{lem:skellat}(T) with $\rho$ maximal such that $\rho^2-4132$, $440$, $2170$, $6095$ ($s=5,\dots,8$), which gives the primes $137$, $443$, $2179$, $6101$. These four numbers are $\Gamma(g^*+1)$ for $\Gamma(B)=B(1+\max_j\|A_j(B)\|_1)$, and $\Gamma$ is increasing on $B\ge g^*+1$ (below), so some $B\ge g^*+1$ has $\Gamma(B)0$, $b\ge0$; and $P(\nu_j+i)=(a\nu_j-b)+i(b\nu_j+a)$, so raising $\nu_j$ by $\Delta\ge0$ raises the real part by $a\Delta$ and the imaginary part by $b\Delta$. All parts stay positive, so every $|A_{jd}|$ is nondecreasing, $\|A_j(B')\|_1\ge\|A_j(B)\|_1$, and $\Gamma(B')>\Gamma(B)$. (The thresholds are also verified in Lean, by a different route: $\Gamma$ is evaluated exactly for $g^*+1\le B0$ and $a_{z_1}=1$, so $\xi_{z_1}\in\{0,1\}$ and $a_{z_2}=-\xi_{z_1}\le0$, a contradiction. The case $\xi_{z_2}=-1$ is symmetric, and $\xi_{z_2}=0$ gives $a_{z_1}=0$, $\xi_{z_1}=0$, $a_{z_2}=0$, contradicting $a\ne0$. Here the opposite signs of $g_{12}$ and $g_{21}$ give the contradiction, so no dependence of $S_{z_i}$ on its own cell is needed. So $M\ge\lambda_\infty(\Lambda_A)-g^*$, and Lemma~\ref{lem:skelred}(i) gives $\delta N^{1/s}\ge(\lambda_\infty(\Lambda_A)-g^*-1)p^{-2/s}$. With $\lambda_\infty\ge\rho-1$ and $g^*=2$ ($s=4$), resp. $\lambda_\infty\ge B+1$ (Lemma~\ref{lem:skellat}), this is the claim. For $s=4$, $\rho\ge5$ iff $p\ge23$, and $(\rho-4)/\sqrt p\to1$. \end{proof} Since $B(p)$ and $\rho(p)$ are nondecreasing, the bounds are positive for every $p\ge p_0$ (Table~\ref{tab:skel}); their limits and the growth of $p_0(s)$ are discussed in Remark~\ref{rem:skelconst}. \begin{remark}[Comparison with $m\equiv0$]\label{rem:skelr0} Seen one digit deeper, the case $m=sk$ fits this picture as the case $\ell=s$: every coordinate is a skeleton coordinate whose skeleton digit is its geometric digit, the occupied fine cells are a translate of all of $p\Z^s$, and the geometric layer $\tilde M$ plays the role of the faults; so $1-K/q$ is the analogue of ``$\lambda_\infty$ minus faults'' (up to the one fine cell more that Lemma~\ref{lem:skelred} loses). Since $p\Z^s$ attains the pigeonhole bound, only the scale of the faults differs: $K/q$ at $r=0$ against $g^*\,p^{-\ell/s}$ for $r\ne0$. This is why the thresholds ($p>2^s$ for $r=s-1$; $p\ge23$, $137$, $443$, $2179$, $6101$ for $r=s-2$, $s=4,\dots,8$) are far below the threshold $5a^*_s+4\le e^{s\log s+O(s)}$ of Theorem~\ref{thm:A}, while at a fixed large $p$ the constants for $r=0$ are larger: the skeletons gain in the range of $p$, not in the constant (heuristic comparison). \end{remark} \subsection{Three or more skeleton coordinates} For $\ell=2$ the frame orbit imposes nothing beyond (N1)--(N3) on skeleton digits of the form in Proposition~\ref{prop:skelcomp}(iii) (Corollary~\ref{cor:skelreach}). For $\ell\ge3$ it constrains the skeleton beyond (N1)--(N3). \begin{lemma}[A Pl\"ucker identity for skeleton digits]\label{lem:frameid} Let $\ell\ge3$, let a graph form be obtained from a frame over $\F_p$ by the moves of Lemma~\ref{lem:box}, and let $d$ be a full coordinate. Let $\kappa_i$ be the coefficient of $S_{z_i}$ on $x_{d,k}$, $g_{ij}$ that on $z_{j,k-1}$, and $\psi_i=1/\kappa_i$. Then, for distinct $i,j,l$, \[ g_{ji}\,g_{lj}\,\psi_j+g_{ij}\,g_{li}\,\psi_i=g_{li}\,g_{lj}\,\psi_l . \] \end{lemma} \begin{proof} The moves only scale the entries of the first offset rows on the last cell level, so by Proposition~\ref{prop:skelcomp}(i) these entries form a Cauchy layer $W_{ce}=\lambda_c\mu_e/D_{ce}$ in the places (Section~\ref{sec:frames}), with $W_{z_iz_j}=g_{ij}$ and $W_{z_ix_d}=\kappa_i$. Lemma~\ref{lem:CH}(X) for the coordinates $z_i,z_j,z_l,x_d$ reads $g_{ji}\kappa_l/(\kappa_jg_{li})+g_{ij}\kappa_l/(\kappa_ig_{lj})=1$; multiplying by $g_{li}g_{lj}/\kappa_l$ gives the identity. \end{proof} \begin{corollary}[Korobov skeleton digits need large faults]\label{cor:frameid} Let $\ell\ge3$, $k\ge1$, let $d$ be a full coordinate, and let $\rho,G\ge1$ be integers with $G^2<\rho$ and $p>G^2(2\rho^2+1)$. Then no graph form obtained from a frame over $\F_p$ by the moves of Lemma~\ref{lem:box} has, in the notation of Lemma~\ref{lem:frameid}, $\kappa_1,\kappa_2,\kappa_3\equiv\rho,\rho^2,\rho^3$ and centred $g_{ij}$ with $|g_{ij}|\le G$ for $i\ne j\le3$. \end{corollary} \begin{proof} Suppose such a graph form exists. The moves multiply every box minor by a unit, so it is a $(0,m,s)$-net, and $g_{ij}\ne0$ by (N2). Multiply the identity for $(i,j,l)=(1,2,3)$ by $\rho^3$: $g_{21}g_{32}\rho+g_{12}g_{31}\rho^2\equiv g_{31}g_{32}\pmod p$. The difference of the two sides is at most $G^2(\rho^2+\rho+1)0.438$ (toroidal). \end{proposition} \begin{proof} Write $\kappa=(4,7,8)$, $\beta=(-2,-1,-1)$ and $\eta=(1,0,0)$ for the coefficients of the $S_{z_i}$ on $x_k$, $x_{k-1}$, $x_{k-2}$, $g_{ij}$ and $H_{ij}$ for those of $S_{z_i}$ on $z_{j,k-1}$ and $z_{j,k-2}$ ($i\ne j$), and $\alpha=(0,1,3)$ for the places of $z_1,z_2,z_3$; all faults $g_{ij}$ are $\pm1$. \emph{Reachability.} By Proposition~\ref{prop:skelcomp}(i), Propositions~\ref{prop:F} and~\ref{prop:Gs} hold for the skeleton rows: column $x$ carries the three rows $S_{z_i}$ and the levels $k$, $k-1$, $k-2$ (the carry depths $0,1,2$), and column $z_j$ carries the two rows $S_{z_i}$, $i\ne j$, and the levels $k-1$, $k-2$; one vector $\lambda'$ of row scalars is shared by all columns. The coefficients of $S_{z_i}$ on its own cell are diagonal entries, free by move (a). By Proposition~\ref{prop:F} (with $w=1$, $y=\alpha_i$ on the column at $\infty$, and $w=y=1/(\alpha_i-\alpha_j)$ on the column $z_j$), the digits are reachable, with all deeper levels $0$, if there are units $\lambda'_i$, $\mu_x$, $\mu_{z_j}$ such that, modulo $13$: (R1) $\kappa_i=\lambda'_i\mu_x$; $\kappa_i+\beta_i=\lambda'_i(c_0+c_1\alpha_i)$ with $c_1\ne0$; and $\kappa_i+\beta_i+\eta_i=\lambda'_iG(\alpha_i)$ with $\deg G=2$; (R2) $g_{ij}=\lambda'_i\mu_{z_j}/(\alpha_i-\alpha_j)$, and $g_{ij}+H_{ij}=\lambda'_i(\alpha_i-\alpha_j)^{-1}\bigl(c'_0+c'_1/(\alpha_i-\alpha_j)\bigr)$ with $c'_1\ne0$, for each $j$ and the two $i\ne j$. Take $\lambda'=\kappa$ and $\mu_x=1$. For (R2) at level $k-1$ each column must give one value of $\mu_{z_j}=g_{ij}(\alpha_i-\alpha_j)/\kappa_i$ for both $i\ne j$: column $z_1$ needs $7\mu=-1$ and $8\mu=-3$, column $z_2$ needs $4\mu=1$ and $8\mu=2$, column $z_3$ needs $4\mu=3$ and $7\mu=2$, and indeed $(\mu_{z_1},\mu_{z_2},\mu_{z_3})=(-2,-3,4)$ solves all six. For (R1), $(\kappa_i+\beta_i)/\kappa_i=(2/4,6/7,7/8)=(7,12,9)=c_0+c_1\alpha_i$ with $(c_0,c_1)=(7,5)$. The remaining conditions say that leading coefficients of interpolation polynomials are nonzero: three values $\lambda'_iG(\alpha_i)$ at distinct $\alpha_i$ always come from a $G$ of degree at most $2$, and since $\det[\lambda',\lambda'\odot(c_0+c_1\alpha),\lambda'\odot G(\alpha)]$ is $\lambda'_1\lambda'_2\lambda'_3\,c_1$ times the leading coefficient of $G$ times a Vandermonde determinant, $\deg G=2$ is equivalent to $\det[\kappa,\beta,\eta]\ne0$; here $\det[\kappa,\beta,\eta]=7\cdot(-1)-8\cdot(-1)=1$. Likewise $c'_1\ne0$ in column $z_j$ is equivalent to $g_{ij}H_{lj}-H_{ij}g_{lj}\ne0$ for $\{i,l\}=\{1,2,3\}\setminus\{j\}$, and these are $1$, $-1$, $1$ for $j=1,2,3$. The places $\infty,0,1,3$ are distinct modulo $13$. So Proposition~\ref{prop:Gs} gives, for every $k\ge1$, moves of Lemma~\ref{lem:box} that turn the graph form $\Phi^F$ of the frame into one with these skeleton digits (truncated for $k=1$). The box minors of $\Phi^F$ are units (Lemma~\ref{lem:Ts}), the moves multiply them by units, and Lemma~\ref{lem:nie} gives the net property. \emph{Separation.} Let $a$ be the centred difference of the fine positions of two distinct points (Lemma~\ref{lem:skelred}). If some $|a_c|\ge13/2$, then $|a|_\infty\ge7$. Otherwise, as $P_x\equiv x_k$ and $P_{z_i}\equiv S_{z_i}\pmod{13}$, Lemma~\ref{lem:skelred}(ii) gives \[ a_{z_i}\equiv\kappa_ia_x+\beta_i\xi_{x,1}+\eta_i\xi_{x,2}+\sum_{j\ne i}\bigl(g_{ij}\xi_{z_j,1}+H_{ij}\xi_{z_j,2}\bigr)-[i=1]\,\xi_{z_1,1}\pmod{13}, \] with sign-consistent prefix patterns $(\xi_{c,1},\xi_{c,2})\in\{(0,0),(\sgn a_c,0),(\sgn a_c,\sgn a_c)\}$ (and $\xi=0$ for missing levels). We enumerate $|a_x|\le6$ and all patterns ($4625$ combinations, with both signs for the patterns of the $z_i$ before the sign check; rerun by the scripts of Appendix~\ref{app:verify}), take the centred $a_{z_i}$ and keep the sign-consistent solutions: the least $|a|_\infty$ of a nonzero solution is $M_{13}=4$. Lemma~\ref{lem:skelred}(i) gives $\delta N^{1/4}\ge(M_{13}-1)\,13^{-3/4}=0.4381\ldots$. \end{proof} Without the faults the occupied fine cells of a coarse block would lie on the Korobov lattice $\{u:u_{z_i}\equiv\kappa_iu_x\pmod{13}\}$, whose minimum is $\lambda_\infty=5$, against the pigeonhole bound $\lfloor13^{3/4}\rfloor=6$; the faults $\pm1$ cost one fine cell. The vector $\kappa$ is not a unit multiple of $(\rho,\rho^2,\rho^3)$ (as $\kappa_2/\kappa_1=5\ne3=\kappa_3/\kappa_2$ modulo $13$), in accordance with Corollary~\ref{cor:frameid}. \section{Dimension three, every resolution}\label{sec:dim3} In dimension three the cells can be cubes only for $m\equiv0\pmod3$, and a frame needs only three places, $0$, $1$ and $\infty$. For $m\equiv0$ one lower-triangular scramble, applied to all three coordinates of this frame, gives $t=0$ in every base $q\ge2$ and the separation $1-1/q$ for every $q\ge8$ (Theorem~\ref{thm:main}). The scramble is the Toeplitz matrix of $(1-t)^{-k}$, followed by a matrix that differs from the identity in three rows. It is one integer matrix whose box minors are all $\pm1$ (Theorem~\ref{thm:t0}), so $t=0$ holds in every base at once. The other resolutions use the other two regimes of the lattice picture (Section~\ref{sec:prelim}): at $m=3$ one geometric digit shared by all three coordinates (regime~III, the column nets, Theorem~\ref{thm:col}), and for $m\equiv1,2\pmod3$ skeleton digits on cells of unequal depths (regime~II, Theorem~\ref{thm:allm}, Remark~\ref{rem:B2}). Corollary~\ref{cor:allm} puts them together. \subsection{\texorpdfstring{$m\equiv0$}{m = 0}: the scramble}\label{sec:construction} Here $s=3$, $k\ge2$, $m=3k$, and $\Psi$, $T$ are the maps of the M\"obius frame. Let $\mathcal K$ be the lower-triangular Toeplitz matrix of $(1-t)^{-k}$: $\mathcal K_{ij}=\binom{i-j+k-1}{k-1}$ $(i\ge j)$, so $\mathcal K\,[W]_m=\bigl[W(1-t)^{-k}\bigr]_m$. The three rows of the second matrix are chosen so that the digit with index $k$, the first offset digit, depends only on the last two cell digits of each coordinate, through two small matrices (Proposition~\ref{prop:cell} and the paragraph after its proof). Let $\mathcal E$ be the identity except in three rows: for a digit vector $d$, \[ (\mathcal Ed)_{k-2}=d_0+d_1+\dots+d_{k-2},\qquad (\mathcal Ed)_{k-1}=-d_{k-1},\qquad (\mathcal Ed)_k=d_{k-1}+d_k, \] and put $L=\mathcal E\mathcal K$, lower-triangular with diagonal $(1,\dots,1,-1,1,\dots,1)$ ($-1$ at position $k-1$). \begin{definition}\label{def:net} $\mathcal P_{k,q}$ is the digital net over $\Z/q\Z$ with generator matrices $(LT^2,\,L,\,LT)$. For $V\in(\Z/q\Z)[t]_{<3k}$ the point has coordinates $\phi_q(d(W))$ for $W=T^2V,V,TV$, where $d(W)=L\,[W]_m$: \begin{align*} d_j&=\coef{j}\,W(1-t)^{-k}\quad(j\notin\{k-2,k-1,k\}),& d_{k-2}&=\coef{k-2}\,W(1-t)^{-k-1},\\ d_{k-1}&=-\coef{k-1}\,W(1-t)^{-k},& d_k&=\coef{k}\,(1+t)\,W(1-t)^{-k}. \end{align*} \end{definition} The same $L$ is applied to all three readings, so $\mathcal P_{k,q}$ inherits the cyclic symmetry of the frame. Definition~\ref{def:net} also makes sense for $k=1$ (row $k-2$ of $\mathcal E$ is absent); then $k=1$ is covered by Theorem~\ref{thm:t0} but not by the separation bound, and Figure~\ref{fig:slab} shows it. \subsection{The net property} \begin{theorem}\label{thm:t0} For every $k\ge1$, every box minor of $(LT^2,L,LT)$ equals $\pm1$. Hence $\mathcal P_{k,q}$ is a $(0,3k,3)$-net in base $q$ for every $q\ge2$. \end{theorem} In words: a box with all its prescribed digits zero asks one binary form of degree $m-1$ to vanish to total order $m$ at $0$, $1$ and $\infty$, which forces it to be zero. After the re-indexing of Proposition~\ref{prop:faure} this is Lemma~\ref{lem:hermite} \cite[Thm.~6.2]{Nie87}; we give the direct proof, which is also the one formalized. \begin{proof} From $(\Psi V)(t)=\sum_iv_i(1-t)^{m-1-i}$ one reads off, over any commutative ring: (F1) if $t^j\mid V$ then $\deg\Psi Vk}(d^c_j\bmod q)\,q^{k-j}\in[0,1). \] \begin{proposition}\label{prop:cell} The geometric digit of each coordinate depends only on the last two cell digits of the three coordinates, and the deeper digits of a coordinate do not depend on its own cell. Precisely, for every $V$, writing $d^x,d^y,d^z$ for the digit vectors of the three coordinates: (a) $d^x_k=-d^x_{k-1}-d^y_{k-1}+d^z_{k-1}-d^z_{k-2}$; (b) for $j>k$, $d^x_j$ is a function of the cells $(n_y,n_z)$ alone; (c) the same holds after the cyclic shift $(x,y,z)\mapsto(y,z,x)$. Consequently $r\equiv Mu+Jv\pmod q$ with \[ M=\begin{pmatrix}-1&-1&1\\1&-1&-1\\-1&1&-1\end{pmatrix},\qquad J=\begin{pmatrix}0&0&-1\\-1&0&0\\0&-1&0\end{pmatrix}, \] and $f_c$ does not depend on $n_c$. \end{proposition} \begin{proof} For $\deg F\le N$ put $\Psi_NF=(1-t)^NF(1/(1-t))$, so $\Psi=\Psi_{m-1}$, and $\Psi_{a+b}(FG)=\Psi_aF\cdot\Psi_bG$ whenever $\deg F\le a$, $\deg G\le b$ (on binary forms, $\Psi_N$ is the $N$-th symmetric power of the matrix $g$ of Section~\ref{sec:frames}). Moreover $\Psi_kt^k=1$, $\Psi_k(t-1)^k=t^k$, $\Psi_k(1-t)^k=(-t)^k$, $\Psi_k1=(1-t)^k$, and for $\deg F1-2/q+3/q^2$ when $|E|^2\ge2$ and $q\ge3$. The own-axis term $1-1/q$ of $c(q)$ is its component along the moving axis. The proof of Theorem~\ref{thm:sep} uses only this component: the transverse parts $\pm1/q$ are of the size of the jitter, which may cancel them, so the certified $c(q)$ is below $\lambda_1(\Lambda_q)$. $\Lambda_q$ has covolume $\det(I+M/q)<1$, so it has slightly more than one site per cell; reducing $Mu$ modulo $q$ skips the extra sites at the slip planes, so that each cell keeps exactly one point. Theorem~\ref{thm:sep} bounds the steps coordinatewise and does not use $\lambda_1(\Lambda_q)$. \end{remark} \begin{figure}[tbp] \centering \arcfig{arc_small} \caption{\arcSmallCaption} \label{fig:arcsmall} \end{figure} \needspace{8\baselineskip} \subsection{The separation bound} \begin{theorem}[Dimension three, $m\equiv0$]\label{thm:main} Let $q\ge 4$ and $k\ge 2$, and let $\mathcal P_{k,q}$ be the digital net of Definition~\ref{def:net}, with $N=q^{3k}$ points. Then: \begin{enumerate} \item $\mathcal P_{k,q}$ is a $(0,3k,3)$-net in base $q$ (Theorem~\ref{thm:t0}; this holds for every $q\ge2$); \item every cell $\prod_i[n_i q^{-k},(n_i+1)q^{-k})$ contains exactly one point, so $h_{\mathcal P}\le\sqrt3\,N^{-1/3}$; \item the toroidal (hence also the Euclidean) minimum distance satisfies $\delta_{\mathcal P}\ge c(q)\,N^{-1/3}$, where \[ c(q)^2=\min\Bigl\{\bigl(1-\tfrac1q\bigr)^2,\ \bigl(1-\tfrac2q\bigr)^2+\bigl(1-\tfrac4q\bigr)^2,\ 3\bigl(1-\tfrac3q\bigr)^2\Bigr\}. \] In particular $c(q)=1-1/q$ for all $q\ge8$ (Theorem~\ref{thm:sep}). \end{enumerate} \end{theorem} The constant is close to sharp for this net: for $q=8,\dots,13$ and $16$ the exact minimum distance at $m=6$ exceeds $c(q)$ by less than $4\cdot10^{-4}$ (Table~\ref{tab:c}), and for odd prime $q$ the first-order term $1-1/q$ cannot be improved by any digital $(0,3k,3)$-net over $\F_q$, $k\ge3$, whose geometric layer has bounded entries and a negative diagonal entry (the first-order ceiling, Corollary~\ref{cor:ceil3}). Compared with Theorem~\ref{thm:A} at $s=3$ ($a^*_3=2$), the net $\mathcal P_{k,q}$ has $\delta N^{1/3}\ge1-1/q$ against the $1-2/q$ of Theorem~\ref{thm:A}, half the loss, and needs no threshold beyond $q\ge4$. Part~(3) is the following theorem, which we state on its own (it is also a separate formal statement) and prove now. \begin{theorem}\label{thm:sep} For $q\ge4$ and $k\ge2$, the toroidal minimum distance of $\mathcal P_{k,q}$ is at least $c(q)\,q^{-k}$, with $c(q)$ as in Theorem~\ref{thm:main}. \end{theorem} \begin{proof} Work in cell units, with two points in cells $n$ and $n+e$ (toroidally). If some $|e_i|\ge2$, the $i$-th coordinates differ by at least $1\ge c(q)$; $e=0$ is impossible. So let $e\in\{-1,0,1\}^3\setminus\{0\}$, and let $\nu$ be the number of moving axes. \emph{The step.} For a moving axis $d$, $u_d$ changes by $e_d$ modulo $q$, and $v_d$ changes by $e_d$ if $u_d$ wraps ($\omega_d=1$; this includes the step across the torus boundary, $n_d$ from $q^k-1$ to $0$ or back, where all cell digits wrap; since $k\ge2$, $v_d$ then still changes by exactly $e_d$) and is unchanged otherwise ($\omega_d=0$). Hence $r'-r\equiv\sigma\pmod q$ with \begin{equation}\label{eq:sigma} \sigma_i=-e_i-e_{i+1}+(1-\omega_{i+2})\,e_{i+2}\qquad(\text{indices mod }3,\ x\to y\to z), \end{equation} $|\sigma_i|\le3$, and $\Delta_i=e_i+((r'_i-r_i)+(f'_i-f_i))/q$, where $r'_i-r_i\in\{\sigma_i,\ \sigma_i-q\sgn\sigma_i\}$ ($\{0\}$ if $\sigma_i=0$), $|f'_i-f_i|<1$, and $f'_i=f_i$ if only axis $i$ moves (Proposition~\ref{prop:cell}(b)). \emph{Per-coordinate bound.} Call a moving coordinate \emph{backward} if $e_i\sigma_i\le0$ and \emph{forward} otherwise. For a backward coordinate, $r'_i-r_i=\sigma_i$ gives $|\Delta_i|\ge1-(|\sigma_i|+1)/q$ and $r'_i-r_i=\sigma_i+qe_i$ gives $|\Delta_i|\ge2-(|\sigma_i|+1)/q$; as $q\ge4>|\sigma_i|$, both give $|\Delta_i|\ge c_{|\sigma_i|+1}$ with $c_j:=1-j/q\ge0$. Forward coordinates, and coordinates that do not move, are only bounded by $0$. \emph{Signs.} From \eqref{eq:sigma}: (i) If $e=\pm e_i$ then $\sigma_i=-e_i$ and there is no jitter, so $\Delta_i\in\{e_i(1-1/q),\ e_i(2-1/q)\}$ and $|\Delta|\ge c_1$. (ii) $e_i\sigma_i=-1-e_ie_{i+1}+(1-\omega_{i+2})e_ie_{i+2}$ is positive iff $e_{i+1}=-e_i$, $e_{i+2}=e_i$ and $\omega_{i+2}=0$. So a forward coordinate exists only if $\nu=3$, and it is unique; then $\sigma_{i+1}=(1-\omega_i)e_i$ (backward, $|\sigma|\le1$) and $\sigma_{i+2}=-(3-\omega_{i+1})e_i$ (backward, $|\sigma|\le3$). (iii) $|\sigma_i|=3$ forces $e_{i+1}=e_i$, $e_{i+2}=-e_i$, $\omega_{i+2}=0$, and then $\sigma_{i+2}=(1-\omega_{i+1})e_i$: backward with $|\sigma|\le1$. (iv) If $\nu=2$ with moving axes $\{i,i+1\}$, both are backward by (ii); either $e_{i+1}=e_i$ and $|\sigma_{i+1}|\le1$, or $e_{i+1}=-e_i$ and $\sigma_i=0$; the other value is at most $2$ in absolute value. \emph{Cases.} $\nu=1$: $|\Delta|\ge c_1$ by (i). $\nu=2$: $|\Delta|^2\ge c_2^2+c_3^2$ by (iv). $\nu=3$ with some backward $|\sigma_i|=3$: $|\Delta|^2\ge c_4^2+c_2^2$ by (iii). $\nu=3$ with a forward coordinate and all backward $|\sigma|\le2$: $|\Delta|^2\ge c_2^2+c_3^2$ by (ii). Otherwise three backward coordinates with $|\sigma|\le2$: $|\Delta|^2\ge3c_3^2$. As $c_3\ge c_4$, the minimum is $c(q)^2$. For $q\ge8$, $c_2^2+c_4^2\ge c_1^2$ (as $1-10x+19x^2\ge0$ for $x=1/q\le1/8$) and $3c_3^2\ge c_1^2$, so $c(q)=1-1/q$. \end{proof} For $s=3$ and the layer $(M,J)$ of $\mathcal P_{k,q}$, the quantity $\sigma$ of \eqref{eq:sigma} is the $\sigma$ of Lemma~\ref{lem:Ss}, and the proof above is that lemma together with the sign analysis (i)--(iv). The three terms of $c(q)$ come from an own-axis step, a twisted diagonal $e=\pm(1,1,-1)$ and the body diagonal; for $q\ge8$ the own-axis step binds. Dimension three is the largest in which one integer family has $t=0$ in every base (Remark~\ref{rem:base2}). \subsection{Resolution \texorpdfstring{$m=3$}{m=3}: column nets}\label{sec:m3} At $m=3$ a cell is a cube of side $q^{-1}$, and each coordinate has one cell digit and two offset digits. The nets of this section give all three coordinates the same two offset digits. Then every point is the corner of its cell moved along the body diagonal of the cell, and the net is a union of $q^2$ lines of points parallel to $\mathbf 1=(1,1,1)$, its \emph{columns} (Figure~\ref{fig:col}). For such nets the minimum distance can be computed exactly. Throughout this subsection $q\ge2$ is an integer, $\mathbf 1=(1,1,1)$, $\Sigma n=n_x+n_y+n_z$ (the coordinate sum, also for steps: $\Sigma E=\sum_cE_c$), and we work on the torus $(\Z/q^3\Z)^3$, the integer scale at which a cell has side $q^2$; so $\delta N^{1/3}$ is the toroidal distance divided by $q^2$. Let $\mathbb T=(\Z/q\Z)^3/\langle\mathbf 1\rangle$, a group with $q^2$ elements; the class $[n]\in \mathbb T$ of a cell $n\in\{0,\dots,q-1\}^3$ is its \emph{column}. A \emph{column point set} is a set of $q^3$ points of the torus, one for each cell, at \[ X(n)=q^2n+\varphi([n])\,\mathbf 1,\qquad \varphi:\mathbb T\to\{0,\dots,q^2-1\} \] (the \emph{heights}). The first base-$q$ digits of $X(n)$ are $n$, so every cell holds exactly one point, whatever the heights. The $q$ points of a column $[n]$ are $q^2(n+t\mathbf 1)+\varphi([n])\mathbf 1$, $t=0,\dots,q-1$: they lie on a closed line of the torus parallel to $\mathbf 1$, $q^2\sqrt3$ apart, and projected along $\mathbf 1$ the columns stand over the triangular lattice $q^2\Pi_\perp\Z^3$, $\Pi_\perp=I-\frac13\mathbf 1\mathbf 1^{\!\top}$ (the projection along $\mathbf 1$). \begin{definition}[Column nets]\label{def:col} Let $q\ge5$ with $\gcd(q,6)=1$, let $w=(-2,1,1)$ if $q\equiv2\pmod3$ and $w=(-1,-1,2)$ if $q\equiv1\pmod3$, and let $a=\frac13(q\mathbf 1+w)$, an integer vector with $a_x+a_y+a_z=q$ ($w$ is forced modulo $3$, $w\equiv-q\mathbf 1$, by the integrality of $a$, and $w\cdot\mathbf 1=0$). The \emph{column net} $\mathcal C_q$ is the digital net over $\Z/q\Z$ whose point with index $n\in(\Z/q\Z)^3$ has, in every coordinate $c$, the three digits \[ n_c,\qquad r(n)=a\cdot n,\qquad j(n)=n_y-n_x, \] i.e. the generator matrices with the rows $\e_c$, $a$ and $b:=(-1,1,0)$. \end{definition} So $a=(\frac{q-2}3,\frac{q+1}3,\frac{q+1}3)$ for $q\equiv2$ and $a=(\frac{q-1}3,\frac{q-1}3,\frac{q+2}3)$ for $q\equiv1\pmod3$, and the geometric layer is $\tilde M=\mathbf 1a^{\!\top}$: one geometric digit, shared by all three coordinates, with entries between $\frac{q-2}3$ and $\frac{q+2}3$. Because all three coordinates read the same two offset digits, each point lies on the main diagonal of its cell, and the distance between two points depends only on their cells and the difference of these shared digits (Theorem~\ref{thm:colmin}). Since $a\cdot\mathbf 1=q\equiv0$ and $b\cdot\mathbf 1=0$, both shared digits are functions of the column, and the integer coordinates are $X(n)=q^2n+\varphi([n])\mathbf 1$ with $\varphi=q\,r+j$: $\mathcal C_q$ is a column point set. The first theorem computes the minimum distance of every column point set. \begin{theorem}[Minimum distance of column point sets]\label{thm:colmin} Let $q\ge2$, let $X$ be a column point set, and for a direction $i\in\{x,y,z\}$ and a cell $n$ let $\Delta_i([n])=\varphi([n+\e_i])-\varphi([n])$. Then the toroidal minimum distance $d$ of $X$ at integer scale $q^3$ satisfies \[ d^2=\frac23q^4+3g^2,\qquad g=\min_{i,[n]}\operatorname{dist}\Bigl(\Delta_i([n]),\ \frac23q^2+q^2\Z\Bigr), \] and it is attained by two points in neighbouring columns $[n]$, $[n+\e_i]$. In particular $\delta N^{1/3}\ge\sqrt{2/3}$. \end{theorem} Here $\frac23q^2$ is the worst height step: two neighbouring columns whose heights differ by $\frac23q^2$ modulo $q^2$ hold points only $\sqrt{2/3}$ cells apart, and $g$ measures how far every step stays from it. \begin{proof} Let $n\ne n'$ be cells and $D$ a representative of $X(n')-X(n)$ modulo $q^3$. Then $D=q^2E+\Delta\mathbf 1$ with an integer vector $E\equiv n'-n\pmod q$ and $\Delta=\varphi([n'])-\varphi([n])$; every such $E$ occurs, and the toroidal distance of the pair is the least $|D|$. With $\Sigma E=E_x+E_y+E_z$, \[ |q^2E+\Delta\mathbf 1|^2=q^4|PE|^2+\tfrac13\bigl(q^2\Sigma E+3\Delta\bigr)^2,\qquad |PE|^2=\tfrac13\bigl((E_x-E_y)^2+(E_y-E_z)^2+(E_z-E_x)^2\bigr). \] The three differences $E_x-E_y$, $E_y-E_z$, $E_z-E_x$ are integers with sum $0$. If they all vanish, $E=t\mathbf 1$ with $t\in\Z$; then $n'\equiv n+t\mathbf 1$ lies in the column of $n$, $\Delta=0$, $t\not\equiv0\pmod q$, and $|D|^2=3t^2q^4\ge3q^4$. If one of them has absolute value at least $2$, the other two sum to its negative, so the three squares sum to at least $6$ and $|D|^2\ge2q^4$. Otherwise they are $\pm(1,-1,0)$ up to order, i.e. $E=\pm\e_i+t\mathbf 1$ for some $i$ and $t\in\Z$. For $E=\e_i+t\mathbf 1$ we get $D=q^2\e_i+U\mathbf 1$ with $U=\Delta+tq^2$, and \[ |q^2\e_i+U\mathbf 1|^2=(q^2+U)^2+2U^2=\tfrac23q^4+3\bigl(U+\tfrac13q^2\bigr)^2 . \] As $t$ runs over $\Z$, $U$ runs over $\Delta+q^2\Z$, so the least such value is $\frac23q^4+3\operatorname{dist}(\Delta,-\frac13q^2+q^2\Z)^2$, and $-\frac13q^2\equiv\frac23q^2\pmod{q^2}$; it is at most $\frac23q^4+3(\frac12q^2)^2<2q^4$. A pair with $E=-\e_i+t\mathbf 1$ is, read backwards, a pair with $E=\e_i-t\mathbf 1$. Every pair of neighbouring columns occurs (take any cell $n$ and $n'=n+\e_i$), so the minimum over all pairs is $\frac23q^4+3g^2$. \end{proof} \begin{theorem}[Resolution $m=3$]\label{thm:col} Let $q\ge5$ with $\gcd(q,6)=1$, and let $\mathcal C_q$ be the column net of Definition~\ref{def:col}, with $N=q^3$ points. \begin{enumerate} \item $\mathcal C_q$ is a digital $(0,3,3)$-net over $\Z/q\Z$; up to sign, its box minors are $1$, $a_x$, $a_y$, $a_z$ and $a_x+a_y$ (the last two modulo $q$; as integers the minors of the boxes with depth $3$ on $x$, $y$ and $z$ are, up to sign, $a_z$, $a_z$ and $a_x+a_y$). Every cell of side $q^{-1}$ holds one point, so $h\le\sqrt3\,N^{-1/3}$. \item Its toroidal minimum distance $d$ at integer scale $q^3$, and also its Euclidean one, is \[ d^2=\frac23q^4+3g^2,\qquad 3g=\begin{cases}q^2-q-3&\text{if }q\equiv2\pmod3,\\ q^2-2q&\text{if }q\equiv1\pmod3,\end{cases} \] with $g$ as in Theorem~\ref{thm:colmin}, that is, $d^2=q^4-\frac23q^3-\frac53q^2+2q+3$ resp.\ $q^4-\frac43q^3+\frac43q^2$, and \[ \delta N^{1/3}=1-\frac1{3q}-\frac8{9q^2}+O(q^{-3})\quad\text{resp.}\quad\delta N^{1/3}=1-\frac2{3q}+\frac4{9q^2}+O(q^{-3}). \] Two points are closest on the torus if and only if they differ by $\pm D$ modulo $q^3$, where $D=\frac13\bigl(-(2q^2-q-3),\,q^2+q+3,\,-(2q^2-q-3)\bigr)$ for $q\equiv2$ and $D=\frac13\bigl(-(2q^2-2q),\,-(2q^2-2q),\,q^2+2q\bigr)$ for $q\equiv1\pmod3$. \item Within each residue class of $q$ modulo $3$, $\delta N^{1/3}$ increases with $q$. It is $0.9059\ldots$ at $q=5$ and $0.9147\ldots$ at $q=7$, so $\delta N^{1/3}>0.9059$ for every such $q$, and $\delta N^{1/3}>0.9823$ for every such $q\ge37$. \end{enumerate} \end{theorem} \begin{proof} (1) The first rows are $\e_x,\e_y,\e_z$, so $\mathcal C_q$ is a graph form with the depths $(1,1,1)$, and we use Lemma~\ref{lem:box}. The box of shape $(1,1,1)$ has determinant $1$ and puts one point in every cell. A box with depth $2$ on $c$, $1$ on $c'$ and $0$ on $c''$ has, up to sign, the determinant $a_{c''}$; as $3a_{c''}=q+w_{c''}\equiv w_{c''}\in\{\pm1,\pm2\}$ and $\gcd(q,6)=1$, it is a unit. A box with depth $3$ on $c$ has, up to sign, the $2\times2$ minor of the rows $a$ and $b$ on the two columns other than $c$. Both rows sum to $0$ modulo $q$, so in each of them column $c$ is minus the sum of the other two modulo $q$, and modulo $q$ all three such minors equal $\pm\mu$ with $\mu=a_xb_y-a_yb_x=a_x+a_y$ (as integers they are $a_x+a_y$ and $\pm a_z$); and $3\mu=2q+w_x+w_y\equiv-w_z\in\{-1,-2\}\pmod q$, a unit. Lemma~\ref{lem:nie} concludes. (2) We apply Theorem~\ref{thm:colmin}. Let $\bar b_i\in\{0,\dots,q-1\}$ be the residue of $b\cdot\e_i$, so $\bar b=(q-1,1,0)$. A step from the cell $n$ to $n+\e_i$ changes $r$ by $a_i$ and $j$ by $\bar b_i$ modulo $q$, so \[ \Delta_i([n])=q\bigl(a_i-q\,[r(n)+a_i\ge q]\bigr)+\bigl(\bar b_i-q\,[j(n)+\bar b_i\ge q]\bigr). \] The map $[n]\mapsto(r(n),j(n))$ is a bijection from $\mathbb T$ onto $(\Z/q\Z)^2$, because $\mu$ is a unit; so the two wraps occur independently, in all four combinations (two if $\bar b_i=0$). Modulo $q^2$ the values of $\Delta_i$ are therefore $s_i$ and, if $\bar b_i\ne0$, $s_i-q$, where $s_i=qa_i+\bar b_i$, and $3\operatorname{dist}(\Delta_i,-\frac13q^2+q^2\Z)$ is the least value of $|3U+q^2|$ over $U\in s_i+\{0,-q\}+q^2\Z$. Here $3s_i+q^2$ lies between $2q^2-q+3$ and $2q^2+2q$, so the least value is attained at $U=s_i-q^2$ or $U=s_i-q^2-q$, where $3U+q^2=-(2q^2-3s_i)$ resp.\ $-(2q^2-3s_i+3q)$; every other $U$ gives at least $2q^2-4q+3>q^2+q$. The values $2q^2-3s_i$ for $i=x,y,z$ are \[ q^2-q+3,\quad q^2-q-3,\quad q^2-q\quad(q\equiv2),\qquad\qquad q^2-2q+3,\quad q^2+q-3,\quad q^2-2q\quad(q\equiv1), \] all positive, and adding $3q$ only increases them. So $3g$ is the least of them, $q^2-q-3$ resp.\ $q^2-2q$, attained only at $i=y$ (resp.\ $i=z$) with $U=s_i-q^2$, and Theorem~\ref{thm:colmin} gives $d^2=\frac23q^4+3g^2$. The closest differences are then $\pm(q^2\e_i+U\mathbf 1)=\pm D$ modulo $q^3$. The Euclidean minimum distance is at least $d$, and it is attained without a wrap: $\varphi([\mathbf 1])=0$ and $\varphi([\e_i])=qa_i+\bar b_i=s_i$, so the points of the cells $\mathbf 1$ and $\e_i$ differ by $q^2(\e_i-\mathbf 1)+s_i\mathbf 1=q^2\e_i+(s_i-q^2)\mathbf 1=D$ exactly. The expansions follow from $(\delta N^{1/3})^2=\frac23+\frac13(3g/q^2)^2$. (3) $3g/q^2=1-\frac1q-\frac3{q^2}$ resp.\ $1-\frac2q$ is positive and increases with $q$. The values at $q=5$ and $7$ are $\sqrt{513}/25$ and $\sqrt{2009}/49$, and at $q=37$ (the least $q\ge37$ coprime to $6$, in the class $q\equiv1$; the least in the other class is $41$, with $0.9913\ldots$) it is $0.9823\ldots$. \end{proof} \begin{figure}[tbp] \centering \IfFileExists{figs/col_net.pdf}{\includegraphics[width=\linewidth,height=0.5\textheight,keepaspectratio]{figs/col_net.pdf}}{\fbox{\parbox{0.9\textwidth}{\centering figure \texttt{col\_net} pending}}} \caption{\colNetCaption\interactive{anatomy.html}} \label{fig:col} \end{figure} \emph{Locally a reflected cube.} In cell units the point of cell $n$ is $n+\frac{\varphi([n])}{q^2}\mathbf 1$: it lies on the body diagonal of its cell. As $a/q=\frac13\mathbf 1+\frac1{3q}w$, two points whose cells differ by a bounded $E$ differ, in cell units, by \[ E+\Bigl(\frac{\Sigma E}3+z+\frac{w\cdot E}{3q}\Bigr)\mathbf 1+\frac{\Delta j}{q^2}\mathbf 1 \] for an integer $z$ (the wrap of $r$) and the change $\Delta j\in(-q,q)$ of the third digit. This is an $O(1/q)$ perturbation, \emph{along} $\mathbf 1$, of the point $E+\frac{z'}3\mathbf 1$, $z'=\Sigma E+3z$, of the set $\{E+\frac{z'}3\mathbf 1:z'\equiv\Sigma E\pmod3\}=H\Z^3$, where $H=I-\frac23\mathbf 1\mathbf 1^{\!\top}$ (for $v=E-\frac{z'+2\Sigma E}3\mathbf 1$ one checks $Hv=E+\frac{z'}3\mathbf 1$). Here $-H$ is the rotation by $180^\circ$ about the main diagonal, so $H\Z^3=-H\Z^3$ is the cubic lattice turned half a turn about $\mathbf 1$; we call it the \emph{reflected cube}. It has covolume $1$, minimum $1$ and minimal vectors $\pm H\e_c=\pm(\e_c-\frac23\mathbf 1)$. \emph{Stacking.} Seen along $\mathbf 1$, both $\Z^3$ and $H\Z^3$ are the triangular lattice $\Pi_\perp\Z^3$ (a copy of $A_2$) with lines of points of period $\sqrt3$ over each site. In $\Z^3$ the line over the neighbour $\Pi_\perp\e_i$ is raised by $\frac13$ of the period, in $H\Z^3$ by $\frac23$, i.e.\ lowered by $\frac13$ (in crystallographic terms, the stackings $ABC$ and $ACB$ of triangular planes). If neighbouring columns had equal heights (the stacking $AA$), points would be only $\sqrt{2/3}$ apart. In the notation of Theorem~\ref{thm:colmin}, the height steps of $\Z^3$ and of $H\Z^3$ are $\Delta\equiv0$ and $\Delta\equiv\frac13q^2\pmod{q^2}$, both at distance $\frac13q^2$ from this worst step $\frac23q^2$; the theorem measures every step by its distance from it. \emph{Drift and jitter.} The two perturbations are a \emph{drift} $\frac{w\cdot E}{3q}$, linear in $E$ and zero on $\mathbf 1$, and a \emph{jitter} $\Delta j/q^2$ of the third digit. Both are along $\mathbf 1$, so the columns are exact and only the heights deviate from the ideal, by $O(1/q)$. Section~\ref{sec:drift} shows that the losses $\frac1{3q}$ and $\frac2{3q}$ of Theorem~\ref{thm:col}(2) are forced, and that $\mathcal C_q$ is optimal among all column nets: going once around the torus along an axis takes $q$ steps, the ideal step of $H\Z^3$ is a third of the period, and $q/3$ is not an integer when $3\nmid q$, so the heights must deviate somewhere. Globally the set is not a lattice, since $qH\e_c\notin\Z^3$: at the wraps of $r$ the points jump back into their cells, and these are the slip planes of the column nets (Figure~\ref{fig:arclarge}). The drift vanishes on $\mathbf 1$ because $a\cdot\mathbf 1=q$, the condition that $q$ steps along $\mathbf 1$ close up; this is what makes $r$ a function of the column. \begin{figure}[tbp] \centering \arcfig{arc_large} \caption{\arcLargeCaption} \label{fig:arclarge} \end{figure} The column structure is special to $m=3$. The following box count, which holds for every net, digital or not, shows that at every other resolution not even two coordinates can share the first offset digit (the geometric digit). \begin{lemma}[Shared digits]\label{lem:shared} Let $P$ be a $(0,m,s)$-net in base $q$, digital or not, $j\ge0$, and $a\ne b$ two coordinates. If $2(j+1)\le m$, then the pair of the digits with index $j$ of $a$ and of $b$ takes every value in $\{0,\dots,q-1\}^2$ at exactly $q^{m-2}$ points of $P$. In particular, if $f$ is a map of digits such that for every point of $P$ the digit with index $j$ of coordinate $b$ equals $f$ of the digit with index $j$ of coordinate $a$, then $2(j+1)>m$. \end{lemma} \begin{proof} Let $2(j+1)\le m$ and choose the depths $d_a=j+1$, $d_b=m-j-1\ge j+1$ and $d_c=0$ for the other coordinates. Prescribing the digits $0,\dots,j$ of $a$ and of $b$ fixes the coordinate $a$ of an elementary box of this shape and the first $j+1$ digits of its coordinate $b$, and leaves $q^{m-2(j+1)}$ choices for the other digits of $b$; by $t=0$ each such box holds exactly one point. So each prescription is met by $q^{m-2j-2}$ points, and each value of the pair of digits with index $j$ by $q^{2j}\cdot q^{m-2j-2}=q^{m-2}$ points. A map $f$ as stated would leave the values $(\alpha,\beta)$, $\beta\ne f(\alpha)$, unmet. \end{proof} For $s=3$ and $m=3k$ the geometric digit has the index $k$, and $2(k+1)>3k$ only for $k=1$; so for $k\ge2$ no two coordinates of a $(0,3k,3)$-net can read the same geometric digit, let alone all three, and column nets exist only at $m=3$. At the other resolutions $m\ge4$ the lemma likewise excludes two coordinates that agree at the digit with index $\lfloor m/3\rfloor$, since $2(\lfloor m/3\rfloor+1)\le m$. So once $2(j+1)\le m$ the lemma excludes every relation between the digits with index $j$ of two coordinates, functional or not; it says nothing about relations among three coordinates, such as the rank-one layers of Section~\ref{sec:wnet} (Remark~\ref{rem:open}(7)). At $m=3$ the third digit does not matter at first order: with the shared second digit of a reflected-cube layer and an arbitrary third digit in each coordinate, $\delta N^{1/3}=1-O(1/q)$. \begin{proposition}[Arbitrary third digits]\label{prop:colfree} Let $q\ge2$, let $w\in\Z^3$ with $w_x+w_y+w_z=0$ and $w_c\equiv-q\pmod3$, $a=\frac13(q\mathbf 1+w)$, and $\beta\in\Z$. Let $P$ be a set of $q^3$ points of the torus $(\Z/q^3\Z)^3$, one for each cell $n\in\{0,\dots,q-1\}^3$, at \[ X(n)=q^2n+q\,r(n)\mathbf 1+j(n),\qquad r(n)\in[0,q),\ \ r(n)\equiv a\cdot n+\beta\pmod q,\qquad j(n)\in[0,q)^3 \] (integers; $j$ arbitrary). Then any two distinct points of $P$ have toroidal distance at least $q^2-\bigl((\|w\|_1+3)q-3\bigr)/\sqrt3$. \end{proposition} \begin{proof} Let $n\ne n'$ and let $D=q^2E+q\,\Delta r\,\mathbf 1+\Delta j$ be a representative of $X(n')-X(n)$ modulo $q^3$, with an integer vector $E\equiv n'-n\pmod q$, $\Delta r\in(-q,q)$ and $\Delta j\in(-q,q)^3$. If some $|E_c|\ge2$, then $|D_c|\ge2q^2-q(q-1)-(q-1)>q^2$. So let $E\in\{-1,0,1\}^3\setminus\{0\}$ and $\Sigma E=E_x+E_y+E_z$. The constant $\beta$ cancels: $\Delta r\equiv a\cdot E\pmod q$, and $3a\cdot E=q\Sigma E+w\cdot E$, so $3\Delta r=w\cdot E+zq$ with an integer $z\equiv\Sigma E\pmod3$. Hence $v=E-\frac{z+2\Sigma E}3\mathbf 1$ is an integer vector with $Hv=E+\frac z3\mathbf 1$, and \[ 3D=3q^2E+q\,(w\cdot E+zq)\mathbf 1+3\Delta j=3q^2Hv+R,\qquad R=q\,(w\cdot E)\mathbf 1+3\Delta j, \] with $|R_c|\le q\|w\|_1+3(q-1)$, since $|w\cdot E|\le\|w\|_1$. If $v\ne0$, then $|D|\ge q^2|Hv|-|R|/3\ge q^2-\bigl((\|w\|_1+3)q-3\bigr)/\sqrt3$, as $H$ is orthogonal. If $v=0$, then $E=t\mathbf 1$ with $3t=z+2\Sigma E=z+6t$, so $z=-3t$ and, as $w\cdot\mathbf 1=0$, $\Delta r=-tq$; then $|\Delta r|q^2$. So let $E\in\{-1,0,1\}^3\setminus\{0\}$, and let $\vartheta\in[0,q)$ be the residue of $a\cdot E$. The constants $\beta_c$ cancel: $\Delta r_c\equiv a\cdot E$ and $|\Delta r_c|0$, so $\xi_y\ge2$, impossible; if $\xi_z=-1$ then $a_z<0$ and $\xi_y\le-2$, impossible. So $\xi_z=0$, $a_y=0$, $\xi_y=0$ and $a_z=0$, contradicting $a\ne0$. Hence $M\ge\rho^2-3$, and Lemma~\ref{lem:skelred}(i) with $\ell=2$ gives $\delta N^{1/3}\ge(\rho^2-4)p^{-2/3}$. \emph{Constants.} Let $c_1(p)=(\rho^2-4)p^{-2/3}$ be the bound with $\rho$ maximal. Then $p<(\rho+1)^2(\rho+2)$, hence \[ c_1(p)^3>G(\rho)=\frac{(\rho-2)^3(\rho+2)}{(\rho+1)^4}. \] $G$ increases for $\rho\ge3$, since $(\log G)'=9/((\rho-2)(\rho+1))-1/((\rho+1)(\rho+2))$ is positive, and tends to $1$. Since $\rho\ge27$ iff $p\ge20412$, and $G(27)^{1/3}=0.9033\ldots$, the bound exceeds $0.9$ from this prime on. \end{proof} The case $u=0$ uses the own-coordinate term $-z_{k-1}$ of $S_z$: with $+z_{k-1}$ or $0$ there and equal signs of the faults this argument does not apply. Like that of Remark~\ref{rem:B2}, the bound is a sawtooth in $p$. \emph{Lattice reading.} For $\mathcal B^1$ the skeleton digits give, with $P_c$ the fine cell of coordinate $c$ (Section~\ref{sec:fine}), \[ P_y\equiv\rho P_x+\lfloor P_z/p\rfloor,\qquad P_z\equiv\rho^2P_x+\lfloor P_x/p\rfloor+\lfloor P_y/p\rfloor-\lfloor P_z/p\rfloor\pmod p, \] so inside each coarse block the occupied fine cells (one in $p^2$) form a coset of $\Lambda_{\mathcal B^1}$, the Korobov lattice with generating vector $(1,\rho,\rho^2)$ and modulus $p$ \cite{Kor59,SJ94}, shifted by unit \emph{faults} between neighbouring coarse blocks (Figure~\ref{fig:Blattice}(c)). The choice $\rho\approx p^{1/3}$ makes the shortest (primal) vector as long as possible for a lattice of index $p^2$, up to a constant (the pigeonhole bound after Lemma~\ref{lem:skelred}). For integration it would be a poor Korobov generator: $(\rho,-1,0)$ lies in the dual lattice, so its Zaremba index is at most $\rho$. Separation (primal) and integration (dual) favour different generators. \subsection{Every resolution}\label{sec:everym} Together, the nets of this section cover every resolution $m\ge3$ for every large prime. \begin{corollary}[Every prime $p\ge37$, every $m\ge3$]\label{cor:allm} For every prime $p\ge37$ and every $m\ge3$ the nets of Theorems~\ref{thm:col} ($m=3$), \ref{thm:main} ($m\equiv0$, $m\ge6$), \ref{thm:allm} ($m\equiv1$) and~\ref{thm:S1} with $s=3$ ($m\equiv2$; the nets $\mathcal B^2$ of Remark~\ref{rem:B2}), with $\rho$ maximal, are explicit digital $(0,m,3)$-nets over $\F_p$ with \[ \delta N^{1/3}>0.2715,\quad\text{and}\quad\delta N^{1/3}>0.4250\ \text{if } p\ge83, \] uniformly in $m$, and the lower bound tends to $1$ as $p\to\infty$. Their covering radius satisfies $hN^{1/3}\le\sqrt3$ for $m\equiv0$, $hN^{1/3}\le p^{1/3}\sqrt{2+p^{-2}}$ for $m\equiv1$, and $hN^{1/3}\le\min\{p^{2/3}\sqrt{1+2p^{-2}},\,4.264\}$ for $m\equiv2\pmod 3$. So for each such $p$ the mesh ratio $h/\delta$ is bounded uniformly in $m$: the nets are quasi-uniform along every $N=p^m$, $m\ge3$. For $m\equiv2$ the mesh ratio is moreover at most $7\sqrt3<12.2$ for every prime $p\ge11$ and every $m\ge5$ (Theorem~\ref{thm:S1}(3)). \end{corollary} \begin{proof}[Proof of Corollary~\ref{cor:allm}] Take $\mathcal C_p$ for $m=3$ (Theorem~\ref{thm:col}; for $p\ge37$, $\delta N^{1/3}>0.9823$, and its covering radius is at most $\sqrt3N^{-1/3}$ because every cell holds a point), $\mathcal P_{k,p}$ for $m=3k\ge6$, $\mathcal B^1_{k,p,\rho}$ for $m=3k+1\ge4$ and $\mathcal B^2_{k,p,\rho}$ for $m=3k+2\ge5$, with $\rho$ maximal: $\rho_1$ and $\rho_2$, the largest integers with $\rho_1^2(\rho_1+1)\le p$ and $\rho_2^30.508$, and $1-1/p>0.97$. If $p\ge83$ then $\rho_1\ge4$ and $\rho_2\ge4$, and the bounds exceed $G(4)^{1/3}=(48/625)^{1/3}>0.4250$ and $(\rho-1)/(\rho+1)=3/5$ at $\rho=4$ (Remark~\ref{rem:B2}). All bounds tend to $1$. For the covering radius: by $t=0$ the elementary box with depths $\mathbf k$ (resp. $(k,k,k)$) around any point of the torus contains a net point, and its diagonal is $p^{-k}\sqrt{2+p^{-2}}=p^{1/3}\sqrt{2+p^{-2}}\,N^{-1/3}$ for $m\equiv1$ and $p^{-k}\sqrt{1+2p^{-2}}=p^{2/3}\sqrt{1+2p^{-2}}\,N^{-1/3}$ for $m\equiv2$. For $m\equiv2$, Theorem~\ref{thm:S1}(3) with $s=3$ gives $hN^{1/3}\le\sqrt3(\mu+1)p^{-1/3}$ and $h/\delta\le\sqrt3(\mu+1)/(\rho-1)$, $\mu=\lceil p/\rho^2\rceil$; over all primes $p\ge11$ the maxima are $4.2633\ldots<4.264$ and $7\sqrt3$, both at $p=23$ (for $\rho\ge4$ use $\mu<\rho(1+1/\rho)^3+1$). \end{proof} For $m\equiv1$ the mesh ratio obtained in this way grows like $p^{1/3}$. Whether a bound uniform in $p$ holds is open (Remark~\ref{rem:open}(5); for $m\equiv2$ it is Theorem~\ref{thm:S1}(3)). (Heuristic for one: the cosets of $\Lambda_{\mathcal B^1}$ have reduced bases of about $\rho^2$ fine cells, i.e. $O(N^{-1/3})$; a proof has to handle the faults between coarse blocks, and we have not given one.) The separation bounds and the lattice reading use only the skeleton digits: the bounds of Theorem~\ref{thm:allm} hold for every graph form over $\F_p$ with the depths $\mathbf k$ and the skeleton digits of Definition~\ref{def:B}, net or not. At $m=3$ the constant can exceed $1$ (Part~II). If two coordinates share the geometric digit and the third reads it with the factor $\frac12$ modulo $p$, the points lie locally on two cosets of a lattice, with the limit $\sqrt{66}/8=1.0155\ldots$ (Proposition~\ref{prop:twocoset}). If all three geometric digits are multiples of one digit, the three coordinates wrap together, and $\delta N^{1/3}>1$ for every prime $p\ge17$, with the limit $\sqrt{281}/16=1.0476\ldots$ (Proposition~\ref{prop:wnet}). For small primes, the per-prime nets of Remark~\ref{rem:perprime} give larger constants and close most of the remaining gaps: \begin{remark}[Per-prime designs]\label{rem:perprime} For small primes, computer-found designs give larger constants. They are graph forms with the depths $\mathbf k$, built by the same moves from the frame, in which the skeleton digits and the next offset digit of each coordinate (which controls the jitter) are prescribed on the last three or four levels; their skeletons are index-$p^2$ Korobov-type ($m\equiv1$) and index-$p$ congruence lattices ($m\equiv2$). A finite computation analogous to Lemma~\ref{lem:Ss} (it enumerates the steps between nearby fine cells, all carry patterns and both representatives of each digit difference) certifies the constants of Table~\ref{tab:perprime} for every $m=3k+1\ge10$ resp. $m=3k+2\ge8$, since the prescribed rows do not depend on $k$; they exceed $1$ for $p=47$, $53$, $59$, $61$, $71$, $73$, $79$ ($m\equiv1$) and $p=41$, $43$, $59$, $61$, $79$ ($m\equiv2$). This separation bound is Proposition~\ref{prop:ppsep}, and it holds for every digital shift as well (Remark~\ref{rem:ppshift}). The existence of such a $(0,m,3)$-net for every $k$ (Propositions~\ref{prop:pp2} and~\ref{prop:pp1}, Appendix~\ref{app:perprime}) rests on quantities that the moves of Lemma~\ref{lem:box} do not change and that do not depend on $k$ (Lemma~\ref{lem:ppinv}: five for $m\equiv1$, two for $m\equiv2$), and on finitely many exact checks per design (proofs in Appendix~\ref{app:perprime}; formalization status in Section~\ref{sec:formal}). \end{remark} Together these results give, for every prime $p\ge5$, explicit digital $(0,m,3)$-nets over $\F_p$ at the following resolutions. At $m=3$, Theorem~\ref{thm:col} gives $\delta N^{1/3}>0.9059$, and Theorem~\ref{thm:main} covers every $m\equiv0$, $m\ge6$. For $m\equiv1$, $m\ge10$, $\delta N^{1/3}>0.4250$ (Theorem~\ref{thm:allm}, and for $p\le79$ the per-prime certificates of Remark~\ref{rem:perprime}). For $m\equiv2$, $m\ge8$, $\delta N^{1/3}>0.5$ (Remark~\ref{rem:B2}, and for $p\le79$ the per-prime certificates); for $p\ge11$ the nets $\mathcal B^2$ of Remark~\ref{rem:B2} also cover $m=5$. Not covered are $m\in\{4,7\}$ for $5\le p\le31$ and $m=5$ for $p\le7$; finitely many resolutions do not matter for the question of Dick, Goda and Suzuki (Remark~\ref{rem:dgsfinite}). In the bases $2$ and $3$ every $m$ is covered (Theorem~\ref{thm:base23}). \section{Bases two and three: the deep-shift lift}\label{sec:base23} The constructions so far need large bases. The net of Theorem~\ref{thm:main} is a $(0,3k,3)$-net in every base, but its separation bound needs $q\ge4$; the other residues need large primes (Corollary~\ref{cor:allm}). The bases two and three, the most common ones in practice, need another idea. \begin{remark}[Why $s=3$, and base two]\label{rem:base2} A $(0,m,s)$-net in base $2$ with $m\ge2$ requires $s\le3$ \cite[Cor.~5.11]{Nie87}; for digital nets over rings see \cite[Thm.~2]{LNS96}. So $s=3$ is the largest dimension in which one set of integer generator matrices can give $t=0$ in every base (for dimension $s$ see the text after Lemma~\ref{lem:Ts}). In the frame language of Section~\ref{sec:frames}: $t=0$ in every base needs places whose pairwise differences are $\pm1$; $\{0,1\}$ is the largest set of integers with this property (in the language of exceptional units \cite{Len77}), and the place $\infty$ gives the third coordinate. \end{remark} In base $b\in\{2,3\}$ we therefore build the nets recursively, $m\to m+3$, from a finite seed. Each step adds one digit at the top of every coordinate and two at the bottom. The three new top digits form a $(0,3,3)$-net in base $b$, so they split the cube into $b^3$ \emph{top boxes} of side $1/b$. Every top box receives a copy of the old net, and the two new bottom digits are jitter. Neighbouring copies differ by digital shifts that change only the lower half of the old digits. The upper half of every old point therefore never changes (it is \emph{frozen}), and the separation that the old net has at the scale of its upper half survives every later step. This is the mechanism of the introduction again, with the old net in the role of the lattice and the shifted lower digits as jitter (Figure~\ref{fig:base23}). \emph{Setting.} Let $b$ be a prime and $m_0\ge1$. A digitally shifted digital $(0,m_0,3)$-net over $\F_b$ is a set $P=\{\xi(n)=Cn+\tau:n\in\F_b^{m_0}\}$ with $C=(C_x,C_y,C_z)$, $C_c\in\F_b^{m_0\times m_0}$, and a shift $\tau\in\F_b^{3m_0}$; a point has the three digit vectors $\xi_c(n)\in\F_b^{m_0}$, most significant digit first. For a pair of coordinates $(c,c')\in\{(y,z),(x,y),(x,z)\}$ let $P_{cc'}=\{(C_cn,C_{c'}n):n\in\F_b^{m_0}\}\subset\F_b^{m_0}\times\F_b^{m_0}$, an $m_0$-dimensional subspace (a digital $(0,m_0,2)$-net). For $1\le e\le m_0$ let $Z_e\subset\F_b^{m_0}\times\F_b^{m_0}$ be the vectors that vanish on the first $e$ digits of $c$ and on the first $m_0+1-e$ digits of $c'$. Since the map from $P_{cc'}$ to the first $e$ digits of $c$ and the first $m_0-e$ digits of $c'$ (the box of shape $(e,m_0-e)$) is injective, $P_{cc'}\cap Z_e=0$, so $H_e=P_{cc'}\oplus Z_e$ is a hyperplane; let $\lambda^{cc'}_e$ be a nonzero linear functional with kernel $H_e$. A point of $P_{cc'}+t\,s$, $t\ne0$, and a point of $P_{cc'}$ fall into the same elementary box of shape $(e,m_0+1-e)$ only if $s\in H_e$; so $\lambda^{cc'}_e(s)\ne0$ says that adding multiples of $s$ to the pair $(c,c')$ creates no collision in boxes of that shape. \begin{definition}[Deep-shift lift]\label{def:lift} Let $P$ be as above, $0\le D$, and let $s_A,s_B\in\F_b^{3m_0}$ vanish on the first $D$ digits of each coordinate, i.e.\ change only the \emph{deep} digits, those of index $>D$. They are \emph{separating} if, for every $1\le e\le m_0$, \[ \lambda^{yz}_e\bigl(s_A^{yz}\bigr)\ne0,\qquad \lambda^{xy}_e\bigl(s_B^{xy}\bigr)\ne0,\qquad \lambda^{xz}_e\bigl((s_B-s_A)^{xz}\bigr)\ne0, \] where $s^{cc'}$ is the restriction of $s$ to the coordinates $c,c'$. The \emph{lift} $L_D(P)$ is the set of $b^{m_0+3}$ points with index $(n,\alpha,\beta_1,\beta_2)\in\F_b^{m_0+3}$ whose coordinates have the $m_0+3$ digits \[ \begin{aligned} x&:\ \alpha\ \bigm|\ (\xi(n)+s_\beta)_x\ \bigm|\ \beta_2,\ \beta_1,\\ y&:\ \alpha+\beta_1\ \bigm|\ (\xi(n)+s_\beta)_y\ \bigm|\ \beta_1+\beta_2,\ \beta_1,\\ z&:\ \alpha+\beta_1+\beta_2\ \bigm|\ (\xi(n)+s_\beta)_z\ \bigm|\ \beta_1,\ \beta_2, \end{aligned} \qquad s_\beta=\beta_1s_A+\beta_2s_B, \] one \emph{top} digit, $m_0$ \emph{block} digits and two \emph{jitter} digits. The shift $\tau$ of $P$ stays in the block digits, so $L_D(P)$ is again a digitally shifted digital net over $\F_b$. \end{definition} \begin{lemma}[Lift]\label{lem:lift} Let $P$ be a digitally shifted digital $(0,m_0,3)$-net over $\F_b$. \begin{enumerate} \item[(A)] If $s_A,s_B$ are separating, then $L_D(P)$ is a digitally shifted digital $(0,m_0+3,3)$-net over $\F_b$. \item[(B)] If $2D\le m_0$, separating $s_A,s_B$ exist. \end{enumerate} \end{lemma} There are $m_0$ conditions for each pair of coordinates, one for each $e$. Each $\lambda_e$ involves a deep digit, and $2D\le m_0$ makes these deep digits distinct, so the conditions can be met independently. \begin{proof} (A) Let $(d_x,d_y,d_z)$ be a box shape with $\sum_cd_c=m_0+3$; by Lemma~\ref{lem:nie} it suffices that the box map, from the index to the prescribed digits, is injective. \emph{All $d_c\ge1$.} The top digits $(\alpha,\alpha+\beta_1,\alpha+\beta_1+\beta_2)$ determine $(\alpha,\beta)$. The remaining depths $d_c-1$ sum to $m_0$ and are at most $m_0$ each, so no jitter digit is prescribed, and what remains is an $m_0$-box of the shifted net $P+s_\beta$, which is injective. \emph{Exactly one $d_c=0$}, say $d_x=0$ (the cases $d_y=0$, $d_z=0$ are the same with the pairs $xz$ and $xy$). The top digits of $y$ and $z$ fix $\alpha+\beta_1$ and $\beta_2$, and the remaining index is $(n,t)$ with $t=\beta_1$; the block digits of $(y,z)$ are those of $P_{yz}+\text{const}+t\,s_A^{yz}$. If $d_y-1=e$ and $d_z-1=m_0+1-e$ with $1\le e\le m_0$, injectivity on $(n,t)$ is injectivity of the projection to the first $e$ resp.\ $m_0+1-e$ digits on $P_{yz}+\F_bs_A^{yz}$; it holds on $P_{yz}$ by the $(e,m_0-e)$ box, and for $t\ne0$ a collision would put $s_A^{yz}$ in $H_e=\ker\lambda^{yz}_e$, which the separating condition excludes. If $d_y=1$ or $d_z=1$, the other coordinate reads all of its block and one jitter digit, $\beta_1$ for $z$ and $\beta_1+\beta_2$ for $y$, which determines $t$, and then the block determines $n$. For $d_y=0$ the pair is $(x,z)$ with the line $s_B-s_A$ and the jitter digits $\beta_2$, $\beta_1$; for $d_z=0$ it is $(x,y)$ with $s_B$ and $\beta_2$, $\beta_1+\beta_2$. \emph{Two $d_c=0$.} One coordinate carries all $m_0+3$ digits: its top digit, its block (a bijection in $n$ for fixed $\beta=(\beta_1,\beta_2)$) and its two jitter digits, which are a bijection of $\F_b^2$ onto $(\beta_1,\beta_2)$ in every coordinate. (B) The functional $\lambda_e=\lambda^{cc'}_e$ vanishes on $Z_e$, so it is supported on the first $e$ digits of $c$ and the first $m_0+1-e$ digits of $c'$. Its coefficient on digit $e$ of $c$ is nonzero: otherwise $\lambda_e$ would live on a window of $e-1$ plus $m_0+1-e$ digits, onto which $P_{cc'}$ projects bijectively (the $(e-1,m_0+1-e)$ box), and vanish on $P_{cc'}$, hence be $0$. Likewise its coefficient on digit $m_0+1-e$ of $c'$ is nonzero. If $e>D$, digit $e$ of $c$ is \emph{deep} (index $>D$); if $e\le D$, digit $m_0+1-e\ge m_0+1-D>D$ of $c'$ is deep. So each $\lambda_e$ has a nonzero coefficient on a deep digit, its own: digit $e$ of $c$ if $e>D$, digit $m_0+1-e$ of $c'$ if $e\le D$. In any vanishing linear combination, the term with the largest $e>D$ (or, if there is none, the smallest $e$) is the only one that sees its own digit, so its coefficient is $0$; by induction all coefficients are $0$. Hence the restrictions of $\lambda_1,\dots,\lambda_{m_0}$ to the deep digits are linearly independent, and the deep parts of $s_A^{yz}$ can be chosen with $\lambda^{yz}_e(s_A^{yz})=1$ for all $e$, those of $s_B^{xy}$ with $\lambda^{xy}_e(s_B^{xy})=1$; the conditions on $(s_B-s_A)^{xz}$ involve the still free deep digits of $(s_A)_x$ and $(s_B)_z$, and are solved in the same way. \end{proof} A \emph{deep-shift chain} is a sequence $P_{m_0}\to P_{m_0+3}\to\cdots$ of lifts $P_{\ell+3}=L_{D_\ell}(P_\ell)$ with $2D_\ell\le\ell$, where $P_\ell$ has $b^\ell$ points (the resolution $\ell$); its \emph{canonical continuation} from a resolution $\ell_0$ on uses $D_\ell=\lfloor\ell/2\rfloor$. For a resolution $L$ write $A=b^{\lfloor L/2\rfloor}$ (the number of values of the top $\lfloor L/2\rfloor$ digits) and $W=b^{L-\lfloor L/2\rfloor}$, read the coordinates of a point of $P_L$ as integers $X_c\in[0,b^L)$, and let $t_c=\lfloor X_c/W\rfloor$ be the integer of its top $\lfloor L/2\rfloor$ digits. Put \[ v^*_L=\min_{p\ne p'\in P_L}\ \sum_{c}\max\bigl(0,\ |t_c(p)-t_c(p')|_{A}-1\bigr)^2,\qquad g_L=W\sqrt{v^*_L}\,b^{-2L/3}, \] with $|\cdot|_A$ the distance to the nearest multiple of $A$; $v^*_L$ is an integer, computable from $P_L$ alone. It is the smallest squared toroidal distance between the top halves of two points of $P_L$, in units of $W$, after one unit per coordinate is subtracted for the unknown lower half; $g_L$ is the same quantity normalized like $\delta N^{1/3}$. \begin{lemma}[Freezing and separation]\label{lem:liftsep} Let $P_L\to P_{L+3}\to\cdots$ be lifts whose depth parameters satisfy $D_{L+3j'}\ge\lfloor L/2\rfloor+j'$ for every $j'\ge0$ (for instance the canonical continuation). Then for every $j\ge0$ the toroidal (hence also Euclidean) minimum distance of $P_{L+3j}$ satisfies \[ \delta N^{1/3}\ \ge\ \min\bigl(g_L,\ b^{L/3}-W\,b^{-2L/3}\bigr). \] \end{lemma} Each lift prepends one digit, so the old top half moves one digit down; this is why the depth must grow by one per step. In the bound, the first term covers two points that come from different points of $P_L$, and the second two copies of the same point, which differ in a top digit. \begin{proof} \emph{Freezing.} By induction on $j$, $P_{L+3j}$ is the set of points whose coordinate $c$ has the digits $\pi_c(u)\,|\,(p+\eta(u))_c\,|\,\mathrm{low}_c(u)$, where $u\in\F_b^{3j}$ collects the new index digits, $u\mapsto\pi(u)=(\pi_c(u))_c\in\F_b^{3j}$ is a bijection ($j$ top digits per coordinate), $p$ runs over $P_L$, $\eta(u)$ is a digitwise shift of the $L$ block digits, and $\mathrm{low}(u)$ has $2j$ digits per coordinate. The lift at resolution $\ell=L+3j'$ shifts only digits of index $>D_\ell$ of its own block, in which the block of $P_L$ occupies the indices $j'+1,\dots,j'+L$; so it touches block digits of index $>D_\ell-j'\ge\lfloor L/2\rfloor$ only, and never the $j'$ top digits before them. Hence $\eta(u)$ vanishes on the top $\lfloor L/2\rfloor$ digits of every coordinate: these are \emph{frozen}, equal to those of $p$. \emph{Separation.} Measure in units of $b^{-(L+3j)}b^{2j}$, so that the torus has period $b^{L+j}$ and a point has coordinates $\pi_c(u)b^L+t_c(p)W+\varepsilon_c$ with $\varepsilon_c\in[0,W)$. Two points with different $p\ne p'$: reducing modulo $b^L$, which divides the period, does not increase toroidal distances, and modulo $b^L=AW$ the difference in coordinate $c$ is $(t_c(p)-t_c(p'))W+(\varepsilon_c-\varepsilon'_c)$, of toroidal size more than $W(|t_c(p)-t_c(p')|_A-1)$; so their distance is at least $W\sqrt{v^*_L}$. Two points with the same $p$ and different $u$: some $\pi_c$ differ, while $t_c$ agree, so coordinate $c$ differs by $(\pi_c-\pi'_c)b^L+(\varepsilon_c-\varepsilon'_c)$ with $\pi_c\not\equiv\pi'_c\pmod{b^j}$, of toroidal size more than $b^L-W$. A distance of $x$ units is $x\,b^{-L-j}$ in $[0,1)^3$, and $N^{1/3}=b^{L/3+j}$, so $\delta N^{1/3}$ is the distance in units times $b^{-2L/3}$. \end{proof} The bound does not depend on $j$, nor on the shifts chosen after resolution $L$: one exact computation at the resolution $L$ certifies every larger $m\equiv L\pmod3$, and finitely many exact computations certify the resolutions between the seed and $L$. For the every-shift part of Theorem~\ref{thm:base23} below we need the same certificate for every digital shift. A digital shift adds digits without carries, so it can change the integer difference of two digit vectors; the following lemma computes the worst case exactly. \begin{lemma}[Every digital shift]\label{lem:shiftcert} For $\mathbf d\in\F_b^n$ let $\mu_n(\mathbf d)$ be the least toroidal distance $|X'-X|_{b^n}$ (the distance to the nearest multiple of $b^n$) over the integers $X\in[0,b^n)$, where $X'$ has the digits of $X$ plus $\mathbf d$, digit by digit modulo $b$. Let $H=\lfloor L/2\rfloor$. \begin{enumerate} \item[(a)] Under the hypothesis of Lemma~\ref{lem:liftsep}, for every $j\ge0$ and every digital shift $\tau$ of $P_{L+3j}$ (Definition~\ref{def:dshift}), \[ \delta(P_{L+3j}\oplus\tau)\,N^{1/3}\ \ge\ \min\bigl(W\sqrt{v^\circ_L}\,b^{-2L/3},\ b^{L/3}-W\,b^{-2L/3}\bigr), \] where $v^\circ_L=\min_\Delta\sum_c\max\bigl(0,\mu_H(\Delta_c)-1\bigr)^2$, the minimum over the differences $\Delta=(\Delta_x,\Delta_y,\Delta_z)$ of the top $H$ digits of two distinct points of $P_L$ ($\Delta=0$ allowed, with weight $0$). \item[(b)] For every digitally shifted digital $(0,m,3)$-net $P$ over $\F_b$, \[ \min_\tau\delta(P\oplus\tau)^2=b^{-2m}\min_\Delta\textstyle\sum_c\mu_m(\Delta_c)^2, \] the minimum over the digital shifts $\tau$ and over the nonzero digit differences $\Delta$ of two points of $P$ (toroidal distances). \end{enumerate} \end{lemma} \begin{proof} For a pair of points with digit difference $\Delta$, coordinate $c$ of the shifted pair has the digits $X_c$ and $X_c+\Delta_c$, where $X_c$ is coordinate $c$ of the first point plus $\tau_c$. As $\tau$ varies, $(X_x,X_y,X_z)$ runs over all triples of digit vectors, independently in the three coordinates, so the minimum over $\tau$ of the squared distance of the pair is $\sum_c\min_{X_c}|X'_c-X_c|^2$. This gives (b). Since $t\mapsto\max(0,t-1)^2$ is nondecreasing, the same argument gives $v^\circ_L=\min_{\tau'}v^*_L(P_L\oplus\tau')$ over the digital shifts $\tau'$ of $P_L$. For (a), write the points of $P_{L+3j}$ as in the proof of Lemma~\ref{lem:liftsep}. A digital shift $\tau$ keeps $u\mapsto\pi(u)+\tau_\pi$ a bijection, where $\tau_\pi$ are its digits on the top $j$ digits, and adds the same vector $\tau_{\mathrm{blk}}$ to the block digits of every point. So $P_{L+3j}\oplus\tau$ has the structure used in that proof, with $P_L$ replaced by the digitally shifted copy $P_L\oplus\tau_{\mathrm{blk}}$: the top $H$ block digits of every point are those of a point of $P_L\oplus\tau_{\mathrm{blk}}$, and the proof applies verbatim. The first term is then $W\sqrt{v^*_L(P_L\oplus\tau_{\mathrm{blk}})}\,b^{-2L/3}\ge W\sqrt{v^\circ_L}\,b^{-2L/3}$, and the second term does not involve the shift. \end{proof} The quantity $\mu_n$ has a simple description. Write $d_k$ for the digit of $\mathbf d$ of weight $b^k$. Each nonzero $d_k$ contributes $d_kb^k$ or $(d_k-b)b^k$ to $X'-X$, according to whether digit $k$ of $X$ is below $b-d_k$ or not, and both occur. So $\mu_n(\mathbf d)$ is the distance, modulo $b^n$, from the integer with the digits $\mathbf d$ to the nearest number $\sum_{k\in S}b^{k+1}$, $S\subseteq\operatorname{supp}\mathbf d$; the numbers $b^{k+1}$, $k\in\operatorname{supp}\mathbf d$, are super-increasing, so the nearest such sum is found greedily. Thus $v^\circ_L$ is the minimum weight, under an additive weight, of a linear code of length $3H$: the top $H$ digits of the differences of points of $P_L$, of dimension $L$ when the top-digit map is injective (as at all resolutions used below). It is computed exactly by enumerating the code words of small weight, in milliseconds even at $2^{30}$ points. It vanishes if and only if two distinct points of $P_L$ share their top $H$ digits or the code contains a nonzero vector whose coordinates are each $0$ or $\pm(0,\dots,0,1,\dots,1)$. The reason is the mechanism of Remark~\ref{rem:shiftshared} in another guise: without a shift, two points whose digits differ by $0\cdots01\cdots1$ are close only if they happen to sit on a carry chain, as $d(b-1)\cdots(b-1)$ and $(d+1)0\cdots0$ do; a digital shift can put any such difference on a carry chain, and then the two coordinates differ by one unit. \begin{theorem}[Bases two and three; computer-assisted]\label{thm:base23} Let $b\in\{2,3\}$. For every $m\ge1$ there is an explicit digital $(0,m,3)$-net $P_m$ over $\F_b$ and a digital shift $\tau_m\in(\F_b^m)^3$ (Definition~\ref{def:dshift}) such that the toroidal (hence also Euclidean) minimum distances satisfy \begin{enumerate} \item $\delta(P_m\oplus\tau_m)\,N^{1/3}\ge0.4129$ for $b=2$ and $\ge0.3538$ for $b=3$; \item $\delta(P_m)\,N^{1/3}\ge0.1718$ for $b=2$ and $\ge0.0770$ for $b=3$; \item $\delta(P_m\oplus\tau)\,N^{1/3}\ge0.1718$ for $b=2$ and $\ge0.0770$ for $b=3$, for every digital shift $\tau\in(\F_b^m)^3$. \end{enumerate} Per residue $m\equiv0,1,2\pmod3$ the constants are those of Table~\ref{tab:base23}. For $m\ge9$ ($b=2$) resp.\ $m\ge6$ ($b=3$) the nets come from a finite seed by deep-shift lifts: along a fixed chain up to the certificate resolution of Table~\ref{tab:base23}, then by any continuation with $D_\ell=\lfloor\ell/2\rfloor$. For smaller $m$ they are explicit nets with exactly computed minimum distances. \end{theorem} \begin{proof} For the lifted resolutions: Lemma~\ref{lem:lift}(A) gives the net property at every resolution, Lemma~\ref{lem:lift}(B) the existence of the continuation, and Lemma~\ref{lem:liftsep} the bound $\delta N^{1/3}\ge g_L$ for every $m\ge L$ in the residue class of a certificate resolution $L$, once the finite certificate $v^*_L$ is known and the proviso $g_L\le b^{L/3}-Wb^{-2L/3}$ holds. The finite part is an exact computation on explicitly given nets: the $t=0$ property of the seeds and of the nets at the certificate resolutions, the integers $v^*_L$ (with the proviso, which holds by a wide margin), the exact minimum distances at the finitely many resolutions between the seed and $L$, and, for $m$ below the seed resolutions, explicit digital $(0,m,3)$-nets with their exact minimum distances (all above the stated constants). The constants are the minima of these exact values, truncated. Without the shift the same matrices are used: Lemmas~\ref{lem:lift} and~\ref{lem:liftsep} never use the shift, and only the certificates change. For (3), Lemma~\ref{lem:shiftcert}(a) gives the bound for every digital shift and every $m\ge L$ from one integer $v^\circ_L$, computed at the same certificate resolution $L$ (the proviso holds again); at the finitely many resolutions below $L$, and for the explicit nets below the seed, the exact minimum of $\delta(P_m\oplus\tau)N^{1/3}$ over all $\tau$ is computed by Lemma~\ref{lem:shiftcert}(b). \end{proof} \begin{table}[tbp] \centering\small\setlength{\tabcolsep}{5pt} \begin{tabular}{@{}ccccccc@{}} \toprule $b$ & $m\bmod3$ & seed resolution & certificate resolution $L$ & (1) shifted & (2) unshifted & (3) every shift\\ \midrule 2 & 0 & 9 & 30 & .4250 & .4050 & .4050\\ 2 & 1 & 10 & 28 & .4129 & .4129 & .4129\\ 2 & 2 & 11 & 20 & .4209 & .1718 & .1718\\ 3 & 0 & 6 & 18 & .4699 & .4699 & .1047\\ 3 & 1 & 7 & 22 & .3538 & .2586 & .1651\\ 3 & 2 & 8 & 20 & .3676 & .0770 & .0770\\ \bottomrule \end{tabular} \medskip \caption{Theorem~\ref{thm:base23}: lower bounds for $\delta N^{1/3}$ per residue class, for all $m$ in the class (including the explicit nets below the seed resolution, whose values are larger), truncated. One certificate resolution $L$ serves all three columns. The certificates of (1) and (2) were reproduced exactly by two independent implementations, one of which the scripts of the software deposit (Appendix~\ref{app:verify}) rerun, and so were the integers $v^\circ_L$ of (3). Columns (2) and (3), apart from the entries $.4699$ and $.2586$, are also checked in the Lean kernel, with the nets and the certificates $v^\circ_L$; column (1) is checked by these programs only (Section~\ref{sec:formal}). The nets at the certificate resolutions have up to $2^{30}$ points; $v^*_L$ is computed by grouping the points by their top digits, using that the groups are digit translates of one linear set (the structure in the proof of Lemma~\ref{lem:liftsep}), and $v^\circ_L$ as a minimum code weight (after Lemma~\ref{lem:shiftcert}).} \label{tab:base23} \end{table} \begin{figure}[tbp] \centering \IfFileExists{figs/base23_lift.pdf}{\includegraphics[width=\linewidth,height=0.5\textheight,keepaspectratio]{figs/base23_lift.pdf}}{\fbox{\parbox{0.9\textwidth}{\centering figure \texttt{base23\_lift} pending}}} \caption{\baseTwoThreeCaption} \label{fig:base23} \end{figure} The shifts in (1) are explicit, one per $m$: the shift of the seed, carried into the block digits by each lift. The definitions of \cite{DGS2} need only a family indexed by $m$, so the shifted nets are well-separated families of digitally shifted digital nets in their sense (Corollary~\ref{cor:dgs}). Part~(3) holds for every digital shift (Lemma~\ref{lem:shiftcert}). Its constants are those of (2) in base two and, for $m\equiv0,1$, lower in base three, where a shift can place a digit difference $0\cdots01\cdots1$ of two points on a carry chain. Digital $(0,m,3)$-nets in base $2$ were classified by Hofer and Suzuki \cite{HS19}; here we need only explicit seeds and the lift. Propagation rules for digital nets \cite[Sec.~4.2, 4.4]{DP10} control the quality parameter $t$, not distances; the lift is a propagation rule $m\to m+3$ at $t=0$ that also carries a separation certificate. \begin{remark}[Why every-shift certificates vanish at some resolutions]\label{rem:shiftparity} Along the chains the every-shift certificate $v^\circ_L$ vanishes at many resolutions (for $b=3$, $m\equiv0$, at every resolution below $18$; computed), and it is positive at the six certificate resolutions of Table~\ref{tab:base23}, all of them even. Two facts explain the pattern. (i) \emph{Parity.} If $L$ is even and $v^\circ_L=0$, then $v^\circ_{L+3}=0$ for every lift $P_{L+3}$ of $P_L$. Indeed, the indices $(n,0,0,0)$ of the lift (Definition~\ref{def:lift}) give points with top digit $0$, block digits those of $P_L$ and jitter digits $0$, so the top $\lfloor(L+3)/2\rfloor=H+1$ digits of their differences are $0$ followed by the top $H$ digits of the differences of $P_L$. For such differences the toroidal $\mu_{H+1}$ of the extended vector is at most the plain (non-toroidal) value of the original one, and a vector $\pm(0,\dots,0,1,\dots,1)$ has plain value $1$; so a vanishing certificate at $L$ gives one at $L+3$. For odd $L$ the window of $P_{L+3}$ gains one more block digit, so a certificate can first become positive only at an even resolution. (ii) \emph{Dichotomy.} If $v^\circ_L=0$, then along every lift chain from $P_L$ every $m\ge L$ in the class has $\min_\tau\delta(P_m\oplus\tau)N^{1/3}<2\sqrt3\,b^{\lceil L/2\rceil-2L/3}$. Indeed, the indices whose new digits are all $0$ embed $P_L$ into $P_m$ with top and jitter digits $0$, and a difference whose top window is $0$ or $\pm(0,\dots,0,1,\dots,1)$ in each coordinate can be shifted to have each coordinate below $2W$ in the units $b^{-(L+j)}$ of the proof of Lemma~\ref{lem:liftsep}, $m=L+3j$. So, for a chain frozen from every resolution, a floor $\kappa N^{-1/3}$, $\kappa>0$, that holds for every shift from some resolution on exists if and only if $v^\circ_L>0$ for some $L$; if the certificate vanished at infinitely many $L$, the every-shift separation would tend to $0$. For the nets of Theorem~\ref{thm:base23} the certificate is positive at every certificate resolution (Table~\ref{tab:base23}), so no shift makes their separation decay. \end{remark} The six seeds are printed in Table~\ref{tab:b23seeds} (Appendix~\ref{app:tables}); the chains, shifts and small-$m$ nets are data of the deposit, whose scripts rerun the certificates of Table~\ref{tab:base23} (Appendix~\ref{app:verify}). \section{The question of Dick, Goda and Suzuki}\label{sec:dgs} This section relates the results of Part~I to the definitions of \cite{DGS2}: low discrepancy, separation in their sense, digital shifts, and the maximum norm. \begin{remark}[Low discrepancy]\label{rem:lowdisc} All nets in this paper are $(0,m,s)$-nets in base $q$: every elementary box of volume $q^{-m}$ contains exactly one point (Lemma~\ref{lem:nie}). This is a property of the point set alone, whatever ring the net is digital over, so Niederreiter's discrepancy bound for $(t,m,s)$-nets in an integer base \cite{Nie87} applies: $D^*_N\le C_{s,q}(\log N)^{s-1}/N+O_{s,q}\bigl((\log N)^{s-2}/N\bigr)$ for $N=q^m$, with $C_{s,q}$ independent of $m$. So all our nets have low discrepancy in the sense of \cite{DGS2,DGLPS1}, and the question of Dick, Goda and Suzuki reduces, for these nets, to separation. The same holds for digitally shifted nets, which are again $(0,m,s)$-nets (Lemma~\ref{lem:shift}). \end{remark} \needspace{8\baselineskip} \subsection{Digital shifts} \begin{lemma}[Shifted graph forms]\label{lem:shift} Let $P$ be a graph form over $\Z/q\Z$ with depths $\mathbf k$ and blocks $\Phi$, and $\tau=(\tau^{\mathrm{cell}}_c,\tau^{\mathrm{off}}_c)_c$ a digital shift (Definition~\ref{def:dshift}), split into its digits on the cell digits and on the offset digits of each coordinate. Then $P\oplus\tau$ is the set of points whose coordinate $c$ has the digits $n_c$ followed by $\sum_d\Phi_{cd}n_d+\gamma_c$, $n\in\prod_c(\Z/q\Z)^{k_c}$, where $\gamma_c=\tau^{\mathrm{off}}_c-\sum_d\Phi_{cd}\tau^{\mathrm{cell}}_d$. In particular $n$ is the cell of the point, every cell holds one point, $P\oplus\tau$ is a $(0,m,s)$-net if $P$ is, and for two points in the cells $n,n'$ every offset digit difference is congruent to the corresponding entry of $\Phi(n'-n)$ modulo $q$, as for $P$. \end{lemma} \begin{proof} Substitute $n=\tilde n-\tau^{\mathrm{cell}}$ and index the shifted point by $\tilde n$. The shift maps every elementary box of shape $d$ bijectively onto one of the same shape, since it shifts the first $d_c$ digits of coordinate $c$. \end{proof} So a shifted net is an \emph{affine} graph form: offsets $\Phi n+\gamma$ with a constant $\gamma$. Lemma~\ref{lem:Ss}, Proposition~\ref{prop:Hk} and the skeleton lemmas of Section~\ref{sec:skel} hold for affine graph forms with the same proofs, since they use only differences of digits. \begin{proposition}[Every digital shift]\label{prop:shift} Lemma~\ref{lem:Ss} holds verbatim, with the same bound, for every digital shift of a graph form that satisfies (H1) and (H2). Consequently the lower bounds on the separation in Theorems~\ref{thm:A}, \ref{thm:main}, \ref{thm:S1}, \ref{thm:S2}, \ref{thm:allm} and \ref{thm:HL} and in Propositions~\ref{prop:D} and~\ref{prop:Hk} hold for every digital shift of the respective nets (in Theorem~\ref{thm:S1}(2) the occupied fine cells form a coset of the lattice), and so does the first-order sharpness in Theorem~\ref{thm:A}(2). \end{proposition} \begin{proof} By Lemma~\ref{lem:shift} the shift only adds the constant $\gamma$ to the offsets. The proofs use the cells (the step and its carries), the residues of digit differences, the facts that a digit difference lies in $(-q,q)$ and vanishes if its residue does, and $|f'-f|<1$; the constant cancels in every residue, and in Lemma~\ref{lem:Ss} the deeper digits of a coordinate whose own axis alone moves are still equal, by (H2). Theorem~\ref{thm:S2} does not even need the re-indexing by the shifted cells: its proof uses only differences of the skeleton digits. The sharpness argument prescribes the cell digits and the first offset digits of a pair; in the shifted net these are digits of the unshifted net prescribed in translated sets of the same size, so the joint uniformity of the digits of the unshifted net, given by its $t=0$, suffices. \end{proof} The nets of Theorem~\ref{thm:base23}(1) carry one explicit shift by construction. Proposition~\ref{prop:shift} does not apply to them, because their bounds come from the lift certificates (Lemma~\ref{lem:liftsep}), not from Lemma~\ref{lem:Ss}; Theorem~\ref{thm:base23}(3) covers every digital shift instead. The per-prime nets of Remark~\ref{rem:perprime} keep their constants under every digital shift as well (Remark~\ref{rem:ppshift}). In dimension three, Proposition~\ref{prop:shift} gives every resolution under every digital shift: \begin{corollary}[Corollary~\ref{cor:allm} under digital shifts]\label{cor:allmshift} Let $p\ge37$ be prime. For every $m\ge3$ and every digital shift, the shifted nets of Corollary~\ref{cor:allm} satisfy its separation bounds, with the same constants $0.2715$ and $0.4250$, and its covering bounds; so they are quasi-uniform along every $N=p^m$, $m\ge3$, as the Larcher--Pillichshammer nets in \cite[Thm.~3.11]{DGS2}. For $m\ge4$ the lower bound still tends to $1$ as $p\to\infty$. At $m=3$ the nets are the column nets $\mathcal C_p$, whose three coordinates share the second and third digits, and the bound depends on whether the shift keeps them shared: \begin{enumerate} \item[(a)] for every shift, $\delta N^{1/3}>0.4693$; \item[(b)] if the shift digits of the three coordinates agree at the second and at the third digit, the exact value of Theorem~\ref{thm:col}(2) holds; \item[(c)] if they agree at the second digit, $\delta N^{1/3}\ge1-(7p-3)/(\sqrt3p^2)$; \item[(d)] otherwise $1/\sqrt3-7/(\sqrt3p)\le\delta N^{1/3}\le1/\sqrt3+5/(\sqrt3p)$. \end{enumerate} \end{corollary} \begin{proof} A digital shift keeps one point in every cell and keeps $t=0$ (Lemma~\ref{lem:shift}), which is all the covering bounds use, except for the mesh-ratio constant $4.264$ of Corollary~\ref{cor:allm}, which comes from Theorem~\ref{thm:S1}(3); at $m=3$ this gives $h\le\sqrt3N^{-1/3}$. In the shifted net the occupied fine cells form a coset $g+\Lambda_A$ (Proposition~\ref{prop:shift}), and the proof of Theorem~\ref{thm:S1}(3) runs with its target residue moved by $g_z-\sum_d\rho^dg_{x_d}$; this is not a translation of the unshifted construction, since the target cell must stay in range. For $m\ge4$ the separation bounds of Theorems~\ref{thm:main}, \ref{thm:allm} and~\ref{thm:S1} used in Corollary~\ref{cor:allm} hold for every shift by Proposition~\ref{prop:shift}. For $m=3$ Proposition~\ref{prop:colshift} gives the lower bound $0.4693>0.4250$, the exact value and the bound for agreeing shift digits, and otherwise the two bounds $\frac1{\sqrt3}(1-\frac{W+3}p+\frac3{p^2})$ and $\frac1{\sqrt3}(1+\frac{3+w_{\max}}p-\frac3{p^2})$ with $W\le4$ and $w_{\max}\le2$, which lie in the stated interval. \end{proof} So only the limit changes, and only at $m=3$: with unequal shift digits at the geometric digit $\delta N^{1/3}$ lies within $7/(\sqrt3p)$ of $1/\sqrt3$. The reason: a digital shift adds a different constant to the shared digit in each coordinate, so the three coordinates no longer wrap at the same cells, and a step across a wrap of one coordinate alone has length about $1/\sqrt3$ (Remark~\ref{rem:glue}). \begin{remark}[Shared digits under shifts]\label{rem:shiftshared} So the exact value of Theorem~\ref{thm:col} does not extend to all shifted nets: if the shift digits of the three coordinates of $\mathcal C_q$ at the second digit are not all equal, then for every $q\ge11$, prime or not, the shifted net has $\delta N^{1/3}\le\frac1{\sqrt3}(1+\frac{3+w_{\max}}q-\frac3{q^2})$, below the value of Theorem~\ref{thm:col} (Proposition~\ref{prop:colshift}(3)). For each coordinate, the set of values of the shared digit at which a step wraps (its \emph{wrap window}) is translated by that coordinate's shift digit; unequal shift digits give distinct windows, so some cell wraps in exactly one coordinate, and its step along that axis gives the pair $qa_c\mathbf 1+\Delta j$ at integer scale. In general a digital shift translates the wrap windows independently in each coordinate. Bounds certified with arbitrary wraps (Remark~\ref{rem:glue}) therefore survive every shift; bounds that use locked wraps, wraps that are correlated across coordinates, need not. The nets $\mathcal T_p$ and $\mathcal W_p$ of Part~II are of the second kind (Remark~\ref{rem:wshift}). \end{remark} \subsection{Well-separated families} In \cite{DGS2} a point set is \emph{$\kappa$-separated} ($\kappa$ as in \cite{DGS2}) if no (toroidal) shifted elementary interval with $\kappa$ digits in every coordinate contains two of its points \cite[Defs.~3.1--3.3]{DGS2}, and a family $(\mathcal P_m)_{m\in I}$ on an infinite set $I$ of resolutions is \emph{well separated} if $\mathcal P_m$ is $\kappa_m$-separated with $\kappa_m\le m/s+O(1)$ \cite[Def.~3.8]{DGS2} (in \cite{DGS2} the dimension is $d$). \begin{corollary}[Well-separated families in the sense of Dick, Goda and Suzuki]\label{cor:dgs} Let $s\ge2$ and $q\ge2$ be integers and $I\subseteq\N$ infinite, and for $m\in I$ let $\mathcal P_m\subset[0,1)^s$ be a $(0,m,s)$-net in base $q$ whose toroidal minimum distance satisfies $\delta\ge c\,q^{-m/s}$, with $c>0$ independent of $m$. Then every $\mathcal P_m$ is $\kappa_m$-separated and toroidally $\kappa_m$-separated in base $q$ \cite[Def.~3.3]{DGS2}, with \[ \kappa_m=\Bigl\lceil\frac ms+\log_q\frac{\sqrt s}{c}\Bigr\rceil\ \le\ \frac ms+\log_q\frac{\sqrt s}{c}+1, \] so $(\mathcal P_m)_{m\in I}$ is a well-separated and toroidally well-separated family on $I$ \cite[Def.~3.8]{DGS2}. In particular: \begin{enumerate} \item the nets $\mathcal P_{k,q}$ of Theorem~\ref{thm:main}, for every integer $q\ge4$, are $(k+1)$-separated, i.e.\ $\kappa_m=m/3+1$ on $I=\{3k:k\ge2\}$; \item the nets of Theorem~\ref{thm:A}(2) are $(k+1)$-separated on $I=\{sk:k\ge1\}$; \item the nets $\mathcal H^s_{k,p}$ of Theorem~\ref{thm:HL} are $(k+1)$-separated for every prime $p\ge\max(2s,2B_s+4)$ (two more than in Theorem~\ref{thm:HL}; see Step~3 of the proof) and every $k\ge1$; \item the nets $\mathcal S^1_{k,p,\rho}$ of Theorem~\ref{thm:S1}, with $m=sk+s-1$, are $(k+1)$-separated, i.e.\ $\kappa_m=\lceil m/s\rceil$, if $(\rho-1)^2\ge s$ (for $\rho=\lfloor p^{1/s}\rfloor$ this holds when $p>(\lceil\sqrt s\rceil+1)^s$), and $(k+2)$-separated otherwise; \item for every prime $p\ge37$ the nets of Corollary~\ref{cor:allm} are $(\lfloor m/3\rfloor+1)$-separated for every $m\ge3$. \end{enumerate} All of these hold toroidally as well, and for every digital shift (Definition~\ref{def:dshift}) of the respective nets. For prime $q$ the nets are digital nets over $\F_q$ in the sense of \cite[Def.~2.1]{DGS2}; this covers (3)--(5) entirely and the prime instances of (1) and~(2). For composite $q$ the nets of (1) and (2) are digital over $\Z/q\Z$, which is not covered by \cite[Def.~2.1]{DGS2}; Definitions~3.1 and~3.2 of \cite{DGS2} are stated for every integer base $b\ge2$, and we read Definitions~3.3 and~3.8 in the same generality. \end{corollary} \begin{proof} \emph{Step 1 (\cite[Lemma~3.7]{DGS2}, with strict inequality).} Let $P\subset[0,1)^s$ be finite and $\kappa\ge0$, and suppose every two distinct points of $P$ have toroidal $\ell^\infty$-distance at least $q^{-\kappa}$. A toroidal shifted elementary interval $\tilde J_{\mathbf c,a,e}$ with $\kappa$ digits in every coordinate, $\mathbf c=(\kappa,\dots,\kappa)$ \cite[Def.~3.2]{DGS2}, is contained in a product of sets $I_j\bmod1$, where $I_j=[\alpha_j,\alpha_j+q^{-\kappa})$ is half-open. If $x,y\in\tilde J_{\mathbf c,a,e}$, then for each $j$ there are $z,z'\in\Z$ with $x_j+z,\ y_j+z'\in I_j$, so $\min(|x_j-y_j|,1-|x_j-y_j|)\le|(x_j+z)-(y_j+z')|2^s$. If $p>(\lceil\sqrt s\rceil+1)^s$, then $\rho=\lfloor p^{1/s}\rfloor\ge\lceil\sqrt s\rceil+1$. (5) $m=3$: $\delta\ge\sqrt{2/3}\,p^{-1}$ (Theorem~\ref{thm:colmin}), and $\kappa=2$ needs $p\sqrt{2/3}\ge\sqrt3$, i.e.\ $p\ge3/\sqrt2$. $m=3k$, $k\ge2$: part~(1). $m=3k+1$: $\delta\ge(\rho^2-4)p^{-(k+1)}$ with $\rho$ maximal subject to $\rho^2(\rho+1)\le p$ (Theorem~\ref{thm:allm}), and $p\ge37$ gives $\rho\ge3$, so $\rho^2-4\ge5\ge\sqrt3$ and $\kappa=k+1$. $m=3k+2$: part~(4) with $s=3$ and $\rho=\lfloor p^{1/3}\rfloor\ge3$, so $(\rho-1)^2=4\ge3$ and $\kappa=k+1$. In every case $\kappa_m=\lfloor m/3\rfloor+1$. \emph{Digital shifts.} By Proposition~\ref{prop:shift} the separation bounds used in (1)--(4) and, for $m\ge4$, in (5) hold for every digitally shifted net. For $m=3$, Proposition~\ref{prop:colshift}(3) gives $\delta\ge\frac1{\sqrt3}(1-\frac{W+3}p+\frac3{p^2})\,p^{-1}$ with $W\le4$, and $\kappa=2$ needs $p^2-(W+6)p+3\ge0$, which holds for $p\ge10$. \end{proof} The nets of Theorem~\ref{thm:base23}, with their shifts, are digitally shifted digital nets over $\F_b$, $b\in\{2,3\}$, in the sense of \cite[Def.~2.2]{DGS2}, so the corollary applies to them as well, with $\kappa_m=\lceil m/3+\log_b(\sqrt3/c)\rceil$: for every digital shift with the constants $c$ of part~(3), and with the larger constants of part~(1) for its explicit shifts. Likewise the per-prime nets of Remark~\ref{rem:perprime}, with the constants of Table~\ref{tab:perprime}, are well separated under every digital shift (Remark~\ref{rem:ppshift}). \begin{remark}[Finitely many resolutions]\label{rem:dgsfinite} Definition~3.8 of \cite{DGS2} constrains only $m\in I$, and finitely many further resolutions can be added at no cost. If $\mathcal Q\subset q^{-m}\Z^s\cap[0,1)^s$ is a $(0,m,s)$-net in base $q$, the elementary boxes of shape $(m,0,\dots,0)$ each hold one point, so the first coordinates of the points are distinct elements of $q^{-m}\Z\cap[0,1)$ and any two of them have toroidal distance at least $q^{-m}$; by Step~1 of the proof of Corollary~\ref{cor:dgs}, $\mathcal Q$ is $m$-separated and toroidally $m$-separated. Adding finitely many such $m$ to $I$ changes the $O(1)$ of \cite[Def.~3.8]{DGS2} by a constant. In dimension three digital $(0,m,3)$-nets over $\Z/q\Z$ exist for every integer $q\ge2$ (over $\F_p$ for every prime $p$) and every $m\ge1$: by Lemma~\ref{lem:Ts}, whose proof does not use $m\equiv0\pmod s$, every box minor of the frame of the places $\infty,0,1$ is $\pm1$, a unit modulo every $q$, so the frame is one (Lemma~\ref{lem:nie}; see also \cite[Thm.~6.2, Cor.~6.3]{Nie87} and \cite[Sec.~8.2]{DP10}), and trivially for $m=0$. Together, Theorems~\ref{thm:main} and~\ref{thm:col}, Corollary~\ref{cor:allm}, Remark~\ref{rem:perprime} and Theorem~\ref{thm:base23} cover all but finitely many $m$ in each prime base with a constant independent of $m$; with the frame nets for the remaining $m$ they give well-separated families on all of~$\N$, in every prime base. The same holds under every digital shift: the constants survive every shift (Proposition~\ref{prop:shift}, Corollary~\ref{cor:allmshift}, Remark~\ref{rem:ppshift} and Theorem~\ref{thm:base23}(3)), and a digitally shifted $(0,m,s)$-net is again a $(0,m,s)$-net on the grid $q^{-m}\Z^s$ (Lemma~\ref{lem:shift}), so the argument above applies to it. \end{remark} \subsection{The maximum norm} \cite{DGS2} measure separation and mesh ratios also in the maximum norm. Since $\Vert x\Vert_\infty\ge\Vert x\Vert_2/\sqrt3$, the Euclidean bound gives $\delta_\infty\ge(1-1/q)q^{-k}/\sqrt3$ for the nets of Theorem~\ref{thm:main}; the sign analysis in the proof of Theorem~\ref{thm:sep} gives more: \begin{proposition}[Maximum norm]\label{prop:linf} Let $q\ge4$, $k\ge2$. For every digital shift $\tau$, any two distinct points of $\mathcal P_{k,q}\oplus\tau$ have toroidal $\ell^\infty$-distance at least $(1-3/q)\,q^{-k}$. Hence the $\ell^\infty$ mesh ratio of $\mathcal P_{k,q}\oplus\tau$ in the sense of \cite{DGS2}, toroidal or not, is at most $2q/(q-3)$, and $\mathcal P_{k,q}\oplus\tau$ is toroidally $(k+1)$-separated \cite[Def.~3.3]{DGS2}. \end{proposition} \begin{proof} By Proposition~\ref{prop:shift} the proof of Theorem~\ref{thm:sep} applies to $\mathcal P_{k,q}\oplus\tau$. In its sign analysis every step has a coordinate with $|\Delta_i|\ge1-3/q$ ($=\beta_3$ in that proof): an own-axis step has $\beta_1$; for $\nu=2$, (iv) gives a backward coordinate with $|\sigma|\le1$ or $\sigma=0$; for $\nu=3$, a forward coordinate or a backward one with $|\sigma|=3$ forces a backward one with $|\sigma|\le1$ by (ii), (iii), and otherwise all three are backward with $|\sigma|\le2$. One point per cell gives $h_\infty\le q^{-k}$, and $(1-3/q)q^{-k}\ge q^{-k-1}$ for $q\ge4$; the separation statement is Step~1 of the proof of Corollary~\ref{cor:dgs}. \end{proof} The constant $3$ is the exact $\ell^\infty$ value of the relaxation of Lemma~\ref{lem:Ss} for this geometric layer, attained at the step $e=\mathbf 1$ with three carries. For a dominant design (Proposition~\ref{prop:D}) every moving coordinate is backward with $|\sigma_i|\le2a$, so the same argument gives $\delta_\infty\ge(1-(2a+1)/q)\,q^{-k}$ and an $\ell^\infty$ mesh ratio at most $2q/(q-2a-1)$ for every $q\ge2a+2$ and every digital shift, in particular below $10/3$ for the nets of Theorem~\ref{thm:A}(2). \emph{The sequence question.} All nets here depend on $m$; they are not prefixes of one sequence, and the question of \cite{DGS2} for sequences (a sequence whose first $q^m$ points are well separated for every $m$, \cite[Def.~3.8]{DGS2}) remains open in dimension $s\ge3$ (Remark~\ref{rem:open}(10)). \part{Separation constants}\label{part:constants} Part~I proves bounds $\delta N^{1/s}\ge c$ with explicit constants $c$. Part~II asks how large $c$ can be. At the resolutions $m=sk$ the bounds of Part~I have the form $1-K/q$, and for a family of nets with $\delta N^{1/s}=1-K/q+O(q^{-2})$ as $q\to\infty$ we call $K$ its \emph{first-order deficit}, or simply its deficit. Sections~\ref{sec:deficit} and~\ref{sec:deficitgrowth} bound the deficit from above for given designs and from below for whole classes of nets. Section~\ref{sec:drift} explains the deficits $\frac13$ and $\frac23$ of the column nets, whose geometric layers have entries of size about $q/3$. Section~\ref{sec:whylat} shows why the cubic value $1$, the value of $\Z^s$ with one point per cell, is a natural barrier when the cells are cubes, and Section~\ref{sec:large} goes beyond it. \section{The first-order deficit: certificates and ceilings}\label{sec:deficit} This section works with designs. Its first subsection turns the one-digit relaxation into a number, the \emph{certified deficit} $K(F)$ of a design $F$, and shows how a finite computation (a \emph{certificate}) proves the bound $1-K(F)/q$ at every admissible base (defined after Proposition~\ref{prop:Dprime}). The second subsection proves \emph{ceilings}: upper bounds on $\delta N^{1/s}$ that hold for a whole class of nets. The remaining subsections give the designs with the smallest certified deficits that we have, in dimension four and at $k=1$. Section~\ref{sec:deficitgrowth} shows why the deficit must grow with $s$. \subsection{The certified deficit of a design} Remark~\ref{rem:K} expands the relaxation $\mathrm{LB}$ of Lemma~\ref{lem:Ss} for a fixed design as $1-K/q+O(q^{-2})$; we now define this $K$ precisely. Let $F=(F_0,\dots,F_{L-1})$ be a design with integer partial sums $M^{(j)}$. A \emph{configuration} $(e,j)$ is a step $e\in\{-1,0,1\}^s\setminus\{0\}$ with carry depths $j_d\in\{0,\dots,L-1\}$ on its moving axes, with $\sigma_i=\sum_de_dM^{(j_d)}[i][d]$ and $\mathrm{jit}_i$ as in Lemma~\ref{lem:Ss}. A moving coordinate is backward or forward as in Remark~\ref{rem:K}: a \emph{backward} coordinate ($e_i\sigma_i\le0$) has its geometric digit move against the step, so its point lands about one cell away; a \emph{forward} one has it move with the step, so its point may land almost on that of its neighbour. \begin{definition}[Certified first-order deficit]\label{def:K} The \emph{cost} of a configuration is $|\sigma_b|+\mathrm{jit}_b$ if exactly one moving coordinate $b$ is backward, $0$ if at least two are, and $+\infty$ if none is. The \emph{certified deficit} $K(F)$ is the maximum of the cost over all configurations, and $\sigma_{\max}(F)$ is the maximum of $|\sigma_i|$ over all configurations and all $i$. \end{definition} The cost counts, in units of $1/q$ of a cell, by how much the step falls short of a full cell at first order. A one-axis step along $a$ with carry depth $j$ has cost $\alpha^{(j)}_a=-M^{(j)}[a][a]$ if this is $\ge0$, and $+\infty$ otherwise. For a dominant design (Proposition~\ref{prop:D}) every moving coordinate is backward, so $K(F)=\max_c\alpha_c$, the number $a$ of Proposition~\ref{prop:D}. Both $K(F)$ and $\sigma_{\max}(F)$ are maxima over the $(2L+1)^s-1$ configurations, and $\sigma_{\max}(F)=\max_i\sum_d\max_j|M^{(j)}[i][d]|$. The certified deficit is an upper bound for the deficit of every family of nets that realizes $F$ (Proposition~\ref{prop:Dprime}); it need not equal it. \begin{proposition}[Certificate for all large $q$]\label{prop:Dprime} Let $F$ be a design with $K=K(F)<\infty$ and $\sigma_{\max}=\sigma_{\max}(F)$, and put \[ q_{\mathrm{tail}}(K,\sigma_{\max})=\max\Bigl(\sigma_{\max}+1,\ \Bigl\lceil\frac{\sqrt2(\sigma_{\max}+1)-K}{\sqrt2-1}\Bigr\rceil\Bigr). \] Then $\mathrm{LB}\ge1-K/q$ for every $q\ge q_{\mathrm{tail}}(K,\sigma_{\max})$, where $\mathrm{LB}$ is the relaxation of Lemma~\ref{lem:Ss} for $F$ or for any truncation $(F_0,\dots,F_{L'-1})$, $L'\le L$. \end{proposition} \begin{proof} Fix a configuration and let $q\ge q_{\mathrm{tail}}$, so $q>\sigma_{\max}\ge|\sigma_i|$ for all $i$. For a moving backward coordinate $i$ the bound $b_i$ of Lemma~\ref{lem:Ss} is $q-|\sigma_i|-\mathrm{jit}_i\ge0$: for $e_i=1$ and $-q<\sigma_i<0$, $\bar\sigma_i=q+\sigma_i$ and the two options give $|q+t|\in\{2q-|\sigma_i|,\,q-|\sigma_i|\}$; for $\sigma_i=0$ the only option gives $q$; $e_i=-1$ is symmetric. If at least two coordinates are backward, $\sum_ib_i^2\ge2(q-\sigma_{\max}-1)^2\ge(q-K)^2$, because $\sqrt2(q-\sigma_{\max}-1)\ge q-K$ is the second condition on $q_{\mathrm{tail}}$ (and $K\le\sigma_{\max}+1\le q$). If exactly one coordinate $b$ is backward, $\sum_ib_i^2\ge(q-|\sigma_b|-\mathrm{jit}_b)^2\ge(q-K)^2$. No backward coordinate is excluded by $K<\infty$. So every configuration gives $(\sum_ib_i^2)^{1/2}/q\ge1-K/q$. The configurations of a truncation are among those of $F$, with the same $\sigma$ (Lemma~\ref{lem:Ss}, capped carry depths), so its $\mathrm{LB}$ is at least that of $F$. \end{proof} For a dominant design $K=a$ and $\sigma_{\max}\le2a$, and $q_{\mathrm{tail}}\le\lceil((2\sqrt2-1)a+\sqrt2)/(\sqrt2-1)\rceil<5a+4$ for $a\ge1$: Proposition~\ref{prop:Dprime} is the proof of Proposition~\ref{prop:D} without the dominance hypothesis. It turns every statement ``$\mathrm{LB}\ge x$ for every admissible $q\ge q_0$'' about a fixed design into a finite computation (a certificate): $K(F)$ and $\sigma_{\max}(F)$, and the exact value of $\mathrm{LB}$ at the finitely many admissible $q<\max(q_{\mathrm{tail}},K/(1-x))$. Here the \emph{admissible} bases of a design are those for which Corollary~\ref{cor:tsk} gives a net, $\gcd(q,DD')=1$ (Definition~\ref{def:Ns}); on the places $\infty,0,1,\dots,s-2$ they satisfy $\operatorname{lpf}(q)\ge s-1$. This is how the thresholds below are certified. The relaxation never certifies a deficit below $1$: \begin{lemma}[The relaxation stops at one]\label{lem:Kone} Let $F$ be a design with some nonzero off-diagonal entry $F_0[a][b]$. Then $K(F)\ge1$. \end{lemma} \begin{proof} Write $M=F_0$ and consider the carry-free steps $e=\e_a+\varepsilon\e_b$, $\varepsilon=\pm1$, along which only $a$ and $b$ move, with $\sigma_a=M_{aa}+\varepsilon M_{ab}$ and $\varepsilon\sigma_b=M_{bb}+\varepsilon M_{ba}$. If for some $\varepsilon$ at most one of $a$, $b$ is backward, that configuration costs at least $\mathrm{jit}=1$, or $+\infty$. Otherwise $M_{aa}+\varepsilon M_{ab}\le0$ for both signs, so $M_{aa}\le-|M_{ab}|\le-1$, and the one-axis step along $a$ without carry costs $-M_{aa}\ge1$. \end{proof} By Lemma~\ref{lem:phase}(ii) (Section~\ref{sec:whylat}) the off-diagonal entries of the geometric layer of a digital $(0,sk,s)$-net are units modulo $q$, hence nonzero. So for every design with $|F_0|2\max|F_0|$. If $\gcd(\det F_0,q)=1$, or if $\mathcal P$ is a $(0,s,s)$-net with $s\ge4$ and $\sum_{u\ne a}|F_0[u][a]|2\max|F_0|$, the centred layer is $F_0$. Take the step $e_a$ from a cell with $r_a\ge K_1$ and $r_u+F_0[u][a]\in[0,q)$ for every $u\ne a$. Then $\Delta r_a=-K_1$ exactly and $\Delta f_a=0$ (the deeper rows of $a$ vanish on $n_a$), so coordinate $a$ moves by exactly $1-K_1/q$ cells, and every $u\ne a$ by $(F_0[u][a]+\Delta f_u)/q$, of absolute value less than $(|F_0[u][a]|+1)/q$. Such a cell exists: if $\gcd(\det F_0,q)=1$, $n\mapsto r=F_0n$ is a bijection of $(\Z/q\Z)^s$, so $r$ can be prescribed. In the second case, for each $u\ne a$ the box with depths $2$ on $a$ and on $u$, $0$ on two further coordinates and $1$ elsewhere (total $s$) makes $(n_a,r_a,r_u)$ uniformly distributed (Lemma~\ref{lem:nie}); among the cells with $n_a\ne q-1$ and $r_a\ge K_1$ the fraction on which $r_u$ wraps is $|F_0[u][a]|/q$, and the union bound leaves a cell as $\sum_{u\ne a}|F_0[u][a]|0$, $\varepsilon\sigma_b<0$ and $\sum_c|\sigma_c|0$ and $\varepsilon\sigma_b>0$, then $c^2<\sum_c(|\sigma_c|+1)^2/q^2\le s(2C+1)^2/q^2$, $C=\max|\tilde M|$. \end{proposition} \begin{definition}\label{def:Ktrue} Let $\tilde M$ be a layer with nonzero diagonal. If some diagonal entry is positive, or some carry-free step $\e_a+\varepsilon\e_b$ has both moving coordinates forward, let $K_{\mathrm{pair}}(\tilde M)=\infty$. Otherwise $K_{\mathrm{pair}}(\tilde M)$ is the largest of \begin{enumerate} \item[(a)] $|\tilde M_{aa}|$ over the coordinates $a$, and \item[(b)] $|\sigma_b|$ over the carry-free steps $\e_a+\varepsilon\e_b$ (as above) in which $a$ is forward and $b$ is the only backward coordinate, with $\varepsilon\sigma_b<0$. \end{enumerate} \end{definition} By Corollary~\ref{cor:ceils}(i), (ii) and Proposition~\ref{prop:P2}, for a layer with nonzero diagonal, $s\ge4$, $k\ge3$, $3k+4\le sk$ and $q>4sC$, $C=\max|\tilde M|$, \begin{equation}\label{eq:Kpair} \delta N^{1/s}<1-\frac{K_{\mathrm{pair}}}q+\frac{6sC^2}{q^2}\quad(K_{\mathrm{pair}}<\infty),\qquad \delta N^{1/s}<\frac{3\sqrt s\,C}q\quad(K_{\mathrm{pair}}=\infty), \end{equation} the latter by Corollary~\ref{cor:ceils}(ii) for a positive diagonal entry and by Proposition~\ref{prop:P2} for two forward coordinates. As $K_{\mathrm{pair}}$ sees only carry-free steps (carry depth $0$) along one or two axes, it bounds the deficit of the nets from below. It is neither their deficit nor the certified deficit $K(F)$ of a design. \begin{theorem}[The deficit grows with $s$]\label{thm:L} Let $\mathcal P$ be a digital $(0,sk,s)$-net over $\Z/q\Z$ with $q\ge2$, $s\ge4$ and $k\ge1$, whose layer $\tilde M$ has no zero diagonal entry. Then $K_{\mathrm{pair}}(\tilde M)\ge\sqrt{(s-3)/8}$ (Definition~\ref{def:Ktrue}). Hence, by \eqref{eq:Kpair} (which includes the case $K_{\mathrm{pair}}=\infty$), a family of such nets with $k\ge3$ and $3k+4\le sk$ whose layers stay bounded, $|\tilde M|\le C$, has $\delta N^{1/s}<1-\sqrt{(s-3)/8}/q+6sC^2/q^2$ for $q>4sC$. \end{theorem} \begin{proof} Let $K=K_{\mathrm{pair}}<\infty$. Every diagonal entry is $-\alpha_a$ with $1\le\alpha_a\le K$. Every pair has a small entry: if $|x|=|\tilde M_{ab}|>K$, put $\varepsilon=\operatorname{sgn}x$; then $\sigma_a=-\alpha_a+|x|>0$, so $b$ is backward, $\varepsilon\sigma_b=\varepsilon y-\alpha_b\le0$ for $y=\tilde M_{ba}$, and if it is negative the definition gives $\alpha_b-\varepsilon y\le K$; so $|y|\le\max(\alpha_b,K-\alpha_b)\le K$. Orient $c\to d$ when $|\tilde M_{cd}|\le K$, and let $\mathrm{out}(c)$ be the number of $d\ne c$ with $c\to d$. Every pair carries at least one arc, so $\sum_c\mathrm{out}(c)\ge\binom s2$, and by convexity $\sum_c\binom{\mathrm{out}(c)}2\ge s\binom{(s-1)/2}2=s(s-1)(s-3)/8$. The left side counts the triples $(c,\{d,d'\})$ with $c\to d$ and $c\to d'$, so averaging over the $\binom s2$ pairs $\{d,d'\}$, some two columns $d,d'$ have at least $(s-3)/4$ common rows $c\notin\{d,d'\}$ with $|\tilde M_{cd}|,|\tilde M_{cd'}|\le K$. By $t=0$ and Lemma~\ref{lem:box}, the box with depths $k+1$ on two such rows $c,c'$, $k-1$ on $d,d'$ and $k$ elsewhere makes every $2\times2$ minor $\tilde M_{cd}\tilde M_{c'd'}-\tilde M_{cd'}\tilde M_{c'd}$ a unit, hence nonzero, and the $1\times1$ boxes make the entries nonzero. So these rows give pairwise non-proportional vectors in $([-K,K]\setminus0)^2$, of which there are at most $2K^2$ directions: $(s-3)/4\le2K^2$. \end{proof} $K_{\mathrm{pair}}$ is an integer $\ge1$ for a nonzero diagonal, so Theorem~\ref{thm:L} first gives $K\ge2$ at $s=12$; its content is that the deficit is unbounded in $s$ for bounded layers. The same count with $K=1$ shows: \emph{for $s\ge5$ no digital $(0,sk,s)$-net over any $\Z/q\Z$ has a geometric layer with all off-diagonal entries in $\{\pm1\}$} (three rows of $\pm1$ entries on two columns contain two proportional rows). Layers with zero diagonal are open, as in Remark~\ref{rem:diag0}. \emph{Inside the frame orbit} (Section~\ref{sec:frames}), for an integer design reachable over $\mathbb Q$ from rational places, the geometric layer $F_0$ is a Cauchy matrix $\lambda_c\mu_d/D_{cd}$, so Lemma~\ref{lem:CH}(C), (H) apply over $\mathbb{Q}$ to $\pi_{cd}=\tilde M_{cd}\tilde M_{dc}$. For instance, for an all-$\pm1$ layer on four rational places identity (H) would read $1=1+1$, so no such layer exists (modulo $q$: Remark~\ref{rem:pm1}). \begin{theorem}[Frame designs have $K_{\mathrm{pair}}\ge2$ for $s\ge5$]\label{thm:O5} Let $s\ge5$ and let $F$ be an integer design (Section~\ref{sec:prelim}) whose residual problem is solvable over $\mathbb{Q}$ for a frame of rational places, with nonzero diagonal of $F_0$. Then $K_{\mathrm{pair}}(F_0)\ge2$; hence for $k\ge3$, $3k+4\le sk$ and every $q>4sC$ coprime to $DD'$ (Definition~\ref{def:Ns}), $C=\max|F_0|$, $\delta N^{1/s}<1-2/q+6sC^2/q^2$ (for $q>2C$ the layer is $\tilde M=F_0$, and \eqref{eq:Kpair} applies, including the case $K_{\mathrm{pair}}=\infty$). \end{theorem} \begin{proof} Suppose $K_{\mathrm{pair}}\le1$. Then every diagonal entry is $-1$, and the definition of $K_{\mathrm{pair}}$ forces the \emph{pair rule}: each pair $(\tilde M_{ab},\tilde M_{ba})$ is in $\{\pm1\}^2$, or one entry $x$ has $|x|\ge2$ and the other is $\operatorname{sgn}x$ (the step $e_a+\varepsilon e_b$ has $\sigma_a=-1+\varepsilon x$, $\sigma_b=y-\varepsilon$ for $(x,y)=(\tilde M_{ab},\tilde M_{ba})$; if $|x|\ge2$ and $\varepsilon=\operatorname{sgn}x$, $a$ is forward, so $b$ is backward with $|\sigma_b|\le1$ whenever $\varepsilon\sigma_b<0$, which with $y\ne0$ forces $y=\operatorname{sgn}x$). So pairs in one class of Lemma~\ref{lem:CH}(C) have $\pi=-1$, and pairs across classes have an integer $\pi=P\ge1$. A class of size $\ge4$ contradicts (H) ($1=1+1$), so for $s=5$ the classes are $X=\{x_1,x_2,x_3\}$ and $Y=\{y_1,y_2\}$. For $\{x_i,x_j,y_1,y_2\}$, (H) gives $1=1/u+1/v$ with integers $u^2=P_{i1}P_{j2}$, $v^2=P_{i2}P_{j1}$; squaring shows $u,v\in\mathbb{Q}$, hence integers, hence $u=v=2$; so $P_{i1}P_{j2}=4$ for all $i\ne j$, which forces $P_{i1}=\alpha$ for all $i$. Then the three pairings of $\{x_1,x_2,x_3,y_1\}$ all have magnitude $\alpha^{-1/2}$, and $1=2$ is false. For $s=6$ a class of size $\ge4$ again contradicts (H), so both classes have size $3$, and any five of the coordinates form classes of sizes $3$ and $2$, to which the case $s=5$ applies; for $s\ge7$ some class has size $\ge4$. Since $K_{\mathrm{pair}}$ is an integer, $K_{\mathrm{pair}}\ge2$. \end{proof} The proof uses that the entries are integers; designs that are Cauchy only modulo $q$ are not covered (an extension to them, for instance for $q$ larger than a polynomial in $C$, is not proved here). The argument needs $s\ge5$: for $s=4$ two classes of size two are consistent with (C) and (H) ($u=v=2$, $P_{11}P_{22}=P_{12}P_{21}=4$), and Remark~\ref{rem:pm1} excludes only layers with all off-diagonal entries $\pm1$. \subsection{Lower bounds for the deficit of frame designs}\label{sec:Lstar} For a design $F$ realized on a frame, $K(F)$ (Definition~\ref{def:K}) is the deficit that the relaxation certifies, for every $k$. We now bound $K(F)$ from below, for every design with $K(F)<\infty$ and any number $L$ of carry depths. \begin{lemma}[Orientation]\label{lem:O} Let $F$ be a design with $K=K(F)<\infty$. Then for every pair $a\ne b$ of coordinates, $|M^{(j)}[a][b]|\le K$ for all $j$, or $|M^{(j)}[b][a]|\le K$ for all $j$. \end{lemma} \begin{proof} One-axis steps give $0\le\alpha^{(j)}_c\le K$ for all $c$ and $j$. Suppose $u=M^{(j_1)}[a][b]$ and $v=M^{(j_2)}[b][a]$ have $|u|,|v|>K$. Take the step along $a$ and $b$ with carry depths $j_a=j_2$, $j_b=j_1$ and $e_ae_b=\varepsilon:=\sgn u$. Then $e_a\sigma_a=-\alpha^{(j_2)}_a+|u|>0$, so $a$ is forward, and $e_b\sigma_b=\varepsilon v-\alpha^{(j_1)}_b$. If $\sgn v=\sgn u$, then $b$ is forward too and the cost is $+\infty$; otherwise $b$ is the only backward coordinate, of cost $\alpha^{(j_1)}_b+|v|+1>K$. Both contradict the definition of $K$. \end{proof} So in every pair of coordinates one of the two entries is small at every carry depth; orienting the pair towards it gives the orientation used below. \begin{proposition}[Constant row scalars]\label{prop:E} Let $s\ge4$ and let $F$ be an integer design with $L\ge1$ levels reachable over $\mathbb Q$ (Proposition~\ref{prop:F}) from the places $\infty,0,1,\dots,s-2$ with row scalars $\lambda'$ that take one value on all finite places ($\lambda'_\infty$ arbitrary). If $K(F)<\infty$, then $K(F)\ge\operatorname{lcm}(1,\dots,s-3)$. \end{proposition} \begin{proof} Let $\nu=s-2$ and $\lambda$ the common value. By Proposition~\ref{prop:F} at carry depth $0$, $M^{(0)}[c][d]=\lambda'_cg_{d,0}w_{cd}$ with $g_{d,0}\ne0$; put $\mu_d=\lambda g_{d,0}$. In column $0$ the finite rows $c=1,\dots,\nu$ give the integers $\mu_0/c$, so $\operatorname{lcm}(1,\dots,\nu)$ divides $\mu_0$ and $|M^{(0)}[1][0]|=|\mu_0|\ge\operatorname{lcm}(1,\dots,\nu)$. In column $1$ the rows $c\in\{0,2,\dots,\nu\}$ give the integers $\mu_1/(c-1)$, $c-1\in\{-1,1,\dots,\nu-1\}$, so $|M^{(0)}[0][1]|=|\mu_1|\ge\operatorname{lcm}(1,\dots,\nu-1)$. Lemma~\ref{lem:O} for the pair $\{0,1\}$ gives $K(F)\ge\operatorname{lcm}(1,\dots,\nu-1)$. \end{proof} The class of Proposition~\ref{prop:E} contains the monomial design $U_s$, the design $U^*_s$ of Theorem~A and the designs of Table~\ref{tab:Kdesigns}; within it the certified deficit grows at least like $\operatorname{lcm}(1,\dots,s-3)=e^{(1+o(1))s}$, by the prime number theorem \cite{Had96,dlVP96}. Elementarily, $\operatorname{lcm}(1,\dots,s-3)\ge2^{s-3}/(s-2)$ (Chebyshev): the central binomial coefficient $\binom{s-3}{\lfloor(s-3)/2\rfloor}\ge2^{s-3}/(s-2)$ divides $\operatorname{lcm}(1,\dots,s-3)$, as each of its prime-power factors is at most $s-3$ by Kummer's theorem. For arbitrary row scalars a divisor count gives the weaker bound $K(F)\ge s^{1-o(1)}$ (Theorem~\ref{thm:Lstar}). Let $Q_K=\{x/y:x,y\in\Z,\ 1\le|x|,|y|\le K\}$, let $\operatorname{div}(j)$ be the number of divisors of a positive integer $j$, and $\operatorname{div}^*(x)=\max_{1\le j\le x}\operatorname{div}(j)$. For finite places $w_{cd}/w_{cd'}=(a_c-a_{d'})/(a_c-a_d)$ is a cross-ratio of $a_c$, $a_d$, $a_{d'}$ and $\infty$, whence the title of the next lemma. \begin{lemma}[A cross-ratio count]\label{lem:C} Take the places $\infty,0,1,\dots,\nu$, two distinct places $d,d'$ and a rational $\varrho\ne0$, and let $w_{cd}$ be as in Proposition~\ref{prop:Ns}. If $\mathcal C$ is a nonempty set of places $c\notin\{d,d'\}$ with $w_{cd}/w_{cd'}\in\varrho\,Q_K$ for all $c\in\mathcal C$, then $|\mathcal C|\le1+2K\operatorname{div}^*(\nu)\operatorname{div}^*(\nu K)^2$. \end{lemma} \begin{proof} A rational lies in $Q_K$ if and only if its numerator and denominator in lowest terms are at most $K$ in absolute value. Put $\varphi(c)=w_{cd}/w_{cd'}$. \emph{One of $d,d'$ is $\infty$.} As $\varphi\mapsto1/\varphi$ and $Q_K=Q_K^{-1}$, we may take $d'=\infty$; then $c$ is finite and $\varphi(c)=1/t$ with $t=c-d\in[-\nu,\nu]\setminus0$, and the condition says $tQ'/P'\in Q_K$ for $\varrho^{-1}=P'/Q'$ in lowest terms. With $e=\gcd(t,P')$ the lowest terms of $tQ'/P'$ are $(tQ'/e)/(P'/e)$, so $t=et'$ with $e\mid P'$ and $|t'|\le K$: at most $2K\operatorname{div}(P')$ values. Some $c_0\in\mathcal C$ gives $\varrho^{-1}\in t_0Q_K$, so $|P'|\le \nu K$. \emph{$d,d'$ finite.} Row $\infty$ has $\varphi=1$ and contributes at most one element. For finite $c$ put $u=c-d$ and $g=d-d'\ne0$, so $\varphi(c)=(u+g)/u$; with $h=\gcd(u,g)$, $u=hu'$, $g=hg'$, this is $(u'+g')/u'$ in lowest terms. Let $\varrho=P/Q$ in lowest terms, $A=\gcd(Q,u')$ and $B=\gcd(P,u'+g')$. Then $\varphi(c)/\varrho=\bigl(\frac QA\cdot\frac{u'+g'}B\bigr)\big/\bigl(\frac PB\cdot\frac{u'}A\bigr)$ is in lowest terms (each pair of factors is coprime, by $\gcd(P,Q)=\gcd(u',u'+g')=1$ and $\gcd(Q/A,u'/A)=\gcd(P/B,(u'+g')/B)=1$), so $u'+g'=B\beta$ with $1\le|\beta|\le K$. For fixed $(h,A,B)$, $\beta$ determines $u'=B\beta-g'$, hence $c=d+hu'$: at most $2K\operatorname{div}(g)\operatorname{div}(Q)\operatorname{div}(P)$ places, with $|g|\le \nu$ and, as above, $|P|,|Q|\le \nu K$. \end{proof} \begin{theorem}[Frame designs]\label{thm:Lstar} Let $s\ge5$ and let $F$ be an integer design with $L\ge1$ levels reachable over $\mathbb Q$ (Proposition~\ref{prop:F}) from the places $\infty,0,1,\dots,s-2$, or from their image under an element of $PGL_2(\mathbb Q)$ that contains $\infty$ (as the places of every frame do, Section~\ref{sec:dim4}), with arbitrary row scalars. If $K=K(F)<\infty$, then \[ \frac{s-3}4\ \le\ 1+2K\,\operatorname{div}^*(s-2)\,\operatorname{div}^*\bigl((s-2)K\bigr)^2 . \] \end{theorem} \begin{proof} By Lemma~\ref{lem:O} at carry depth $0$ every pair $\{c,d\}$ has an entry $|M^{(0)}[c][d]|\le K$; orient $c\to d$ for such an entry. The convexity count in the proof of Theorem~\ref{thm:L} gives columns $d\ne d'$ and a set $\mathcal C$ of at least $(s-3)/4$ rows $c\notin\{d,d'\}$ with $1\le|M^{(0)}[c][d]|,|M^{(0)}[c][d']|\le K$ (entries at carry depth $0$ are nonzero integers). By Proposition~\ref{prop:F}, $M^{(0)}[c][d]=\lambda'_cg_{d,0}w_{cd}$, so $w_{cd}/w_{cd'}=(g_{d',0}/g_{d,0})\,M^{(0)}[c][d]/M^{(0)}[c][d']$ lies in $\varrho\,Q_K$ with $\varrho=g_{d',0}/g_{d,0}$, and Lemma~\ref{lem:C} with $n=s-2$ bounds $|\mathcal C|$. For the image of the places under $g\in PGL_2(\mathbb Q)$, in normalized homogeneous coordinates every $D_{cd}$, hence every $w_{cd}$, is multiplied by a factor depending only on $c$ times one depending only on $d$; the first cancels in $w_{cd}/w_{cd'}$ and the second changes only $\varrho$. \end{proof} By Wigert's divisor bound $\log\operatorname{div}(j)=O(\log j/\log\log j)$ \cite[Thm.~317]{HW08}, Theorem~\ref{thm:Lstar} gives $K(F)\ge s^{1-O(1/\log\log s)}$. The proof of Theorem~\ref{thm:L} finds a small entry in every pair from the ceilings, so the same count bounds $K_{\mathrm{pair}}(F_0)$ for integer designs reachable over $\mathbb Q$ from these places (or a $PGL_2(\mathbb Q)$-image that contains $\infty$) with nonzero diagonal, under the hypotheses of Theorem~\ref{thm:L} and for $q>2\max|F_0|$; there the bound $\sqrt{(s-3)/8}$ improves to $s^{1-o(1)}$. Theorem~\ref{thm:Lstar} uses the integer places: it says nothing about other rational frames, split frames (Proposition~\ref{prop:Gsplit}), or layers that are Cauchy only modulo $q$. So on these places the least certified deficit of the frame designs lies between $s^{1-o(1)}$ and $a^*_s\le e^{s\log s+O(s)}$ (Theorem~A). Proposition~\ref{prop:E} shows that row scalars constant on the finite places force a certified deficit at least $\operatorname{lcm}(1,\dots,s-3)$, so a certified deficit polynomial in $s$ would need row scalars that differ between the finite places. Whether such designs exist is open. \section{Drift and jitter}\label{sec:drift} The deficits of Section~\ref{sec:deficit} are at least $1$: Corollary~\ref{cor:ceil3} bounds the true distance for bounded layers with negative diagonal, and Lemma~\ref{lem:Kone} shows that the one-digit relaxation of Lemma~\ref{lem:Ss} never certifies a deficit below $1$. The column nets of Section~\ref{sec:m3} have the deficits $\frac13$ and $\frac23$ (Theorem~\ref{thm:col}); their layer $\mathbf 1a^{\!\top}$ has entries between $\frac{q-2}3$ and $\frac{q+2}3$, and their local configuration is the reflected cube $H\Z^3$ ($\Z^3$ reflected in the plane $\mathbf 1^\perp$, Section~\ref{sec:m3}), not a slightly rotated $\Z^3$. This section explains these two numbers. Going once around the torus along an axis takes $q$ steps from column to column. The reflected cube wants every step to raise the height by a third of a cell, and $q$ thirds of a cell is not a whole number of cells when $3\nmid q$. So the steps of each such cycle must deviate from the ideal by a forced total, the \emph{holonomy} of the cycle. The \emph{drift} is this forced deviation spread evenly over the cycle, and the \emph{jitter} is the further deviation caused by the third digit. Theorem~\ref{thm:O} shows that the resulting deficits $\frac13$ and $\frac23$ are optimal among all column point sets that are $(0,3,3)$-nets, digital or not. Section~\ref{sec:halfms} applies the same bookkeeping to the half-integer layers of dimension five, where it lowers the base from which the constants exceed $1$. The two words are used in the same sense in the rest of Part~II. The \emph{drift} of a step is the part of the change of the offsets that comes from the geometric digits, beyond their limit as $q\to\infty$: it is of order $1/q$ of a cell per step, so the points deviate from the limit configuration by a ramp that is linear in the cell. The \emph{jitter} is the part that comes from the deeper digits (Section~\ref{sec:prelim}). For the column nets the limit configuration is the reflected cube, and the drift of a step along $\e_i$ is $w_i/(3q)$ of a cell (Lemma~\ref{lem:drift}). \subsection{Excess, drift and jitter}\label{sec:excess} Let $X$ be a column point set in base $q$ (Section~\ref{sec:m3}). On a step from the column $[n]$ to $[n+\e_i]$ the height changes by $\Delta_i([n])=\varphi([n+\e_i])-\varphi([n])$ (Theorem~\ref{thm:colmin}); let $u$ be its representative modulo $q^2$ in $(-\frac16q^2,\frac56q^2]$, and call $\xi=u-\frac13q^2$ the \emph{excess} of the step. In $H\Z^3$ each step to a neighbouring column raises the height by a third of a cell, $\frac13q^2$ at integer scale $q^3$, so the excess measures the deviation from the reflected cube. If every excess lies in $[-\frac16q^2,\frac13q^2]$, then $\operatorname{dist}(u,\frac23q^2+q^2\Z)=\frac13q^2-\xi$, so in Theorem~\ref{thm:colmin} $g=\frac13q^2-\eta$ with $\eta$ the largest excess, and \[ d^2=q^4-2q^2\eta+3\eta^2,\qquad \delta N^{1/3}=1-\frac\eta{q^2}+O\Bigl(\frac{\eta^2}{q^4}\Bigr). \] The excesses are integers at scale $q^3$, where a cell has side $q^2$: an excess $\eta$ costs $\eta/q^2$ of a cell, that is, a first-order deficit $\eta/q$. For the digital column nets whose shared second digit is $a\cdot n$ with $a=\frac13(q\mathbf 1+w)\approx\frac q3\mathbf 1$, as in Definition~\ref{def:col}, the excess splits into two terms. \begin{lemma}[Drift and jitter]\label{lem:drift} Let $q\ge2$ with $3\nmid q$, let $w\in\Z^3$ with $w_x+w_y+w_z=0$ and $w_c\equiv-q\pmod3$, $a=\frac13(q\mathbf 1+w)$, and $b\in\Z^3$ with $b_x+b_y+b_z\equiv0\pmod q$ (the third-digit row, as in Definition~\ref{def:col}); let $X$ be the digital net over $\Z/q\Z$ whose point of cell $n$ has, in every coordinate $c$, the digits $n_c$, $a\cdot n$, $b\cdot n$ (a \emph{digital column net}; $\mathcal C_q$ is the case of Definition~\ref{def:col}), and let $\bar b_i\in\{0,\dots,q-1\}$ be the residue of $b_i$ (a third-digit step, not a geometric digit). Put $\varepsilon_i=\frac13qw_i+\bar b_i$. \begin{enumerate} \item A step along $\e_i$ has $\Delta_i\equiv\frac13q^2+\varepsilon_i$ if the third digit $b\cdot n$ does not wrap (does not pass from $q-1$ to $0$) and $\Delta_i\equiv\frac13q^2+\varepsilon_i-q$ if it wraps, modulo $q^2$; if $\bar b_i\ne0$ both occur. \item Along each cycle of $q$ steps in the direction $\e_i$ the third digit wraps exactly $\bar b_i$ times. If moreover $|\varepsilon_i|<\frac12q^2$ and $|\varepsilon_i-q|<\frac12q^2$ (for instance if $|w_i|\le2$ and $q\ge4$), the excesses of the cycle are $\varepsilon_i$ and $\varepsilon_i-q$, and they sum to $\frac13q^2w_i$. \item If $q\ge10$ and $|w_c|\le2$, then $\eta=\max_i\varepsilon_i$, so $d^2=q^4-2q^2\eta+3\eta^2$. \item If $X$ is a $(0,3,3)$-net, then $\varepsilon_x+\varepsilon_y+\varepsilon_z\ge q$, and $\max_i\varepsilon_i\ge\frac13q$; if moreover $q\equiv1\pmod3$, then $\max_i\varepsilon_i\ge\frac23q$. \end{enumerate} \end{lemma} \begin{proof} (1) The digit $a\cdot n$ changes by $a_i$ and $b\cdot n$ by $\bar b_i$ modulo $q$, so $\Delta_i\equiv qa_i+\bar b_i-q\omega=\frac13q^2+\frac13qw_i+\bar b_i-q\omega$ modulo $q^2$, with $\omega\in\{0,1\}$ the wrap of the third digit (a wrap of $a\cdot n$ changes $\Delta_i$ by $q^2$). (2) Along a cycle the third digit takes the values $b\cdot n+t\bar b_i\bmod q$, $t=0,\dots,q-1$, and returns after a total increase $q\bar b_i$, so it wraps exactly $\bar b_i$ times and does not wrap $q-\bar b_i\ge1$ times; this also shows that both cases of (1) occur if $\bar b_i\ne0$. Under the range condition the representatives $\frac13q^2+\varepsilon_i$ and $\frac13q^2+\varepsilon_i-q$ of (1) lie in the window $(-\frac16q^2,\frac56q^2]$, so the excesses are $\varepsilon_i$ and $\varepsilon_i-q$, and those of the cycle sum to $q\varepsilon_i-q\bar b_i=\frac13q^2w_i$. (Without it the identity can fail: for $q=5$, $w=(-8,4,4)$ and $b=(0,1,-1)$ the excesses of a cycle in the direction $\e_x$ sum to $\frac{175}3$, not $-\frac{200}3$.) (3) Then $-\frac53q\le\varepsilon_i-q<\varepsilon_i\le\frac53q$, and $\frac53q\le\frac16q^2$ for $q\ge10$, so every excess lies in $[-\frac16q^2,\frac13q^2]$ and the largest one is $\max_i\varepsilon_i$ (attained, since the third digit does not wrap on some step of each cycle, by (2)). (4) As $w_x+w_y+w_z=0$, $\sum_i\varepsilon_i=\sum_i\bar b_i\equiv\sum_ib_i\equiv0\pmod q$, and $\bar b\ne0$: otherwise $b\equiv0$ and the box with depth $3$ on a coordinate, whose minor is a $2\times2$ minor of $(a,b)$ (Lemma~\ref{lem:box}), would be singular. So $\sum_i\bar b_i\ge q$ and $\max_i\varepsilon_i\ge\frac13q$. For $q\equiv1\pmod3$, $w_c\equiv2\pmod3$ and $\sum_cw_c=0$, so some $w_c\ge2$ and $\varepsilon_c\ge\frac23q$. \end{proof} We call $\frac13qw_i$, the mean excess per step of a cycle in the direction $\e_i$, the \emph{drift} of that direction, and $\bar b_i$, the step of the third digit, its \emph{jitter}: on the $q-\bar b_i$ steps of a cycle where the third digit does not wrap the excess is the drift plus $\bar b_i$, and on the $\bar b_i$ steps where it wraps it is the drift plus $\bar b_i-q$. By (3) and (4), the digital column nets with $|w_c|\le2$ have $\eta\ge\frac13q$, so their deficit is at least $\frac13$, or $\frac23$ for $q\equiv1\pmod3$: by (4) the net property forces the third-digit steps $\bar b_x+\bar b_y+\bar b_z$ to add up to at least $q$, which the three directions share, and for $q\equiv1\pmod3$ the integrality of $a$ alone forces some direction to have drift at least $\frac23q$. For $\mathcal C_q$ (Definition~\ref{def:col}), $\bar b=(q-1,1,0)$ and \begin{align*} \varepsilon&=\bigl(\tfrac13q-1,\ \tfrac13q+1,\ \tfrac13q\bigr)&&(q\equiv2,\ w=(-2,1,1)),\\ \varepsilon&=\bigl(\tfrac23q-1,\ 1-\tfrac13q,\ \tfrac23q\bigr)&&(q\equiv1,\ w=(-1,-1,2)), \end{align*} so $\eta=\frac13q+1$ resp.\ $\frac23q$, in agreement with Theorem~\ref{thm:col}. Both nets put the residue $q-1$ (a third digit that wraps on $q-1$ of the $q$ steps of each cycle and adds $q-1$ to the excess on the remaining step, Lemma~\ref{lem:drift}(2)) on a direction with the most negative drift, $w_x=-2$ resp.\ $w_x=-1$, where the excess stays below the maximum, and the residue $0$ on a direction with the largest drift. Figure~\ref{fig:drift}(a) shows the excesses along the three cycles through one column. \begin{figure}[tbp] \centering \IfFileExists{figs/col_drift.pdf}{\includegraphics[width=\linewidth,height=0.5\textheight,keepaspectratio]{figs/col_drift.pdf}}{\fbox{\parbox{0.9\textwidth}{\centering figure \texttt{col\_drift} pending}}} \caption{\colDriftCaption\interactive{anatomy.html}} \label{fig:drift} \end{figure} \subsection{The holonomy, and the optimum among column nets} No column point set beats the cubic value $\delta N^{1/3}=1$, and among those that are $(0,3,3)$-nets the column nets of Definition~\ref{def:col} are optimal. \begin{corollary}[Column ceiling]\label{cor:colceil} Every column point set in base $q\ge2$ has $g\le\frac13q^2$ in Theorem~\ref{thm:colmin}, i.e. toroidal minimum distance at most $q^2$ at integer scale $q^3$: $\delta N^{1/3}\le1$. \end{corollary} \begin{proof} If $g>\frac13q^2$, every step value $\Delta_i([n])$ has distance more than $\frac13q^2$ from $\frac23q^2+q^2\Z$, so its residue modulo $q^2$ lies in the open interval $(0,\frac13q^2)$. The three steps $[n]\to[n+\e_x]\to[n+\e_x+\e_y]\to[n+\mathbf 1]=[n]$ have step values that sum to $0$, so their residues sum to a multiple of $q^2$; but that sum lies in $(0,q^2)$. \end{proof} So a net at $m=3$ with $\delta N^{1/3}>1$ cannot be a column point set. Proposition~\ref{prop:twocoset} shares the geometric digit between two coordinates only, and Proposition~\ref{prop:wnet} uses a layer of rank one, not of the column form $\mathbf 1a^{\!\top}$, under which all three coordinates still wrap together (Section~\ref{sec:wnet}). \begin{theorem}[Column optimum]\label{thm:O} Let $q\ge2$. \begin{enumerate} \item A column point set, with the set of columns $\mathbb T$ and the heights $\varphi$ of Section~\ref{sec:m3}, is a $(0,3,3)$-net in base $q$ if and only if $\varphi$ is a bijection from $\mathbb T$ onto $\{0,\dots,q^2-1\}$ and $\lfloor\varphi/q\rfloor$ is injective on every line of $\mathbb T$ in the directions $[\e_x],[\e_y],[\e_z]$. \item If $q$ is even, no column point set is a $(0,3,3)$-net in base $q$. \item If $q\ge5$ is odd and $3\nmid q$, every column point set that is a $(0,3,3)$-net in base $q$ has toroidal minimum distance $d$ with $d^2\le d^2(\mathcal C_q)$, the value of Theorem~\ref{thm:col}(2); equivalently $\eta\ge\frac13q+1$ for $q\equiv2$ and $\eta\ge\frac23q$ for $q\equiv1\pmod3$, where $\eta=\frac13q^2-g$. \end{enumerate} \end{theorem} \begin{proof} (1) The point of cell $n$ has the digits $n_c$, $\lfloor\varphi/q\rfloor$, $\varphi\bmod q$ in every coordinate. The box of shape $(1,1,1)$ holds one point per cell. A box with depth $3$ on $c$ prescribes $n_c$ and $\varphi$; the $q^2$ cells with a given $n_c$ meet every column exactly once ($n+t\mathbf 1$ for the one $t$ with $n_c+t$ given), so these boxes hold one point each if and only if $\varphi$ is a bijection. A box with depth $2$ on $c$, $1$ on $c'$ and $0$ on $c''$ prescribes $n_c$, $n_{c'}$ and $\lfloor\varphi/q\rfloor$; the $q$ cells with given $n_c,n_{c'}$ form a line in the direction $\e_{c''}$, whose columns form a line of $\mathbb T$ in the direction $[\e_{c''}]$, and every such line of $\mathbb T$ arises. These are all box shapes. (2) Identify $\mathbb T$ with $(\Z/q\Z)^2$ by $[n]\mapsto(n_x-n_z,n_y-n_z)$; the lines in the directions $[\e_x]$, $[\e_y]$, $[\e_z]$ become the horizontal lines, the vertical lines and the lines of slope $1$ of $(\Z/q\Z)^2$. A level set $\{\lfloor\varphi/q\rfloor=v\}$ has $q$ elements and meets each of these lines once, by (1). So it is the graph of a permutation $\pi$ of $\Z/q\Z$ (once on every horizontal and every vertical line) such that $y\mapsto\pi(y)-y$ is a permutation too (once on every line of slope $1$). Then $\sum_y(\pi(y)-y)\equiv0$, while the sum over all residues is $\frac q2\not\equiv0\pmod q$ for even $q$. In other words, $y\mapsto\pi(y)-y$ is a complete mapping of $\Z/q\Z$, and this is the classical argument that a cyclic group of even order has none \cite[Thm.~1 and its corollary]{Pai51}; more generally, no finite group with a nontrivial cyclic Sylow $2$-subgroup has one \cite[Thm.~5]{HP55}. In the language of Latin squares, (1) says that each level set is a transversal of the cyclic Latin square $(a,b)\mapsto b-a$, and that the array $\lfloor\varphi/q\rfloor$ on $(\Z/q\Z)^2$ is a Latin square orthogonal to it. (3) Put $\eta=\frac13q^2-g$. By Corollary~\ref{cor:colceil}, $0\le\eta\le\frac13q^2$, and by Theorem~\ref{thm:colmin} $d^2=q^4-2q^2\eta+3\eta^2$, which decreases in $\eta$ on this interval and equals $d^2(\mathcal C_q)$ at $\eta_0=\frac13q+1$ ($q\equiv2$) resp.\ $\eta_0=\frac23q$ ($q\equiv1$). So it suffices to show $\eta\ge\eta_0$. Suppose $\eta<\eta_0$. \emph{Then $\eta<\frac1{12}q^2$.} For $q\ge8$ this holds because $\eta_0\le\frac1{12}q^2$. For $q=5$ and $7$: $3g=|3U+q^2|$ for an integer $U$, which is not divisible by $3$ since $q^2\equiv1\pmod3$, so $3\eta=q^2-3g\equiv0$ or $2\pmod3$; hence $\eta<\frac83$ gives $\eta\le2<\frac{25}{12}$ at $q=5$, and $\eta<\frac{14}3$ gives $\eta\le4<\frac{49}{12}$ at $q=7$. \emph{Windows and triangles.} Every step value has distance at least $\frac13q^2-\eta$ from $\frac23q^2+q^2\Z$, so it has a unique representative $u$ in the window $[-\eta,\frac13q^2+\eta]$ modulo $q^2$ (narrower than the window of Section~\ref{sec:excess}). As $[\e_x]+[\e_y]+[\e_z]=0$ in $\mathbb T$, for every column and every order of the three directions the three steps around the triangle $[n]\to[n+\e_i]\to[n+\e_i+\e_{i'}]\to[n]$ have values summing to $0$, so their representatives sum to a multiple of $q^2$ in $[-3\eta,q^2+3\eta]$, i.e. to $0$ or $q^2$. In a triangle with sum $0$ every $u$ lies in $[-\eta,2\eta]$, and in one with sum $q^2$ every $u$ lies in $[\frac13q^2-2\eta,\frac13q^2+\eta]$; as $\eta<\frac1{12}q^2$ these ranges are disjoint, so two triangles that share a step have the same sum. The step $([n],i)$ shares a triangle with $([n+\e_i],i')$ for both $i'\ne i$, and two such moves give $([n+\e_i+\e_{i'}],i)=([n-\e_{i''}],i)$; and the classes $[\e_{i''}]$, $i''\ne i$, generate $\mathbb T$; so all steps are connected through shared triangles, and all triangles have the same sum. Sum $q^2$ means that every step is close to $\frac13q^2$, the configuration of the reflected cube $H\Z^3$; sum $0$ means that every step is close to $0$, the configuration of $\Z^3$. \emph{Sum $q^2$ (the reflected cube).} Put $\xi=u-\frac13q^2\in[-2\eta,\eta]$, the excess; then $\operatorname{dist}(u,\frac23q^2+q^2\Z)=\frac13q^2-\xi$, so $\eta$ is the largest excess, and every triangle has excess sum $0$. The class $[\e_i]$ has order $q$ in $\mathbb T$, so the steps in the direction $\e_i$ form $q$ cycles $\gamma$ of length $q$. Along a cycle the step values telescope to $0$, so $\sum_{\gamma}u\in q^2\Z$, and the \emph{cycle excess} $\Xi_\gamma=\sum_\gamma\xi\in q^2\Z-\frac13q^3$. Summing the excess over the triangles $[n]\to[n+\e_x]\to[n+\e_x+\e_y]\to[n]$ for all $[n]\in \mathbb T$ counts every step once, so the $3q$ cycle excesses sum to $0$. Also $-2q\eta\le\Xi_\gamma\le q\eta$. If $q\equiv1\pmod3$, then $\Xi_\gamma=q^2z_\gamma-\frac13q^2$ with integers $z_\gamma$ (as $\frac13q^3\equiv\frac13q^2$), and the cycle excesses sum to $0$ only if $\sum_\gamma z_\gamma=q$. So some $z_\gamma\ge1$, $\Xi_\gamma\ge\frac23q^2$, and $q\eta\ge\frac23q^2$: $\eta\ge\frac23q$, a contradiction. If $q\equiv2\pmod3$, then $\Xi_\gamma=q^2z_\gamma-\frac23q^2$ and $\sum_\gamma z_\gamma=2q$. Every excess lies in $\Z+\frac23$ (as $u\in\Z$ and $\frac13q^2\in\Z+\frac13$), and so does $\frac13q$; hence $\eta<\frac13q+1$ gives $\eta\le\frac13q$, and then every $\Xi_\gamma\in[-\frac23q^2,\frac13q^2]$, i.e. $\Xi_\gamma\in\{-\frac23q^2,\frac13q^2\}$. A cycle excess $\frac13q^2\le q\eta$ forces $\xi=\eta=\frac13q$ on each step of its cycle, and a cycle excess $-\frac23q^2\ge-2q\eta$ forces $\xi=-2\eta=-\frac23q$ on each step. So every step has $u=\frac13q(q+1)$ or $\frac13q(q-2)$, a multiple of $q$, and $\varphi$ is constant modulo $q$ on $\mathbb T$. This contradicts (1): $\varphi$ takes every value in $\{0,\dots,q^2-1\}$. \emph{Sum $0$ (the cube).} Now $u\in[-\eta,2\eta]$ and $\operatorname{dist}(u,\frac23q^2+q^2\Z)=u+\frac13q^2$, so $\min u=-\eta$. The cycle sums $\sum_\gamma u$ lie in $q^2\Z$ and add up to $0$. If one of them is nonzero, one is at most $-q^2$; it is at least $-q\eta$, so $\eta\ge q$. Otherwise every cycle closes: the partial sums $S_t$ of $u$ along a cycle are integers with $S_t\equiv\varphi(c_t)-\varphi(c_0)\pmod{q^2}$, where $c_t$ are its columns. By (1) the $q$ values $\varphi(c_t)$ lie in $q$ different blocks $[kq,kq+q)$, so the $q$ values $\varphi(c_0)+S_t$ meet all $q$ blocks modulo $q^2$; an interval of integers of length $\mathrm{rg}$ meets at most $\mathrm{rg}/q+2$ of them, so the range of the walk is $\mathrm{rg}\ge q(q-2)$. Here $\eta>0$, since $\eta=0$ would make every step value $0$ and $\mathrm{rg}=01$ for every prime $p\ge37$ (Proposition~\ref{prop:half5}). \section{One point per cube: cube tilings and the value 1}\label{sec:whylat} The lattice picture uses only that the geometric digit is linear in the cell digits. This section makes precise what that linearity forces: offsets that are affine in the cell between wraps, so locally a lattice, but broken at the slip planes where a geometric digit wraps, so globally not a lattice. It also shows what the cube cells of the resolutions $m=sk$ cost. Call the position of a point inside its cell its \emph{phase}; for a digital net it is, to first order, $r/q$ with $r$ the geometric digits. By the cube-tiling theorems of Haj\'os and Keller, configurations with one point per unit cube whose phases vary independently are no better than $\Z^s$ (Proposition~\ref{prop:cube}). In Lemma~\ref{lem:hom} and Corollary~\ref{cor:linoff} only, the digits of an integer $0\le P0$ such that $f$ is constant on $x-[0,\varepsilon)^s$: $S$ is locally finite, so we may choose $\varepsilon$ below every positive difference $x_c-\sigma_c$ and $x_c+1-\sigma_c$ for the finitely many $\sigma\in S$ near the cube, and then shifting the cube by $-t$, $t\in[0,\varepsilon)^s$, neither adds nor removes a point. If $f\equiv1$, the cubes $\sigma+(-1,0]^s$, $\sigma\in S$, tile $\R^s$, since $x\in\sigma+(-1,0]^s$ if and only if $\sigma\in x+[0,1)^s$. (i) Density gives a point of $\Z^s+L$ in every $x-[0,\varepsilon)^s$, so $f\equiv1$, and the cubes form a lattice tiling. By Haj\'os's theorem (Minkowski's conjecture) \cite{Haj42} two of them share a facet, so $L$ contains some $\e_c$. (ii) For each $c$, $\pi_c(L)+\Z$ is a finitely generated subgroup of $\mathbb Q$ containing $\Z$, hence equal to $\frac1{e_c}\Z$ for an integer $e_c\ge1$. For $x_c\in(\frac{j-1}{e_c},\frac j{e_c}]$ and $\sigma\in L$ the condition $x_c\le\sigma_cm$. For nets that are not digital only these constraints and the other box conditions remain (Remark~\ref{rem:open}(11)). \section{Beyond the cubic value: Euclidean skeletons and half-integer layers}\label{sec:large} By Proposition~\ref{prop:cube}, a set with one point per unit cube can have minimum distance above the cubic value $1$ only if the phases of its points are correlated (coherent, in the idealized setting of that proposition). For an invertible layer every pattern of wraps occurs (Lemma~\ref{lem:phase}(i)), so a step to a neighbouring cell keeps its full length only if its glue is $\frac12$ in every coordinate that does not move (Remark~\ref{rem:glue}). Three ways above $1$ occur in this paper. \begin{enumerate} \item[(i)] \emph{Euclidean skeletons} (Section~\ref{sec:S1E}). At $m\not\equiv0\pmod s$ the cells are boxes, longer in the skeleton coordinates, not cubes, so the cube constraint does not apply (Section~\ref{sec:skel}). In the Euclidean norm the certified constants exceed $1$ at $m\equiv-1\pmod s$ at the large primes of Table~\ref{tab:S1E}; whether this persists for all large $p$ is open (Remark~\ref{rem:S1Elim}). \item[(ii)] \emph{Locked wraps at $m=3$} (Sections~\ref{sec:twocoset} and~\ref{sec:wnet}). A singular layer can force coordinates to wrap together. In $\mathcal T_p$ two coordinates share one geometric digit, and the points lie on a union of two lattice cosets; in $\mathcal W_p$ the layer has rank one, and all three coordinates wrap together. \item[(iii)] \emph{Half-integer layers} (Sections~\ref{sec:Pnohalf}--\ref{sec:halfms}). Layers whose entries are half-integers, read modulo $q$ via $\frac12\equiv(q+1)/2$, so that each centred entry is about $\pm q/2$ plus a bounded amount (regime III, Figure~\ref{fig:arclarge}), put the points locally on the packing $D_s^+$. For $k\ge2$ the frame orbit cannot provide them (Proposition~\ref{prop:P}: for rational designs, and over $\F_p$ for large $p$), but split frames give them at every resolution $m=sk$ (Theorem~\ref{thm:HL}); at $k=1$ no frame is needed (Section~\ref{sec:halfms}). \end{enumerate} \subsection{Euclidean skeletons}\label{sec:S1E} \emph{Euclidean form.} The bound of Theorem~\ref{thm:S1}(1) measures the centred difference $a$ of the fine-cell positions of two points (Lemma~\ref{lem:skelred}) in the maximum norm. Every skeleton digit $S_z=\sum_d\kappa_dx_{d,k}$ with all $\kappa_d\ne0$ can be realized (Proposition~\ref{prop:skelcomp}(ii)), and in the Euclidean norm the best choice is not $\kappa_d=\rho^d$ (Remark~\ref{rem:codes}(c)). \begin{proposition}[Euclidean form of Theorem~\ref{thm:S1}(1)]\label{prop:S1E} Let $s\ge3$, let $p\ge s-1$ be a prime, $k\ge0$, $m=sk+s-1$, and let $\kappa_1,\dots,\kappa_{s-1}$ be units of $\F_p$. The recipe of Proposition~\ref{prop:skelcomp}(ii) on the frame of Definition~\ref{def:S1} gives a digital $(0,m,s)$-net over $\F_p$ with the skeleton digit $S_z=\sum_d\kappa_dx_{d,k}$, and every graph form with the depths $(k+1,\dots,k+1,k)$ and this skeleton digit satisfies, for the toroidal distance, \[ \delta N^{1/s}\ \ge\ p^{-1/s}\lambda_{\mathrm{jit}}(\kappa)\ \ge\ p^{-1/s}\bigl(\lambda_2(\Lambda_\kappa)-\sqrt s\bigr),\qquad \lambda_{\mathrm{jit}}(\kappa)=\min_{0\ne a\in\Lambda_\kappa}\bigl\|(|a|-\mathbf 1)_+\bigr\|_2, \] where $\Lambda_\kappa=\{a\in\Z^s:a_z\equiv\sum_d\kappa_da_{x_d}\pmod p\}$ and $(|a|-\mathbf 1)_+$ is taken coordinatewise. (So $\lambda_{\mathrm{jit}}$ is the length of the shortest nonzero lattice vector after one fine cell of jitter is subtracted from each coordinate.) \end{proposition} \begin{proof} The net property is Proposition~\ref{prop:skelcomp}(ii); the $s$ places of Definition~\ref{def:S1} are distinct because $p\ge s-1$. As in the proof of Theorem~\ref{thm:S1}(1), $P_z\equiv\sum_d\kappa_dP_{x_d}\pmod p$ for every point, and the centred difference $a\ne0$ of the fine positions of two distinct points lies in $\Lambda_\kappa$. By Lemma~\ref{lem:skelred}(i) in each coordinate $c$, the two points are at toroidal distance at least $(|a_c|-1)_+\varpi$ in coordinate $c$, so their distance is at least $\varpi\|(|a|-\mathbf 1)_+\|_2\ge \varpi\lambda_{\mathrm{jit}}(\kappa)$, and $\varpi N^{1/s}/N=p^{-1/s}$. Finally $|a|\le(|a|-\mathbf 1)_++\mathbf 1$ coordinatewise gives $\|(|a|-\mathbf 1)_+\|_2\ge\|a\|_2-\sqrt s$. \end{proof} For $\kappa_d=\rho^d$ some $|a_c|$ is at least $\rho$, so $\lambda_{\mathrm{jit}}\ge\rho-1$ and the proposition contains Theorem~\ref{thm:S1}(1). Every index-$p$ sublattice of $\Z^s$ without unit vectors is a $\Lambda_\kappa$ after a permutation of the coordinates: it is the kernel of a functional $h$ modulo $p$, it contains no unit vector exactly when $h$ has full weight (all entries nonzero modulo $p$), and then $h$ can be scaled to $h_z=-1$. So the proposition can use the best of these lattices. Table~\ref{tab:S1E} lists certified values for searched $\kappa$; $\lambda_{\mathrm{jit}}^2$ is an integer, computed exactly by lattice reduction and complete enumeration and also checked in Lean (Appendix~\ref{app:formal}). At $p=1009$ they already beat Table~\ref{tab:skel} for $s=3,4,5$. The limit of the certified values as $p\to\infty$ is discussed in Remark~\ref{rem:S1Elim}. \begin{table}[tbp] \centering\small\setlength{\tabcolsep}{4pt} \begin{tabular}{@{}ccccccccc@{}} \toprule & \multicolumn{4}{c}{searched $\kappa$, at $p=$} & \multicolumn{2}{c}{rounded root lattice, $p'\approx$} & & \\ $s$ & $1009$ & $10007$ & $10^6{+}3$ & $10^8{+}7$ & $10^6$ & $10^{10}$ & $\sqrt{\gamma_s}$ & root lattice\\ \midrule 3 & .940 & 1.027 & 1.090 & 1.097 & 1.106 & 1.121 & 1.122 & $A_3$\\ 4 & .813 & .948 & 1.034 & 1.110 & 1.128 & 1.183 & 1.189 & $D_4$\\ 5 & .752 & .823 & .989 & 1.069 & 1.098 & 1.208 & 1.231 & $D_5$\\ 6 & .631 & .714 & .938 & 1.036 & 1.047 & 1.235 & 1.290 & $E_6$\\ 7 & .372 & .599 & .867 & .984 & .974 & 1.238 & 1.345 & $E_7$\\ 8 & .421 & $.547^*$ & .814 & .953 & .881 & 1.219 & 1.414 & $E_8$\\ \bottomrule \end{tabular} \medskip \caption{Certified lower bounds $\lambda_{\mathrm{jit}}(\kappa)p^{-1/s}$ for $\delta N^{1/s}$ (Proposition~\ref{prop:S1E}; toroidal, every $k\ge0$), truncated to three digits; every entry except those of the last two columns is such a certified bound. Left: $\kappa$ found by a random and local search; compare Table~\ref{tab:skel} at $p=1009$, $10007$, $10^6{+}3$. ${}^*$Here the search is weaker than $\kappa_d=\rho^d$, which gives $.632$. Right: $\kappa$ obtained by rounding a randomly rotated copy of the root lattice to an integer matrix of prime determinant $p'$; the primes are $p'=996001$, $1006301$, $1025909$, $1033493$, $1138273$, $1071337$ and $10002435263$, $10040205517$, $10123347709$, $10136355487$, $10660100951$, $11556622703$ for $s=3,\dots,8$. The last two columns give $\sqrt{\gamma_s}$ (truncated) and the lattice that attains it. The $\kappa$ are listed in Table~\ref{tab:S1Ekappa} (Appendix~\ref{app:tables}); they are also fixed data of the Lean formalization (Appendix~\ref{app:formal}).} \label{tab:S1E} \end{table} \subsection{Resolution \texorpdfstring{$m=3$}{m=3}: a layer of two cosets}\label{sec:twocoset} The column nets of Section~\ref{sec:m3} share the geometric digit among all three coordinates, and no column point set beats the cubic value (Corollary~\ref{cor:colceil}). Sharing it between two coordinates only, and letting the geometric digit of the third coordinate have coefficients $h\equiv-\frac12$, so that it moves by about half a cell between neighbouring cells, produces a local configuration that is not a lattice and whose minimum exceeds $1$, the minimum of the reflected cube $H\Z^3$ of the column nets (Section~\ref{sec:m3}; Figure~\ref{fig:arctwocoset}). \begin{proposition}[Two-coset nets at $m=3$]\label{prop:twocoset} Let $p\ge5$ be a prime, $h=(p-1)/2$, let $r$ be the integer nearest to $p/8$, and let $\theta=8r-p\in\{\pm1,\pm3\}$. Let $\mathcal T_p$ be the digital net over $\F_p$ whose generator matrices $C_x,C_y,C_z$ have the rows \[ C_x:\ \e_x,\ (0,h,h),\ (0,1,2);\qquad C_y:\ \e_y,\ (r,-1,1),\ (3,0,1);\qquad C_z:\ \e_z,\ (r,-1,1),\ (1,5,0). \] \begin{enumerate} \item $\mathcal T_p$ is a $(0,3,3)$-net if and only if $p\ne23$, and every cell of side $p^{-1}$ holds one point. \item For $p\ge41$ its toroidal (hence also Euclidean) minimum distance satisfies \[ \delta N^{1/3}\ \ge\ \sqrt{\frac{33}{32}-\frac{41}{16p}}-\frac{\sqrt3}p. \] The bound exceeds $\sqrt{66}/8-3.014/p$ for $p\ge83$, and it exceeds $1$ for $p\ge197$. \item For $p\ge7$ the points of the cells $0$ and $(p-1)\mathbf 1$ have toroidal distance $d_*$ at integer scale $p^3$, where $d_*^2=(p^2-2p+3)^2+((r-1)p+4)^2+((r-1)p+6)^2$. So $\delta N^{1/3}\le d_*/p^2$, and $d_*/p^2<\sqrt{66}/8-2/p$ for $p\ge53$. In particular $\delta N^{1/3}\to\sqrt{66}/8=1.0155\ldots$ as $p\to\infty$. \end{enumerate} \end{proposition} So $y$ and $z$ read the same geometric digit $rn_x-n_y+n_z$ (here $r$ is a parameter, not a geometric digit), and $x$ reads $h(n_y+n_z)$, with $h\equiv-\frac12$. The third rows only add jitter: part (2) holds for any third rows, and only the minors in part (1) use them. In this section and the next, $g(n)$ denotes the vector of geometric digits of the cell $n$ and $f(n)$ that of its third digits, both in $\{0,\dots,p-1\}^3$. \begin{proof} \emph{Net property.} The first rows are $\e_x,\e_y,\e_z$, so $\mathcal T_p$ is a graph form with the depths $(1,1,1)$, and we use Lemma~\ref{lem:box}: the box of shape $(1,1,1)$ has determinant $1$ and puts one point in every cell. A box with depth $2$ on $a$, $1$ on $b$ and $0$ on $c$ has, up to sign, the determinant $(C_a)_{2,c}$, the entry of the second row of $C_a$ in column $c$; these off-diagonal entries of the second rows are $h,h,r,1,r,-1$. The box with depth $3$ on $a$ has, up to sign, the $2\times2$ minor of the second and third rows of $C_a$ off column $a$; it is $2h-h=h$, $r-3$ and $5r+1$ for $a=x,y,z$. Here $1\le r2p^2$. So let $E\in\{-2,\dots,2\}^3\setminus\{0\}$, and put $S=E_y+E_z$ and $T=E_y-E_z$ (so $|S|\le4$, and $|S|\le3$ if $T$ is odd). As $C_y$ and $C_z$ have the same second row, $g_y(n)=g_z(n)$ for every cell, so $\Delta g=(\xi,\eta,\eta)$ with integers $\xi,\eta\in(-p,p)$: \emph{the $y$- and $z$-digits wrap together}. Since $\xi\equiv hS$, $2h\equiv-1$, $\eta\equiv rE_x-T$ and $8r=p+\theta$, there are integers $i$, $j$ with \[ 2\xi=jp-S,\qquad 8\eta=ip+\theta E_x-8T. \] Reducing modulo $2$ and modulo $8$, with $p$ odd and $\theta\equiv-p\pmod8$, gives $j\equiv S\pmod2$ and $i\equiv E_x\pmod8$. As $|\xi|0$. If $|T|\le1$, then $|\theta E_x-8T|\le140$, and $i=0$ if $E_x=T=0$. In cell units $D/p^2=V+\Delta f/p^2$, with $|\Delta f|<\sqrt3\,p$ and $V=E+(\xi,\eta,\eta)/p$. So $|D|/p^2>|V|-\sqrt3/p$, and it suffices to show $|V|^2\ge\frac{33}{32}-\frac{41}{16p}$. With the half-integer $\alpha=E_x+\frac j2$ and the quarter-integer $\beta=S+\frac i4$ (the limits of $V_x$ and $V_y+V_z$ as $p\to\infty$), \[ V_x=\alpha-\frac S{2p},\qquad V_y+V_z=\beta+\frac{\theta E_x-8T}{4p},\qquad V_y-V_z=T,\qquad |V|^2=V_x^2+\tfrac12\bigl((V_y+V_z)^2+T^2\bigr). \] The terms of order $1/p$ are the drift (Section~\ref{sec:drift}): the deviation of the geometric digits from their limit values. Each numerical estimate below increases with $p$ and is evaluated at $p=41$. \emph{The cases.} (a) $|T|\ge2$: $|V|^2\ge T^2/2\ge2$. (b) $T=0$, $E_x=0$: then $S\ne0$ is even and $i=0$, so $V_y=V_z=E_y$ and $|V|^2\ge S^2/2\ge2$. (c) $|T|=1$, $E_x=0$: $j$ is odd, so $|\alpha|=\frac12$ and $|V_x|\ge\frac12-\frac3{2p}$; $i\in\{0,\pm8\}$, so $\beta$ is an odd integer and $|V_y+V_z|\ge1-\frac2p$. Hence $|V|^2\ge(\frac12-\frac3{2p})^2+\frac12(1-\frac2p)^2+\frac12>1.16$. (d) $|E_x|=2$. If $|T|=1$, then $j=\pm1$, $|\alpha|\ge\frac32$ and $|V|^2\ge(\frac32-\frac3{2p})^2>2$. If $T=0$, then $S$ and hence $j$ are even, so $\alpha$ is an integer, and $\alpha\ne0$ (which would need $j=\mp4$); so $|V_x|\ge1-\frac2p$; and $4\beta=4S+i\equiv E_x\equiv2\pmod4$, so $|\beta|\ge\frac12$ and $|V_y+V_z|\ge\frac12-\frac3{2p}$. Hence $|V|^2\ge(1-\frac2p)^2+\frac12(\frac12-\frac3{2p})^2=\frac98-\frac{19}{4p}+\frac{41}{8p^2}$, which is at least $\frac{33}{32}-\frac{41}{16p}$ for $p\ge24$. (e) $|E_x|=1$, $T=0$: $\alpha\in\Z$, $i\in\{E_x,-7E_x\}$, and $4\beta=4S+i\equiv E_x\pmod8$, so $|\beta|\in\{\frac14,\frac74,\frac94,\dots\}$; moreover $|V_x-\alpha|\le\frac2p$ and $|V_y+V_z-\beta|\le\frac3{4p}$. If $\alpha=0$, then $j=-2E_x$, so $E_xS\le-2$ and $E_x\beta=E_xS+E_xi/4\le-\frac74$; then $|V|^2\ge\frac12(\frac74-\frac3{4p})^2>1.4$. If $|\alpha|\ge2$, or $|\alpha|=1$ and $|\beta|\ge\frac74$, then also $|V|^2>1.4$. There remains $|\alpha|=1$, $|\beta|=\frac14$. Since $S=\beta-\frac i4$ is an even integer, $\beta=\frac{E_x}4$ and $S\in\{0,2E_x\}$, and if $S=2E_x$ then $\alpha=E_x$ (as $|j|\le2$). With $(a-b)^2\ge a^2-2ab$, \[ |V|^2\ \ge\ \alpha^2-\frac{\alpha S}p+\frac12\Bigl(\beta^2+\frac{\beta\theta E_x}{2p}\Bigr)=\frac{33}{32}-\frac{\alpha S-\theta/16}p\ \ge\ \frac{33}{32}-\frac{35}{16p}, \] as $\alpha S\in\{0,2\}$ and $\theta\ge-3$. (f) $|E_x|=1$, $|T|=1$: $j=\pm1$, so $\alpha\in\{\pm\frac12,\pm\frac32\}$; $i\in\{E_x,-7E_x\}$, and $4\beta=4S+i\equiv4+E_x\pmod8$, so $|\beta|\in\{\frac34,\frac54,\frac{11}4,\dots\}$; moreover $|V_x-\alpha|\le\frac3{2p}$ and $|V_y+V_z-\beta|\le\frac{11}{4p}$, as $|\theta E_x-8T|\le11$. If $|\alpha|=\frac32$, then $|V|^2\ge(\frac32-\frac3{2p})^2>2$. If $|\beta|\ge\frac54$, then $|V|^2\ge(\frac12-\frac3{2p})^2+\frac12(\frac54-\frac{11}{4p})^2+\frac12>1.41$. There remains $|\alpha|=\frac12$, $|\beta|=\frac34$. Then $\alpha=\frac{E_x}2$ and, from $4\beta\equiv4+E_x\pmod 8$, $\beta=-\frac34E_x$; and $S=\beta-\frac i4=\pm E_x$. As in (e), \[ |V|^2\ \ge\ \alpha^2-\frac{\alpha S}p+\frac12\Bigl(\beta^2+\frac{\beta(\theta E_x-8T)}{2p}+T^2\Bigr)=\frac{33}{32}-\frac{\frac12E_xS-\frac32E_xT+\frac3{16}\theta}{p}\ \ge\ \frac{33}{32}-\frac{41}{16p}, \] as $E_xS,E_xT\in\{\pm1\}$ and $\theta\le3$. So $|V|^2\ge\frac{33}{32}-\frac{41}{16p}$ in every case, which proves (2). The bound increases with $p$; it is $1.0002\ldots$ at $p=197$ and below $1$ at $p=193$. For $p\ge83$ we have $\frac{33}{32}-\frac{41}{16p}\ge1$, and $\sqrt a-\sqrt{a-x}=x/(\sqrt a+\sqrt{a-x})\le x/2$ gives the bound $\sqrt{66}/8-(\frac{41}{32}+\sqrt3)/p$, with $\frac{41}{32}+\sqrt3<3.014$. \emph{The pair.} For $n'=(p-1)\mathbf 1\equiv-\mathbf 1$ one has $g(n')=(1,p-r,p-r)$ and, as the entries of the third rows sum to $3$, $4$, $6$, $f(n')=(p-3,p-4,p-6)$; also $g(0)=f(0)=0$. The representative with $E=-\mathbf 1$ is $D_*=-\bigl(p^2-2p+3,\,(r-1)p+4,\,(r-1)p+6\bigr)$. Since $|D_{*,c}|3$ in this form. Other mechanisms that correlate the wraps at $k\ge2$, not through equal rows, are not ruled out (Remark~\ref{rem:open}(7)). \emph{Computed values.} At eight primes $101\le p\le1009$ the exact minimum distance of $\mathcal T_p$ is attained by the pair of (3), from $\delta N^{1/3}=0.9949\ldots$ at $p=101$ to $1.0132\ldots$ at $p=1009$ (computed; Figure~\ref{fig:arctwocoset}(c), Appendix~\ref{app:verify}). The column nets have, exactly, $0.9966\ldots$ at $p=101$ and $0.9978\ldots$ at $p=307$ (Theorem~\ref{thm:col}). Compared at every prime $7\le p\le199$ other than $23$ (computed), $\mathcal T_p$ is ahead at $103$, $109$ and at every prime $127\le p\le199$, and $\mathcal C_p$ at the other primes, all of them $p\le113$. For $p\ge197$ the bound of Proposition~\ref{prop:twocoset}(2) exceeds $1$, so by Corollary~\ref{cor:colceil} $\mathcal T_p$ is ahead of every column point set at every prime $p\ge197$. The rank-one nets $\mathcal W_p$ of the next subsection are ahead of every column point set already at every prime $p\ge17$, and of $\mathcal T_p$ (Section~\ref{sec:wnet}). \begin{figure}[tbp] \centering \arcfig{arc_twocoset} \caption{\arcTwoCosetCaption} \label{fig:arctwocoset} \end{figure} \subsection{Resolution \texorpdfstring{$m=3$}{m=3}: a rank-one layer}\label{sec:wnet} In $\mathcal T_p$ the coordinates $y$ and $z$ wrap together, and $x$ wraps independently of them. Such a coordinate must move by exactly one cell along its own axis, and its glue must be $\frac12$ (Remark~\ref{rem:glue}; see \emph{Why locking pays} below). Making all three coordinates wrap together removes this restriction and gives a larger constant. We first say what locked wraps are, for any layer at $m=3$. In this section the \emph{layer} is the matrix $B$ over $\F_p$ of the geometric digits, $r(n)=Bn\bmod p$. \emph{Locked wraps.} Write a digital $(0,3,3)$-net over $\F_p$ in graph form with depths $(1,1,1)$, as in Lemma~\ref{lem:phase}(i): coordinate $c$ of the point of cell $n$ has the digits $n_c$, $r_c(n)=(Bn)_c$ and a third digit. The phase of a cell is given by its geometric digits $r(n)$, which run exactly over the image $W=B\F_p^3$, and by Lemma~\ref{lem:phase}(i) coordinate $c$ wraps on the step $E$ if and only if $r_c(n)\ge p-\beta_c$. So the wrap vectors $\omega$ that occur on the step $E$ are exactly those for which $W$ meets the box $\prod_cI_c$, with $I_c=[p-\beta_c,p)$ if $\omega_c=1$ and $I_c=[0,p-\beta_c)$ if $\omega_c=0$. If $B$ is invertible, $W=\F_p^3$ and every wrap vector occurs: the free wraps of Lemma~\ref{lem:phase}(i). If $B$ is singular, every integer vector $\mathbf h$ with $\mathbf h^{\!\top}B\equiv0\pmod p$ (a \emph{relation}) gives $\mathbf h\cdot r(n)\equiv0$ at every cell, and wrap vectors can be excluded: the wraps are \emph{locked}. These statements are exact for every $p$. \emph{The limit model (heuristic).} To explain the constants we also use a limit model, which no proof below uses. Suppose $B=pA+O(1)$ with $A$ fixed, with a fixed finite set of relations $\mathbf h$ of bounded $|\mathbf h|_\infty$, every other relation having $|\mathbf h|_\infty\to\infty$ with $p$. Then the phases $r(n)/p$ become dense in the subtorus $Y\subset(\R/\Z)^3$ cut out by the bounded relations (it is connected: if $k\mathbf h$ is a relation and $0<|k|1$ for every prime $p\ge17$, and $\delta N^{1/3}>\sqrt{66}/8$ for every prime $p\ge53$. \end{enumerate} \end{proposition} Since $2(h-r)\equiv-\rho$, the second rows of $C_x$ and $C_y$ are twice that of $C_z$ modulo $p$: $z$ reads the digit $\zeta(n)=u\cdot n\bmod p$, $u=(h-r,h-r,\rho)$, and $x$ and $y$ both read $2\zeta(n)\bmod p$. The layer is $(2,2,1)u^{\!\top}$, of rank one, and as $u/p\to(\frac{23}{48},\frac{23}{48},\frac1{24})$, in cell units the point of cell $n$ sits near $n+\{\vartheta_n(2,2,1)\}$ with $\vartheta_n=\frac{23}{48}(n_x+n_y)+\frac1{24}n_z$, up to the drift and the jitter. Part (3) bounds $\delta$ from above by one explicit pair, which is not a closest pair: at every prime $53\le p\le199$ and at $p=307$, $353$ the exact $\delta N^{1/3}$ is smaller than $d_*/p^2$ (computed). The proof of (2) rests on an exact description of the wraps of the shared digit. \begin{lemma}[Wraps of a doubled digit]\label{lem:wwrap} Let $p\ge5$ be a prime, $h=(p-1)/2$, $u\in\F_p^3\setminus\{0\}$, and $\zeta(n)=u\cdot n\bmod p\in\{0,\dots,p-1\}$ for $n\in\F_p^3$. For $E\in\Z^3$ and $n'\equiv n+E$ put $\Delta\zeta=\zeta(n')-\zeta(n)$ and $\omega=[2\zeta(n')\ge p]-[2\zeta(n)\ge p]$, so that the digit $2\zeta\bmod p$ changes by $2\Delta\zeta-p\omega$. Let $\sigma\in\{0,\dots,p-1\}$ be the residue of $u\cdot E$. As $n$ runs over $\F_p^3$, the pairs $(\Delta\zeta,\omega)$ that occur are exactly: $(0,0)$ if $\sigma=0$; and if $\sigma\ne0$, for each $\Delta\zeta\in\{\sigma,\sigma-p\}$ the pair $(\Delta\zeta,\sgn\Delta\zeta)$, and also $(\Delta\zeta,0)$ if $|\Delta\zeta|\le h$. \end{lemma} \begin{proof} As $u\ne0$, $\zeta$ takes every value, and $\zeta(n')\equiv\zeta(n)+\sigma$; so $\Delta\zeta=\sigma$ for $\zeta(n)\le p-1-\sigma$ and $\Delta\zeta=\sigma-p$ otherwise, and both occur if $\sigma\ne0$. Reversing the pair exchanges $(\Delta\zeta,\omega)$ and $(-\Delta\zeta,-\omega)$, so let $\Delta\zeta=d>0$ and $\zeta=\zeta(n)\in[0,p-1-d]$. As $2\zeta\ge p$ if and only if $\zeta\ge h+1$, $\omega\in\{0,1\}$, and $\omega=1$ if and only if $\zeta\le h<\zeta+d$; such $\zeta$ form the interval $[\max(0,h+1-d),\min(h,p-1-d)]$, which is not empty, as $1\le d\le p-1$ and $h+2\le p$. And $\omega=0$ if and only if $\zeta+d\le h$, possible if and only if $d\le h$, or $\zeta\ge h+1$, possible if and only if $h+1\le p-1-d$, i.e. $d\le h-1$. \end{proof} The lemma holds for $u=0$ as well: then $\sigma=0$ for every $E$, $\zeta\equiv0$, and only $(0,0)$ occurs. We call each pair $(\Delta\zeta,\omega)$ that occurs for a step $E$ an \emph{option} of $E$. \begin{table}[tbp] \centering\footnotesize\setlength{\tabcolsep}{4pt}\renewcommand{\arraystretch}{1.3} \begin{tabular}{@{}cccccccc@{}} \toprule row & $A$ & steps $E$ & $(\nu,\omega)$ & $Q$ & $|V|^2$ & at $\varepsilon=\frac1{48}$ & slope\\ \midrule (a) & $(1,0,-\frac12)$ & $(1,0,-1)$ & $(\frac12,1)$ & $3$ & $\frac{5}{4}-9\varepsilon+81\varepsilon^2$ & $\frac{281}{256}$ & $-\frac{45}{8}$\\ (b) & $(0,0,1)$ & $(0,0,1)$ & $(0,0)$ & $-2$ & $1+4\varepsilon+36\varepsilon^2$ & $\frac{211}{192}$ & $\frac{11}{2}$\\ & & $(1,1,2)$ & $(-1,-1)$ & & & & \\ (c) & $(1,0,\frac12)$ & $(1,0,0)$ & $(\frac12,1)$ & $1$ & $\frac{5}{4}-5\varepsilon+9\varepsilon^2$ & $\frac{883}{768}$ & $-\frac{37}{8}$\\ (d) & $(1,0,-\frac12)$ & $(1,0,0)$ & $(-\frac12,-1)$ & $1$ & $\frac{5}{4}-3\varepsilon+9\varepsilon^2$ & $\frac{305}{256}$ & $-\frac{21}{8}$\\ & & $(0,-1,-1)$ & $(\frac12,0)$ & & & & \\ (e) & $(1,0,\frac12)$ & $(1,0,1)$ & $(-\frac12,-1)$ & $-1$ & $\frac{5}{4}+5\varepsilon+9\varepsilon^2$ & $\frac{1043}{768}$ & $\frac{43}{8}$\\ (f) & $(1,1,0)$ & $(1,1,0)$ & $(0,0)$ & $2$ & $2-16\varepsilon+36\varepsilon^2$ & $\frac{323}{192}$ & $-\frac{29}{2}$\\ & & $(0,0,-1)$ & $(1,1)$ & & & & \\ (g) & $(1,-1,0)$ & $(1,-1,0)$ & $(0,0)$ & $0$ & $2$ & $2$ & $0$\\ \bottomrule \end{tabular} \medskip \caption{The short options in the proof of Proposition~\ref{prop:wnet}(2): every step $E$ and every $(\nu,\omega)$ allowed by the rules of the proof with $|A|^2\le\frac94$, up to the reversal $(E,\nu,\omega,A,Q)\mapsto(-E,-\nu,-\omega,-A,-Q)$ and the exchange of $x$ and $y$. Here $E$ is a cell step, $Q=E_x+E_y-2E_z$, $\nu\in\frac{E_x+E_y}2+\Z$ the integer part of the change of $\zeta/p$, $\omega$ the wrap of the digit $2\zeta\bmod p$, $A=E+(2\nu-\omega)(1,1,0)+\nu\e_z$ the limit step, $\mathbf w=(2,2,1)$, $V=A-Q\varepsilon\mathbf w$ the step in cell units without the jitter, $\varepsilon=\rho/(2p)$, and the last columns give $|V|^2=|A-Q\varepsilon\mathbf w|^2$, its value and its derivative in $\varepsilon$ at the limit $\varepsilon=\frac1{48}$. Row (a) is the diagonal contact, row (b) the step along $\e_z$, and the second step of row (f) the step along $-\e_z$ at a wrap of $z$, where $x$ and $y$ wrap too.} \label{tab:wcontacts} \end{table} \begin{proof}[Proof of Proposition~\ref{prop:wnet}] (1) The first rows are $\e_x,\e_y,\e_z$, so $\mathcal W_p$ is a graph form with the depths $(1,1,1)$, and we use Lemma~\ref{lem:box}: the box of shape $(1,1,1)$ has determinant $1$ and puts one point in every cell. A box with depth $2$ on $a$, $1$ on $b$ and $0$ on $c$ has, up to sign, the determinant $(C_a)_{2,c}$; these off-diagonal entries of the second rows are $-\rho,2\rho,-\rho,2\rho,h-r,h-r$. The box with depth $3$ on $a$ has, up to sign, the $2\times2$ minor of the second and third rows of $C_a$ off column $a$: $\det\left(\begin{smallmatrix}-\rho&2\rho\\1&2\end{smallmatrix}\right)=-4\rho$, $\det\left(\begin{smallmatrix}-\rho&2\rho\\1&1\end{smallmatrix}\right)=-3\rho$ and $\det\left(\begin{smallmatrix}h-r&h-r\\0&1\end{smallmatrix}\right)=h-r$ for $a=x,y,z$. As $1\le\rho\le\frac p{24}+1|V|-\sqrt3/p$ with $V=E+\Delta g/p$, and by Lemma~\ref{lem:wwrap} $\Delta g=(2\Delta\zeta-p\omega,\,2\Delta\zeta-p\omega,\,\Delta\zeta)$. It suffices to show $|V|^2\ge\frac{281}{256}-\frac{345}{128p}$. \emph{Parametrization.} Put $S=E_x+E_y$, $Q=S-2E_z$ and $\varepsilon=\rho/(2p)$. As $2(h-r)=p-\rho$, $2u\cdot E=pS-\rho Q$, so $\Delta\zeta=p(\nu-Q\varepsilon)$ with $\nu\in\frac S2+\Z$, and \[ V=A-Q\varepsilon\mathbf w,\qquad A=E+(2\nu-\omega)(1,1,0)+\nu\e_z,\qquad \mathbf w=(2,2,1), \] \[ |V|^2=|A|^2-2Q(A\cdot\mathbf w)\,\varepsilon+9Q^2\varepsilon^2 . \] With $\theta=48r-p$, an odd integer in $[-47,-1]$ (here $\theta$ is not that of Proposition~\ref{prop:twocoset}), we have $\rho=(p+\theta)/24+1$, so $\varepsilon=\frac1{48}+\frac{\theta+24}{48p}$ and $|\varepsilon-\frac1{48}|\le\frac{23}{48p}$; for $p\ge53$, $\varepsilon\le\frac{19}{636}<\frac3{100}$. The drift of the layer is this deviation of $\varepsilon$ from its limit. \emph{Which $(\nu,\omega)$ occur.} Let $p\ge53$. Then $|Q|\le8$ and $4\rho\le\frac p6+40$; if $\nu=\pm1$, the step occurs only if $\nu Q>0$, and then $|\Delta\zeta|>h$ and $\omega=\sgn \nu$; and $|\nu|\ge\frac32$ gives $|\Delta\zeta|\ge\frac32p-4\rho>p$, which does not occur. None of these rules depends on $p$ or on its class. \emph{Long options.} $A$ lies in the lattice $\Lambda_{\mathcal W}=\{A:A_x,A_y\in\Z,\ 2A_z\in\Z,\ A_x+A_y\equiv2A_z\pmod2\}$, as $A_x+A_y\equiv S\equiv2\nu\equiv2A_z$. By the rules $|2\omega-2\nu|\le2$, and $Q=(A_x+A_y-2A_z)-2\nu+2\omega$, so $|Q|\le\sqrt6\,|A|+2$. Hence $|V|\ge|A|-3|Q|\varepsilon\ge(1-\frac{9\sqrt6}{100})|A|-\frac9{50}>1.05$ if $|A|^2\ge3$, and $1.05^2>\frac{281}{256}$. \emph{Short options.} Since $2A_z\in\Z$, $|A|^2$ is an integer or an integer plus $\frac14$, and $|A|^2=\frac94$ would need $A=(\pm1,\pm1,\pm\frac12)$ or $\pm\frac32\e_z$, which violate the parity condition; so the vectors of $\Lambda_{\mathcal W}$ with $0<|A|^2<3$ are those with $|A|^2\le2$: $\pm\e_z$, $(\pm1,0,\pm\frac12)$, $(0,\pm1,\pm\frac12)$ and $(\pm1,\pm1,0)$. For each of them $E=A-(2\nu-\omega)(1,1,0)-\nu\e_z$, with $\nu\equiv A_z\pmod1$ and $|\nu|\le1$, so at most three values of $\nu$, and checking the rules for these values and their $\omega$ leaves the options of Table~\ref{tab:wcontacts}; $A=0$ does not occur (it would need $E=\omega(1,1,0)$ with $\nu=0$, $\omega=-\sgn Q=-\sgn(2\omega)$, or $E=-\nu\mathbf 1$ with $|\nu|=1$ and $Q=0$). The squared length $q(\varepsilon)=|V|^2$ of each row is a convex quadratic in $\varepsilon$, so it lies above its tangent line at $\frac1{48}$, and as $|\varepsilon-\frac1{48}|\le\frac{23}{48p}$, $q(\varepsilon)\ge q(\frac1{48})-|q'(\frac1{48})|\cdot\frac{23}{48p}=:T(p)$. For the diagonal contact (a), $T(p)=\frac{281}{256}-\frac{345}{128p}$, as $\frac{45}8\cdot\frac{23}{48}=\frac{345}{128}$. For the step (b) along $\e_z$, $T(p)=\frac{211}{192}-\frac{253}{96p}$, which is larger, as $\frac{211}{192}>\frac{281}{256}$ and $\frac{253}{96}<\frac{345}{128}$. For the rows (c)--(g), $q(\frac1{48})>\frac{281}{256}$, so $p\,(T(p)-\frac{281}{256})=p\,(q(\frac1{48})-\frac{281}{256})-\frac{23}{48}|q'(\frac1{48})|$ is affine in $p$ with positive slope, and at $p=53$ it is positive (for row (c), $53\cdot\frac5{96}-\frac{23}{48}\cdot\frac{37}8>0$, and the other rows have more room); so these options have $|V|^2>\frac{281}{256}$. This proves (2). The constant $\frac{345}{128}$ is the slope of the diagonal contact (a) times the largest drift $\frac{23}{48p}$, which is attained for $p\equiv1\pmod{48}$; exactly, with $x=\varepsilon-\frac1{48}=\frac{\theta+24}{48p}$, \[ |V_{\mathrm a}|^2=\frac{281}{256}-\frac{45}8\,x+81x^2 . \] The bound increases with $p$; it is $1.0024\ldots$ at $p=67$, below $1$ at $p=61$, $1.0164\ldots$ at $p=97$ and below $\sqrt{66}/8=1.01550\ldots$ at $p=89$. (3) Here $\zeta(1,0,0)=h-r$ and $\zeta(2,0,p-1)=p-2-4r$, so the second digits of the two points are $(p-1-2r,\,p-1-2r,\,h-r)$ and $(p-4-8r,\,p-4-8r,\,p-2-4r)$ (as $p\ge8r+4$), and the third digits are $(0,1,0)$ and $(p-2,1,0)$. Hence $D=(p^2-(6r+2)p-2,\,-(6r+3)p,\,-(h+3r+2)p)$ is a representative of the difference with $|D_c|1$ for every prime $p\ge67$ and $>\sqrt{66}/8$ for every prime $p\ge97$. The remaining primes are settled by the exact minimum distances (Appendix~\ref{app:verify}): at every prime $17\le p\le61$, $\delta N^{1/3}\ge1.0009\ldots$ (least at $p=47$), and at every prime $53\le p\le89$, $\delta N^{1/3}\ge1.0257\ldots$ (least at $p=89$). \end{proof} \begin{figure}[tbp] \centering \IfFileExists{figs/w_net.pdf}{\includegraphics[width=\linewidth,height=0.5\textheight,keepaspectratio]{figs/w_net.pdf}}{\fbox{\parbox{0.9\textwidth}{\centering figure \texttt{w\_net} pending}}} \caption{\wNetCaption\interactive{beyond.html}} \label{fig:wnet} \end{figure} \emph{The balance.} In the limit the two shortest contacts are rows (a) and (b) of Table~\ref{tab:wcontacts}: the diagonal contact, lifted by the glue $\frac12+3\varepsilon$, and the step along $\e_z$, lengthened to $1+2\varepsilon$ in $z$. Their squared lengths \[ q_a(\varepsilon)=\tfrac54-9\varepsilon+81\varepsilon^2,\qquad q_b(\varepsilon)=1+4\varepsilon+36\varepsilon^2 \] are equal where $45\varepsilon^2-13\varepsilon+\frac14=0$. The smaller root is $\varepsilon^*=(13-2\sqrt{31})/90=0.02071\ldots$, and there both equal $\mu^{*2}$, where $\mu^*=\sqrt{(72-8\sqrt{31})/25}=1.0480\ldots$; on $[0,\frac1{18}]$, $q_a$ decreases and $q_b$ increases. The limit $\varepsilon=\frac1{48}=0.02083\ldots$ of $\mathcal W_p$ is not a balance: it lies just above $\varepsilon^*$, on the side of the diagonal contact, with $q_a(\frac1{48})=\frac{281}{256}2$ if some $|E_c|\ge3$. The step $\e_z$ gives $V=\e_z$ if $\psi_z\equiv0$, and otherwise, with $\xi=\{\psi_z\}-1$ and $\omega=-1$, it gives $V=(2\xi+1,2\xi+1,1+\xi)$, whose entries lie in $(-1,1)$. Hence $F\le3$, and only the steps $E\in\{-2,\dots,2\}^3\setminus\{0\}$ matter. (1) At integer scale $p^3$ the point of cell $n$ is $p^2n+p\,g(n)+f(n)$ with the third digits $f(n)$. As in the proof of Proposition~\ref{prop:wnet}(2), the representatives of the difference of the points of the cells $n$ and $n'$ are $D=p^2E+p\,\Delta g+\Delta f$ with $E\equiv n'-n\pmod p$ and $|\Delta f_c|0$; $\nu=\pm1$ occurs only if $\nu Q>0$, and then $\omega=\sgn \nu$; and $|\nu|\ge\frac32$ does not occur. The rest of that proof uses only $\varepsilon\le\frac3{100}$: the options with $|A|^2\ge3$ have $|V|>1.05$, and the others are the rows of Table~\ref{tab:wcontacts}. On $(0,\frac3{100}]$ the rows (c), (d) and (f) decrease, to $1.1081$, $1.1681$ and $1.5524$, and the rows (e) and (g) are at least $\frac54$; all of these, and $1.05^2=1.1025$, exceed $\mu^{*2}$, while $\min(q_a(\varepsilon),q_b(\varepsilon))\le\mu^{*2}$, as $q_a$ decreases and $q_b$ increases through $\varepsilon^*$. The rows (a) and (b) occur, since $(\nu,\omega)=(\frac12,1)$ and $(0,0)$ are always allowed. So $F(\psi)=\min(q_a(\varepsilon),q_b(\varepsilon))$. (3) Let $\psi\in(\R/\Z)^3$ with $F(\psi)\ge\mu^{*2}$, and represent $\psi_x,\psi_y,\psi_z$ by numbers in $[0,1)$. We use only options with $\omega=\sgn\xi$, which always occur, and options with $\omega=0$ and $|\xi|<\frac12$. The numbers $\frac{246}{225}=1.0933\ldots$, $\frac{17}{16}$ and $\frac9{25}$ are below $\mu^{*2}=1.0983\ldots$. \emph{The step $\e_z$.} If $\psi_z=0$, then $V=\e_z$ and $|V|^2=1$. Otherwise $\xi=\psi_z-1$, $\omega=-1$ gives $V=(2\psi_z-1,2\psi_z-1,\psi_z)$ with $|V|^2=9\psi_z^2-8\psi_z+2$, a convex function with the values $\frac{246}{225}$ at $\psi_z=\frac2{15}$ and $\frac{17}{16}$ at $\psi_z=\frac34$; so $\psi_z\notin[\frac2{15},\frac34]$. If $\psi_z>\frac34$, then $\xi=\psi_z-1$, $\omega=0$ gives $|V|^2=8(1-\psi_z)^2+\psi_z^2\le\frac{17}{16}$. So $0<\psi_z<\frac2{15}$, and $\xi=\psi_z$, $\omega=0$ gives $V=(2\psi_z,2\psi_z,1+\psi_z)$ with $|V|^2=q_b(\psi_z/2)$; as $q_b$ increases, $\psi_z\ge2\varepsilon^*$. \emph{The steps $\e_x$ and $\e_y$.} If $\psi_x=0$, then $|V|^2=1$. Otherwise $\xi=\psi_x$, $\omega=1$ gives $V=(2\psi_x,2\psi_x-1,\psi_x)$ with $|V|^2=9\psi_x^2-4\psi_x+1$, a convex function with the values $1$ at $\psi_x=0$ and $\frac{246}{225}$ at $\psi_x=\frac7{15}$; so $\psi_x>\frac7{15}$. If $\psi_x>\frac12$, then $\xi=\psi_x-1$, $\omega=0$ gives $V=(1-2t,-2t,-t)$ with $t=1-\psi_x$, and $|V|^2=9t^2-4t+1$ is the same function, so $t>\frac7{15}$. Hence $\alpha=\frac12-\psi_x\in(-\frac1{30},\frac1{30})$, and likewise $\beta=\frac12-\psi_y\in(-\frac1{30},\frac1{30})$. Put $\gamma=\psi_z$. \emph{The step $\mathbf 1$ (the lock).} Here $\psi\cdot\mathbf 1\equiv\gamma-\alpha-\beta=:\tilde\sigma\in(-\frac1{15},\frac15)$. If $\tilde\sigma>0$, then $\xi=\tilde\sigma-1$, $\omega=-1$ gives $V=(2\tilde\sigma,2\tilde\sigma,\tilde\sigma)$ with $|V|^2=9\tilde\sigma^2<\frac9{25}$. So $\alpha+\beta\ge\gamma$. \emph{The diagonal steps $(1,0,-1)$ and $(0,1,-1)$.} For $E=(1,0,-1)$, $\psi\cdot E\equiv\psi_x-\psi_z=\frac12-\lambda$ with $\lambda=\alpha+\gamma\in(2\varepsilon^*-\frac1{30},\frac16)\subset(0,\frac16)$, and $\xi=\frac12-\lambda$, $\omega=1$ gives $V=(1-2\lambda,-2\lambda,-\frac12-\lambda)$ with $|V|^2=\frac54-3\lambda+9\lambda^2=q_a(\lambda/3)$: row (a) of Table~\ref{tab:wcontacts} with $\varepsilon=\lambda/3$. Likewise $E=(0,1,-1)$ gives $q_a(\lambda'/3)$ with $\lambda'=\beta+\gamma$. As $q_a$ decreases on $[0,\frac1{18}]$ and $q_a(\varepsilon^*)=\mu^{*2}\le F(\psi)$, we get $\lambda\le3\varepsilon^*$ and $\lambda'\le3\varepsilon^*$. But $\lambda+\lambda'=\alpha+\beta+2\gamma\ge3\gamma\ge6\varepsilon^*$, by the lock and the step $\e_z$. So equality holds throughout: $\lambda=\lambda'=3\varepsilon^*$, $\alpha+\beta=\gamma$ and $\gamma=2\varepsilon^*$, hence $\alpha=\beta=\varepsilon^*$, i.e.\ $\psi=\psi^*$. Hence $F(\psi)\ge\mu^{*2}$ only at $\psi=\psi^*$, where $F(\psi^*)=\mu^{*2}$ by (2): $F\le\mu^{*2}$ everywhere, with equality only at $\psi^*$. As $\varepsilon^*$ is irrational, $u/p\ne\psi^*$, so $F_p(u)=F(u/p)<\mu^{*2}$, and by (1) $\delta N^{1/3}<\sqrt{F_p(u)}+\sqrt3/p<\mu^*+\sqrt3/p$. \end{proof} \emph{How six contacts fix the optimum.} The proof uses six steps. Along $\e_x$, $\e_y$ and $\e_z$ a coordinate can wrap alone, which shortens the step. This forces $\psi_x$ and $\psi_y$ to stay within $\frac1{30}$ of $\frac12$ ($|\alpha|,|\beta|<\frac1{30}$), and $z$ to move by at least $1+2\varepsilon^*$ along $\e_z$ ($\gamma\ge2\varepsilon^*$), or the step along $\e_z$ is short. The step $\mathbf 1$ is the one along which all three coordinates can wrap together, and these offsets must not shorten it: $\alpha+\beta\ge\gamma$. The diagonal contacts then balance against the step along $\e_z$, at an irrational point. The optimum lies on the plane $\psi\cdot\mathbf 1\equiv0$, i.e.\ $u\cdot\mathbf 1\equiv0$, where the step $\mathbf 1$ has no phase change; $\mathcal W_p$ lies on it, as $2(h-r)+\rho=p$. Proposition~\ref{prop:wfamily} covers only the image $\F_p(2,2,1)$ and its permutations; other rank-one images, layers of rank two and non-digital point sets are not covered (Remark~\ref{rem:open}(7)). \begin{remark}[Reaching $\mu^*$]\label{rem:wstar} Let $\mathcal W^*_p$ be the net of Proposition~\ref{prop:wnet} with $r=\lfloor p\varepsilon^*\rfloor$ in place of $\lfloor p/48\rfloor$. Then $\rho=2r+1$ is the odd integer nearest to $2p\varepsilon^*$, and $\varepsilon_p=\rho/(2p)$ satisfies $|\varepsilon_p-\varepsilon^*|<\frac1{2p}$ (both use that $p\varepsilon^*$ is never an integer, as $\varepsilon^*$ is irrational). $\mathcal W^*_p$ is a $(0,3,3)$-net for every prime $p\ge5$, since the proof of Proposition~\ref{prop:wnet}(1) needs only $1\le\rho\ \sqrt{\mu^{*2}-\frac Kp}-\frac{\sqrt3}p,\qquad K=\frac{|q_a'(\varepsilon^*)|}2=\frac{9\sqrt{31}-36}5=2.8219\ldots, \] for every prime $p\ge17$. With Proposition~\ref{prop:wfamily}(3), $\delta N^{1/3}(\mathcal W^*_p)\to\mu^*$: the optimum of the family is attained in the limit. The bound exceeds $1$ from $p=67$ on and $\sqrt{66}/8$ from $p=97$ on, as that of Proposition~\ref{prop:wnet}(2). It exceeds $\sqrt{281}/16$ from $p=9791$ on, and the upper bound $\sqrt{281}/16+5/(2p)$ of Proposition~\ref{prop:wnet}(3) from $p=17737$ on; only from there on is $\mathcal W^*_p$ provably ahead of $\mathcal W_p$. The optimum is reached only asymptotically: at moderate $p$, $\mathcal W^*_p$ is not ahead of $\mathcal W_p$. $\mathcal W^*_p=\mathcal W_p$ for $p<193$ (computed), and $\mathcal W^*_p\ne\mathcal W_p$ for $p>8548$: as $\frac1{48}-\varepsilon^*=\frac{96\sqrt{31}-534}{4320}>\frac1{8549}$, for $p\ge8549$ we have $p/48-p\varepsilon^*>1$, so $\lfloor p/48\rfloor>\lfloor p\varepsilon^*\rfloor$, and the two nets have different $\rho\varepsilon^*$ and its minimum is the diagonal contact (a). A heuristic reading of the computed values is that the third rows of Proposition~\ref{prop:wnet} cost the step (b) more than the contact (a): at moderate $p$ the jitter, not the drift, decides, and the value that $\mathcal W_p$ gives up by its limit $\varepsilon\to\frac1{48}$ instead of $\varepsilon^*$ is small, $\mu^{*2}-\frac{281}{256}<7\cdot10^{-4}$. \end{remark} \emph{Computed values.} The exact $\delta N^{1/3}$ of $\mathcal W_p$ is $1.0087\ldots$ at $p=17$, $1.0262\ldots$ at $p=53$, $1.0362\ldots$ at $p=101$ and $1.0461\ldots$ at $p=353$, and below $1$ for $p\le13$ (computed). By (4) and Corollary~\ref{cor:colceil}, $\mathcal W_p$ is ahead of every column point set at every prime $p\ge17$. It is ahead of $\mathcal T_p$ at every prime $17\le p\le199$ other than $23$ (computed), and for $p\ge53$ also by (4) and Proposition~\ref{prop:twocoset}(3), as $\delta(\mathcal T_p)N^{1/3}\le d_*(\mathcal T_p)/p^2<\sqrt{66}/8$. Nor is $\sqrt{66}/8$ the best value for layers in which two coordinates share a geometric digit: in the limit model the layer $A=\frac1{15}\left(\begin{smallmatrix}0&6&6\\2&-1&-1\\2&-1&-1\end{smallmatrix}\right)$ with $Y=\{\psi\in(\R/\Z)^3:\psi_y=\psi_z\}$ has the value $\sqrt{233}/15=1.0176\ldots$ (computed), and its realization $B_x=(0,a,a)$, $B_y=B_z=(2r',-r',-r')$, $a=\operatorname{round}(2p/5)$, $r'=\operatorname{round}(p/15)$, with the third rows of $\mathcal T_p$, is a $(0,3,3)$-net at $p=307$ with $\delta N^{1/3}=1.0160\ldots>\sqrt{66}/8$ (computed). Its free coordinate $x$ carries the glue $\frac25$. So Remark~\ref{rem:glue}, which concerns the length of a single step under free wraps, does not decide which glue is optimal for the minimum, even for free coordinates. \emph{Only at $m=3$.} At $m=3k$, $k\ge2$, the geometric digits of any two coordinates of a $(0,m,3)$-net are jointly uniformly distributed (Lemma~\ref{lem:shared}), so no coordinate can read half, or any function, of the geometric digit of another: the lock of $\mathcal W_p$, like that of $\mathcal T_p$, exists only at $m=3$ (Remark~\ref{rem:open}(7)). \begin{remark}[Digital shifts of $\mathcal T_p$ and $\mathcal W_p$]\label{rem:wshift} As for the column nets (Remark~\ref{rem:shiftshared}), a digital shift translates the wrap windows of the coordinates independently, so bounds that use locked wraps need not survive it. The bound of Proposition~\ref{prop:twocoset}(2) does not extend to shifts whose second digits differ in $y$ and $z$, the two coordinates that share their geometric digit, for instance at $p=41$, $53$, $101$ (computed). If they agree, $\tau_{y,2}=\tau_{z,2}$ (all other shift digits arbitrary), the bound holds for the shifted net, for every prime $p\ge41$. Indeed, by Lemma~\ref{lem:shift} the shifted net, indexed by its own cells, is $\mathcal T_p$ with constants added to the offset digits. As $C_y$ and $C_z$ have the same second row, the geometric digits of $y$ and $z$ of each cell differ by the constant $\tau_{y,2}-\tau_{z,2}=0$, so $g_y=g_z$ at every cell; and the differences of the geometric digits of two cells satisfy the same congruences $\xi\equiv hS$, $\eta\equiv rE_x-T$ as for $\mathcal T_p$, since the constants cancel. The proof of Proposition~\ref{prop:twocoset}(2) uses the geometric digits only through these congruences and through $g_y=g_z$, and the third digits only through $|\Delta f_c|0$ and $\zeta=\zeta(n)\in[0,p-1-d]$ we have $\omega\in\{0,1\}$; $\omega=1$ if and only if $\zeta\le h-1<\zeta+d$, which holds for $\zeta=\max(0,h-d)$, so it occurs for every $d$; and $\omega=0$ if and only if $\zeta+d\le h-1$, possible if and only if $d\le h-1$ (both digits below the wrap), or $\zeta\ge h$, possible if and only if $h\le p-1-d$, i.e.\ $d\le h$ (both at or above it). So $\omega=0$ occurs if and only if $d\le h$, as in the lemma. The proof of Proposition~\ref{prop:wnet}(2) uses the second digits only through these pairs and allows any third digits, so its bound holds for the shifted net whenever $t\equiv2t'$ or $t\equiv2t'+1$; the exact value need not be kept, since the third digits move. Other second digits break the lock in every case computed: at $p=101$ the shifts $(\tau_{x,2},\tau_{y,2},\tau_{z,2})=(0,0,1)$ and $(7,7,7)$ give $\delta N^{1/3}=.1355\ldots$, against $1.0362\ldots$ without a shift, and every shift $(t,t,0)$ with $2\le t\le100$ gives $\delta N^{1/3}<0.1356$, while $(1,1,0)$, $(3,3,1)$ and $(5,5,2)$ keep $1.0362\ldots$ (computed). That only $t\equiv2t'$ and $t\equiv2t'+1$ keep the lock is computed, not proved. In the computed cases where the lock breaks, the step along $\e_z$ at a wrap of $z$ alone shrinks to $(4\varepsilon,4\varepsilon,2\varepsilon)$ up to the third digits. Proposition~\ref{prop:wnet} is a statement about unshifted nets. \end{remark} \subsection{No half-integer layer in the frame orbit}\label{sec:Pnohalf} By Section~\ref{sec:frames}, every net obtained from a frame by per-coordinate lower-triangular scrambles has a geometric layer of Cauchy form, $\tilde M_{cd}=\lambda_c\mu_d/D_{cd}$ ($c\ne d$), with units $\lambda_c,\mu_d$ and the homogeneous place differences $D_{cd}$; this holds over $\mathbb{Q}$, and verbatim over $\F_p$ for places in $\mathbb P^1(\F_p)$. The deepest carry depth $s-2$ of row~$0$ is free by Proposition~\ref{prop:F} up to one condition per column $d$ (the polynomial $g_{d,s-2}$ there, which the $s-1$ off-diagonal entries of the column determine, must have a unit leading coefficient) and can be made half-integer. But the geometric layer $F_0$, which every carry-free step sees, sits at carry depth $0$, where the Cauchy form leaves only the row and column scalars free. \begin{proposition}[No half-integer frame layer]\label{prop:P} Let $\tilde M$ be a layer of this Cauchy form over a field $\mathbb K$, and let $a,b,c,d$ be four distinct coordinates. \begin{enumerate} \item If $\mathbb K=\F_p$, $p$ prime, and $2\tilde M_{ij}\equiv h_{ij}\pmod p$ for all twelve ordered pairs $i\ne j$ in $\{a,b,c,d\}$, with odd integers $|h_{ij}|\le\Omega$, then $p$ divides a nonzero odd integer $Z$ with $|Z|\le13\Omega^{16}$. In particular this is impossible for $p>13\Omega^{16}$. \item If $\mathbb K=\mathbb{Q}$, then the twelve entries $\tilde M_{ij}$, $i\ne j$ in $\{a,b,c,d\}$, do not all lie in $\frac12+\Z$. \end{enumerate} \end{proposition} \begin{proof} By Lemma~\ref{lem:CH}(P) over $\mathbb K$, with $\pi_{ij}=\tilde M_{ij}\tilde M_{ji}$, the numbers $\chi^2=\pi_{ab}\pi_{cd}/(\pi_{ac}\pi_{bd})$ and $\psi^2=\pi_{ab}\pi_{cd}/(\pi_{ad}\pi_{bc})$ satisfy $(\chi^2-\psi^2-1)^2=4\psi^2$. (1) The entries of a layer of Cauchy form are units, so for odd $p$ the $h_{ij}\equiv2\tilde M_{ij}$ are units modulo $p$ (for $p=2$ the hypothesis fails, the $h_{ij}$ being odd), and $4\pi_{ij}\equiv h_{ij}h_{ji}$. Let $A_1=|h_{ab}h_{ba}h_{cd}h_{dc}|$, $A_2=|h_{ac}h_{ca}h_{bd}h_{db}|$, $A_3=|h_{ad}h_{da}h_{bc}h_{cb}|$, odd integers at most $\Omega^4$. Then $\chi^2\equiv\varepsilon_2A_1/A_2$ and $\psi^2\equiv\varepsilon_3A_1/A_3$ with signs $\varepsilon_2,\varepsilon_3\in\{\pm1\}$, and multiplying the last relation by $A_2^2A_3^2$ shows that $p$ divides \[ Z=\bigl(\varepsilon_2A_1A_3-\varepsilon_3A_1A_2-A_2A_3\bigr)^2-4\varepsilon_3A_1A_2^2A_3 . \] The bracket is a sum of three odd numbers, hence odd, so $Z$ is odd and in particular nonzero, and $|Z|\le(3\Omega^8)^2+4\Omega^{16}=13\Omega^{16}$. (2) Over $\mathbb{Q}$ the same computation with $h_{ij}=2\tilde M_{ij}$, odd integers by assumption, gives $Z=0$ exactly, contradicting that $Z$ is odd. \end{proof} So a rational design never has a half-integer geometric layer in the frame orbit (on four coordinates, hence for $s\ge4$), and a design over $\F_p$ with $|2\tilde M|\le\Omega$ has none once $p>13\Omega^{16}$. Proposition~\ref{prop:P} reads the Pl\"ucker relation of Lemma~\ref{lem:CH}(P) modulo powers of $2$: it is the $2$-adic counterpart of Lemma~\ref{lem:CH}(H), which reads it in absolute value, and of Remark~\ref{rem:pm1}, which reads it modulo $q$. \subsection{The split frame}\label{sec:split} The ceilings of Theorems~\ref{thm:L} and~\ref{thm:O5} concern bounded layers. A geometric layer with half-integer entries escapes them: locally the points form the packing $D_s^+=D_s\cup(D_s+\frac12\mathbf 1)$ \cite[Ch.~4]{CS99}, whose minimum $\min(\sqrt s/2,\sqrt2)$ exceeds $1$ for $s\ge5$ and equals the $E_8$ value $\sqrt2$ at $s=8$. At the resolution $m=s$ no frame is needed (Section~\ref{sec:halfms}). For $k\ge2$ the layer sits at carry depth $0$ of a design, and there a Pl\"ucker relation between the places excludes rational half-integer layers in the frame orbit (Proposition~\ref{prop:P}). To escape it, the split frame lets each coordinate $c$ read its cell digits at one point $v_c$ and its offset digits at a different point $u_c$. Its box minors are still confluent Vandermonde determinants, its geometric layer is the non-symmetric Cauchy matrix $\lambda'_c\mu_d/(u_c-v_d)$ in two unrelated point sets, and with $u_c$ even and $v_d$ odd it can be half-integer at every carry depth. \begin{definition}\label{def:split} Let $v_0,\dots,v_{s-1}$ (\emph{cell points}) and $u_0,\dots,u_{s-1}$ (\emph{offset points}) be $2s$ pairwise distinct rational numbers, let $D$ be the product of their denominators and of the numerators of their pairwise differences, $R=\Z[1/D]$ and $m=sk$. For $V\in R[t]_{2\max|\sigma_c|$ (over all $E$, $j$ and $c$), then $\delta N^{1/s}\ge\mathrm{LB}$ (toroidal distance), for every choice of the rows $\ge1$. \item Suppose moreover that $F_i[c][c]=0$ for $i\ge1$ and $F_0[c][c]=-\alpha_c$ with \[ \alpha_c\ \ge\ A_c:=\sum_{d\ne c}\max_{j2B$, then $\delta N^{1/s}\ge\min\bigl(\sqrt s(\frac12-\epsilon_p),\sqrt2(1-\epsilon_p)\bigr)$. \item Let $C=\max_{c,d}|F_0[c][d]|$. If $p>4sC+2$, then $\delta N^{1/s}\le\min(\sqrt s/2,\sqrt2)+\sqrt s\,(2C+1)/p$. \end{enumerate} \end{proposition} \begin{proof} (1) In cell units, coordinate $c$ of the point of cell $n$ is $n_c+(r_c+f_c)/p$ with $r_c$ the first offset digit and $f_c\in[0,1)$. Let $n\ne n'$ and let $E$ be a torus representative of $n'-n$ (modulo $p^k$ in each coordinate). If some $|E_c|\ge3$, then the two points differ by $|D_c|>|E_c|-1\ge2$ cells in coordinate $c$, with $D$ their difference in cell units, since $(r'_c+f'_c)-(r_c+f_c)\in(-p,p)$. Otherwise adding $E_d$ to the cell integer $n_d$ changes its level $k-1$ by $E_d$ and the levels $k-1-i$, $1\le i\le j_d$, by $\sgn(E_d)$ (carries or borrows; a carry is at most $1$ since $|E_d|\le2|pE_c+\Delta r_c|-1$, i.e. $p|D_c|\ge b_c$. (2) First $|\sigma_c|\le2\alpha_c+2A_c\le B$, since the own term is $-E_c\alpha_c$ and the term of an axis $d\ne c$ is $E_dM^{(j_d)}[c][d]$ ($|E_d|=1$) or $\sgn(E_d)(M^{(0)}+M^{(j_d)})[c][d]$ ($|E_d|=2$); so (1) applies. Let $e=|E|_\infty\le2$ and $|E_c|=e$. Then $E_c\sigma_c\le-e^2\alpha_c+e\cdot eA_c\le0$: every coordinate of maximal size is backward. If $\Sigma E$ is odd, every coordinate has $|pE_c+t|\ge p|E_c\pm\frac12|-|\sigma_c|\ge p/2-|\sigma_c|$ for $t\in O_c$, so $|D_c|\ge\frac12-\epsilon_p$ for all $s$ coordinates. If $\Sigma E$ is even, a backward coordinate $c$ with $|E_c|=e$ has $\sgn\sigma_c\ne\sgn E_c$, so both options give $|pE_c+t|\ge pe-|\sigma_c|$ and $|D_c|\ge e-\epsilon_p$. For $e=2$ this is at least $2-\epsilon_p\ge\sqrt2(1-\epsilon_p)$; for $e=1$ an even $\Sigma E$ needs at least two nonzero coordinates, each of maximal size, so $|D|\ge\sqrt2(1-\epsilon_p)$. (3) Since $|2F_0[c][d]|\le2Cx$ implies $\delta N^{1/s}>x$ for every prime $p\ge p_1$ (as long as the net exists modulo $p$ and the hypothesis of (1) holds). In (1) the hypothesis implies $p\ge7$: the step $E_d=2\sgn F_0[c][d]$ ($d=0,\dots,s-1$) without carries has $\sigma_c=2\sum_d|F_0[c][d]|\ge s\ge3$. Parts (1)--(3) are formally verified as stated, (3) for the Euclidean distance; for the split-frame nets (1) and (2) also over $\Z/q\Z$ with composite $q$ (Appendix~\ref{app:formal}). \needspace{10\baselineskip} \begin{theorem}[Beyond the cubic lattice]\label{thm:HL} Let $s\ge4$. For the explicit design $H_s$ on the split frame $u_c=2c$, $v_d=2d+1$ (Lemma~\ref{lem:Hs}) there are constants $B_s=e^{O(s^2\log s)}$ and $C_s$ such that for every prime $p\ge\max(2s,2B_s+2)$ and every $k\ge1$ the net $\mathcal H^s_{k,p}$ is a digital $(0,sk,s)$-net over $\F_p$ with one point in every cell, and \[ \min\Bigl(\sqrt s\bigl(\tfrac12-\epsilon_p\bigr),\ \sqrt2\,(1-\epsilon_p)\Bigr)\ \le\ \delta N^{1/s}\qquad\Bigl(\epsilon_p=\frac{B_s+1}p\Bigr), \] toroidal, hence Euclidean; if also $p>4sC_s+2$, then $\delta N^{1/s}\le\min(\sqrt s/2,\sqrt2)+\sqrt s\,(2C_s+1)/p$. So $\delta N^{1/s}\to\min(\sqrt s/2,\sqrt2)$ as $p\to\infty$, uniformly in $k$. \end{theorem} Theorem~\ref{thm:A} already gives non-decaying separation along $N=q^{sk}$; Theorem~\ref{thm:HL} adds the value: above the cubic value $1$ for $s\ge5$, at every resolution $m=sk$, beyond the first-order ceiling $1-2/q$ that Theorem~\ref{thm:O5} proves for bounded layers of frame designs. Unlike $a^*_s$, the constant $B_s$ is enormous ($B_4=1787594$, $B_5=1413079078321$); searched designs have much smaller constants (Section~\ref{sec:halfms}). For $s=4$ nothing is gained, since $D_4^+$ is a rotated copy of $\Z^4$. \begin{proof}[Proof of Theorem~\ref{thm:HL}] Let $H_s$ be the design of Lemma~\ref{lem:Hs} on $u_c=2c$, $v_d=2d+1$, let $\mathcal H^s_{k,p}=\mathcal P^{u,v,H_s}_{k,p}$, $B_s=\max_c(2\alpha_c+2A_c)$ and $C_s=\max_c\alpha_c$, which is the largest entry of $F_0$ in absolute value, since $|F_0[c][d]|\le A_c\le\alpha_c$. For a prime $p\ge2s$ and every $k\ge1$, Lemma~\ref{lem:Hs} and Proposition~\ref{prop:Gsplit} make $\mathcal H^s_{k,p}$ a digital $(0,sk,s)$-net with one point in every cell, whose row $0$ is $H_s$ truncated to $L=\min(k,s-1)$ levels and $0$ below. It satisfies (Half) and the dominance hypothesis of Proposition~\ref{prop:Hk}(2) (with $A_c$ over all $j\le s-2\ge L-1$), so Proposition~\ref{prop:Hk}(2) gives the lower bound (as $p\ge2B_s+2>2B_s$; for $p\le2B_s+2$ it would not be positive) and Proposition~\ref{prop:Hk}(3) the upper bound, both independent of $k$. Finally $|\vartheta_{cd}|\le2s-1$, so $|\Lambda_c|\le(2s-1)^{(s-1)^2}$, $|M^{(j)}[c][d]|\le\frac32|\Lambda_c|$ (absolute values, as $\Lambda_c$ is negative for some $c$ when $s$ is even), $\alpha_c\le A_c+1$ and $B_s\le6(s-1)\max_c|\Lambda_c|+2=e^{O(s^2\log s)}$. \end{proof} By Proposition~\ref{prop:P}, the nets of Theorem~\ref{thm:HL} lie outside the frame orbit of every frame with rational places (they cannot be obtained from it by per-coordinate lower-triangular scrambles and gauge moves), and outside that of every frame over $\F_p$ with $|2\tilde M|\le\Omega$ once $p>13\Omega^{16}$. \subsection{One resolution, and searched designs}\label{sec:halfms} \emph{One resolution.} At $m=s$ ($k=1$) no frame is needed: for a layer $F_0$ in $(\frac12+\Z)^{s\times s}$ (with $\frac12\equiv(p+1)/2$) and any deeper rows, Proposition~\ref{prop:Hk} with $k=1$ gives the bound, and the point of cell $n$ sits, up to the drift, which grows like $|n-n_0|/p$ away from a reference cell $n_0$, and the jitter, at $n$ or at $n+\frac12\mathbf 1$ according to the parity of $\sum_cn_c$: locally the classical packing $D_s^+=D_s\cup(D_s+\frac12\mathbf 1)$ \cite[Ch.~4]{CS99}, with one point per unit cube and minimum $\min(\sqrt s/2,\sqrt2)$ (a lattice for even $s$; $D_4^+\cong\Z^4$, $D_8^+=E_8$). As for the frameless layers of Proposition~\ref{prop:k1designs}, deeper rows that give $t=0$ exist for all but finitely many $p$ when the minors of $F_0$ with disjoint row and column sets are nonzero. For $s=5$ an explicit such layer gives digital $(0,5,5)$-nets with $\delta N^{1/5}>1$ for every prime $p\ge37$ (Proposition~\ref{prop:half5} below). These limits are not the best packing values for $4\le s\le7$. The best known lattices $D_4,D_5,E_6,E_7$ have normalized minima $1.189$, $1.231$, $1.290$, $1.345$, and $D_s^+$ reaches the fractions $.84$, $.90$, $.94$, $.98$ of them; the Cohn--Elkies bound \cite{CE03} caps every periodic configuration (hence the toroidal separation of any point set) at about $1.204$, $1.261$, $1.315$, $1.366$. At $s=8$, $D_8^+=E_8$ is optimal among all packings \cite{Via17}, so there the limit $\sqrt2$ of Theorem~\ref{thm:HL} is sharp for the toroidal separation of every point set. \emph{Drift and jitter at one resolution.} The bookkeeping of Section~\ref{sec:drift} decides how small the base can be. Let $s=5$, $k=1$ and $F_0\in(\frac12+\Z)^{5\times5}$. Call a step $E\in\{-1,0,1\}^5$ with odd $\Sigma E$ \emph{binding}: in the limit these steps land on the minimal vectors of $D_5^+$, of length $\sqrt5/2$, so they decide the separation. For a binding step Proposition~\ref{prop:Hk}(1) offers each coordinate the two options $\beta_c\pm\frac p2$, $\beta=F_0E$, and in the limit every coordinate has length $\frac12$. At finite $p$ one option is shorter than $\frac p2$: for $E_c\ne0$ the one with $|pE_c+t|\approx\frac p2$, for $E_c=0$ the one of sign $\varsigma_c(E):=-\operatorname{sgn}\beta_c$; for $E_c\ne0$ put $\varsigma_c(E)=E_c$. In units of $1/p$ of a cell it is shorter by the \emph{drift} (the drift of Section~\ref{sec:drift} in this setting: the first-order shortening of a binding step caused by the bounded part of $F_0$) \[ \mathrm{dr}_c(E)=|\beta_c|\ \ (E_c=0),\qquad \mathrm{dr}_c(E)=-E_c\beta_c\ \ (E_c\ne0),\qquad \mathrm{dr}(E)=\sum_c\mathrm{dr}_c(E)=-\varsigma(E)^{\!\top}F_0E, \] and $\mathrm{dr}(F_0)$ is the maximum of $\mathrm{dr}(E)$ over the binding steps. Every entry of $F_0$ is $\equiv\frac12(p+2F_0[c][d])\pmod p$: the ideal step $\frac p2$ plus an excess of at least $\frac12$ in absolute value, since $p$ steps of $\frac p2$ do not add up to a multiple of $p$ when $p$ is odd. This is the closing-up obstruction of Section~\ref{sec:drift}, here a parity (the \emph{parity holonomy}), and it is why $|\beta_c|\ge\frac12$ when $E_c=0$. A second offset row $D_1$ (integer) adds the \emph{jitter}: with $v=D_1E$, coordinate $c$ is \emph{harmful} for $E$ if $v_c\ne0$ and $\operatorname{sgn}v_c=\varsigma_c(E)$, since then the second digit can wrap in the direction that shortens coordinate $c$ by a further $1/p$, whereas otherwise the second digit costs $O(p^{-2})$. Put $\mathrm{dj}(F_0,D_1)=\max_E\bigl(\mathrm{dr}(E)+\#\{c\text{ harmful}\}\bigr)$ over the binding steps. Here the first-order constants are measured on the squared value, $(\delta N^{1/5})^2=\frac54-\mathrm{dr}/p+\dots$, equivalently $\delta N^{1/5}=\frac{\sqrt5}2\bigl(1-\frac{2\,\mathrm{dr}}{5p}\bigr)+\dots$; they are not deficits in the sense of Definition~\ref{def:K}. Proposition~\ref{prop:Hk}(1) charges a jitter of $1$ to every coordinate, so its constant is $\mathrm{dr}+5$. The second offset digit can be included in the relaxation: \begin{lemma}[Two-digit relaxation]\label{lem:R2} Let $p$ be an odd prime, $s\ge3$, and let a graph form over $\F_p$ (a digital net or not) with the depths $(1,\dots,1)$ have first offset rows $F_0\in(\frac12+\Z)^{s\times s}$ and second offset rows $D_1\in\Z^{s\times s}$; the deeper rows are arbitrary. For $0\ne E\in\{-2,\dots,2\}^s$ let $\beta=F_0E$, $v=D_1E$, let $O_c$ be the options of Proposition~\ref{prop:Hk} (with $\sigma_c=\beta_c$), $U_c=\{v_c,\,v_c-p\operatorname{sgn}v_c\}$ ($U_c=\{0\}$ if $v_c=0$), and \[ b_c=\min_{t\in O_c,\ u\in U_c}\max\bigl(0,\ |p^2E_c+pt+u|-1\bigr),\qquad \mathrm{LB}_2(p)=\min\Bigl(2,\ \min_E|b|/p^2\Bigr). \] If $p>\max|2\beta_c|$ and $p>\max|v_c|$ over these $E$ and $c$, then $\delta N^{1/s}\ge\mathrm{LB}_2(p)$ for the toroidal distance. \end{lemma} \begin{proof} In cell units coordinate $c$ of the point of cell $n$ is $n_c+r_c/p+(d_c+g_c)/p^2$, with $r_c$ and $d_c$ the first and second offset digits and $g_c\in[0,1)$. For two distinct points and a torus representative $E$ of their cell difference with some $|E_c|\ge3$, coordinate $c$ differs by more than $2$, as in the proof of Proposition~\ref{prop:Hk}(1). Otherwise, as there (at $k=1$ there are no carries), $\Delta r_c\in O_c$. As $d=D_1n\bmod p$ is linear in the cell, $\Delta d_c\equiv v_c\pmod p$ with $\Delta d_c\in(-p,p)$ and $|v_c|1$ (toroidal), and $\delta N^{1/5}\to\sqrt5/2=1.1180\ldots$ as $p\to\infty$. \item For every prime $p\ge53$, every graph form over $\F_p$ with the depths $(1,\dots,1)$ and first offset rows $F_0^*$ (deeper rows arbitrary) has $\delta N^{1/5}>1$ (toroidal). \end{enumerate} \end{proposition} \begin{proof}[Proof (computer-assisted)] (1) The generator matrix of coordinate $c$ has the rows $\e_c$ and row $c$ of $F_0^*$, $D_1$, $D_2$, $D_3$. By Lemma~\ref{lem:box} its box minors are the minors of the stacked offset rows with disjoint row and column sets; each of the $126$ of them, computed in exact rational arithmetic, is a nonzero rational whose numerator and denominator have no prime factors other than $2$, $3$ and $5$. Lemma~\ref{lem:nie} concludes. (2) Over $E\in\{-2,\dots,2\}^5$, $\max|2(F_0^*E)_c|=22$ and $\max|(D_1E)_c|=24$, so Lemma~\ref{lem:R2} applies for $p\ge29$. Evaluated exactly, $\mathrm{LB}_2(p)>1$ for every prime $37\le p<200$ (the least value is $1.0127\ldots$, at $p=37$; and $\mathrm{LB}_2(31)=0.9918\ldots$), and $\mathrm{LB}_2^*(200)^2=1.1877\ldots>1$, so $\mathrm{LB}_2(p)\ge\mathrm{LB}_2^*(p)\ge\mathrm{LB}_2^*(200)>1$ for every $p\ge200$. For the limit, with the row-sum norm $\|F_0^*\|_\infty=\max_c\sum_d|F_0^*[c][d]|=\frac{11}2$ every option of Proposition~\ref{prop:Hk}(1) is at least $p\,\ell_c(E)-(2\|F_0^*\|_\infty+1)$, where \[ \ell_c(E)=\min\bigl(|E_c+\tfrac12|,|E_c-\tfrac12|\bigr)\ (\Sigma E\text{ odd}),\qquad\ell_c(E)=\min_{\omega}|E_c-\omega|\ (\Sigma E\text{ even}), \] $\omega$ over $\{0,\operatorname{sgn}\beta_c\}$ (over $\{0\}$ if $\beta_c=0$). So $\delta N^{1/5}\ge\lambda-\sqrt5(2\|F_0^*\|_\infty+1)/p$ with $\lambda=\min_E|\ell(E)|$. If $\Sigma E$ is odd, every coordinate gives at least $\frac12$, with equality exactly on the binding steps; if $\Sigma E$ is even, $|\ell(E)|^2$ is an integer, and a finite check over the $5^5-1$ vectors shows that it is at least $2$. So $\lambda=\sqrt5/2$, and Proposition~\ref{prop:Hk}(3) with $C=\frac32$ gives $\delta N^{1/5}\le\sqrt5/2+4\sqrt5/p$ for $p>32$. (3) Proposition~\ref{prop:Hk}(1) with $k=1$ and $F_0=F_0^*$ applies for $p>22$. Evaluated exactly, $\mathrm{LB}>1$ for every prime $53\le p<200$ ($\mathrm{LB}(47)=0.9906\ldots$, $\mathrm{LB}(53)=1.0049\ldots$), and the nondecreasing bound $\mathrm{LB}^*$ after Proposition~\ref{prop:Hk} has $\mathrm{LB}^*(200)^2=1.1689\ldots>1$. (The monotone bounds already exceed $1$ from $p=50$ for $\mathrm{LB}_2^*$ and from $p=63$ for $\mathrm{LB}^*$, so the exact scans up to $200$ in (2) and (3) are more than needed.) \end{proof} The threshold $37$ is that of the net itself: its exact normalized minimum distances are $0.9928$ at $p=31$ and $1.0134$ at $p=37$ (computed), and the relaxation of Lemma~\ref{lem:R2} follows the exact values closely (Figure~\ref{fig:half5}). The second row $D_1$ is \emph{lean}: $\mathrm{dj}(F_0^*,D_1)=\mathrm{dr}(F_0^*)=\frac{17}2$, i.e. the harmful residues of $D_1$ lie on steps whose drift leaves room for them. Whether some $(0,5,5)$-net over $\F_p$ with $p\le31$ has $\delta N^{1/5}>1$ is open. \begin{remark}[First-order accounting]\label{rem:accounting} Let $F_0\in(\frac12+\Z)^{5\times5}$ with $\det F_0\ne0$ satisfy \[ \text{(Gap)}\qquad |\ell(E)|>\sqrt5/2\ \text{for every } 0\ne E\in\{-2,\dots,2\}^5 \text{ that is not a binding step}, \] with $\ell$ as in the proof of Proposition~\ref{prop:half5}, let $D_1,D_2,D_3$ be fixed integer matrices, and let $K_{\mathrm{eff}}(p)=\frac{5p}2\bigl(1-\delta N^{1/5}/\frac{\sqrt5}2\bigr)$ for the digital net with these rows over $\F_p$, normalized so that $K_{\mathrm{eff}}\to\mathrm{dr}$ when the second row is lean. Then \[ \mathrm{dr}(F_0)\ \le\ \liminf_{p\to\infty}K_{\mathrm{eff}}(p)\ \le\ \limsup_{p\to\infty}K_{\mathrm{eff}}(p)\ \le\ \mathrm{dj}(F_0,D_1). \] In particular, if $\mathrm{dj}(F_0,D_1)=\mathrm{dr}(F_0)$, then $\delta N^{1/5}=\frac{\sqrt5}2\bigl(1-\frac{2\,\mathrm{dr}(F_0)}{5p}\bigr)+O(p^{-2})$. For $\mathcal Q^*_p$ the finite checks give $\mathrm{dr}=\mathrm{dj}=\frac{17}2$, $\det2F_0^*=-880$ and $|\ell(E)|^2\ge2$ on the non-binding steps, so $\delta N^{1/5}=\frac{\sqrt5}2\bigl(1-\frac{17}{5p}\bigr)+O(p^{-2})$. \emph{Proof.} Upper bound for $K_{\mathrm{eff}}$: on a binding step each option of Lemma~\ref{lem:R2} is $p^2\bigl(\frac12-x/p\bigr)+O(1)$ in absolute value, and the smallest has $x=\mathrm{dr}_c(E)+[c\text{ harmful}]$: the shorter option of the geometric digit $r$ shortens by $\mathrm{dr}_c(E)$, a second-digit wrap of the sign $\varsigma_c(E)$ by $1$ more, a residue of the other sign by $O(1)$ only, and for $E_c=0$ the longer $r$-option combined with a harmful wrap is not shorter, as $|\beta_c|\ge\frac12$. So these steps give $|b|^2/p^4=\frac54-(\mathrm{dr}(E)+\#\text{harmful})/p+O(p^{-2})$. By (Gap) the other steps of $\{-2,\dots,2\}^5$ have $|b|/p^2\to|\ell(E)|>\sqrt5/2$. Hence $\mathrm{LB}_2(p)=\frac{\sqrt5}2\bigl(1-\frac{2\,\mathrm{dj}}{5p}\bigr)+O(p^{-2})$, and $\delta N^{1/5}\ge\mathrm{LB}_2(p)$ gives $\limsup K_{\mathrm{eff}}\le\mathrm{dj}$. Lower bound: fix a binding $E$ with $\mathrm{dr}(E)=\mathrm{dr}(F_0)$. For $p$ large ($p\nmid\det2F_0$, $p$ larger than the entries of $D_1$ and than $2|2\beta_c|+2$) $F_0$ is invertible modulo $p$, so $n\mapsto r=F_0n\bmod p$ is a bijection, and the cells whose five first offset digits all take the shorter option for $E$ form a box in the $r$-coordinates with at least $(\frac p2-O(1))^5$ cells. A second-digit wrap $\Delta d_c\ne v_c$ needs $d_c(n)$ in a set of $|v_c|$ residues, which happens on $|v_c|p^4$ cells if $v_c\ne0$ (as $d_c$ is a nonzero linear form) and never if $v_c=0$. So for $p$ large some cell has the shorter options and $\Delta d=v$, and its step $E$ gives $|D|^2=\frac54-\mathrm{dr}(F_0)/p+O(p^{-2})$, i.e. $\liminf K_{\mathrm{eff}}\ge\mathrm{dr}(F_0)$. In the same way Proposition~\ref{prop:Hk}(1) has $\mathrm{LB}=\frac{\sqrt5}2\bigl(1-\frac{2(\mathrm{dr}(F_0)+5)}{5p}\bigr)+O(p^{-2})$. (Gap) is a genuine hypothesis. The layer \[ 2F_0=\begin{psmallmatrix}-3&1&-1&-3&-3\\-3&-3&3&1&-1\\-3&1&-3&3&1\\1&-1&3&-3&1\\1&1&1&1&-1\end{psmallmatrix} \] satisfies (B2) and (MIN), defined before Theorem~\ref{thm:D}, and has $\det2F_0=560$, but the even step $E=(1,-1,-2,-1,-1)$ has $\beta=(2,-3,-1,-1,-1)$ and $|\ell(E)|=1$: its coordinates with $E_c=\pm1$ move forward, so the step is short (length $1$). Each half of the remark uses only one hypothesis: (Gap) only the upper bound $\limsup K_{\mathrm{eff}}\le\mathrm{dj}$, and $\det F_0\ne0$ only the lower bound; the deeper rows are arbitrary in both. With (Gap) the accounting is an identity for the relaxations and a two-sided bound for the nets, exact when the second row is lean. Cauchy--Schwarz, $\sum_c(\frac12-x_c/p)^2\ge\frac15(\frac52-K/p)^2$, makes $\delta N^{1/5}>1$ plausible once $p\gtrsim\frac{2K}{5(1-2/\sqrt5)}\approx3.79\,K$; this is a first-order estimate, not a threshold. It predicts $51$ for $F_0^*$ with Proposition~\ref{prop:Hk}(1) ($K=\frac{17}2+5$) and $32$ for $\mathcal Q^*_p$ ($K=\frac{17}2$), against the certified $53$ and $37$. \end{remark} How small can the drift be? Two conditions on the layer are natural. Every minor of $F_0$ with disjoint row and column sets is a box minor at $s=5$ (Lemma~\ref{lem:box}); its denominator is a power of $2$, so a zero minor kills $t=0$ at every odd $p$. At $s=5$ these minors are the off-diagonal entries and the $2\times2$ minors on two rows $\{a,b\}$ and two columns outside $\{a,b\}$, and we require \begin{align*} &\text{(MIN)}&&\text{all these minors are nonzero;}\\ &\text{(B2)}&&\alpha_c:=-F_0[c][c]\ \ge\ m_c:=\max_{d\ne c}|F_0[c][d]|\ \text{for every }c. \end{align*} (B2) is the backward condition of Remark~\ref{rem:K}: it says that on every two-axis step $\e_c\pm\e_d$ coordinate $c$ is backward (its geometric digit does not move with its cell step); if $\alpha_c<|F_0[c][d]|$, coordinate $c$ moves forward on one of the steps $\e_c\pm\e_d$. \begin{theorem}[The least drift]\label{thm:D} Every $F_0\in(\frac12+\Z)^{5\times5}$ with (B2) and (MIN), with entries of any size, has $\mathrm{dr}(F_0)\ge\frac{17}2$, and $\mathrm{dr}(F_0^*)=\frac{17}2$. The part of the proof for layers whose off-diagonal part is not skew-symmetric is computer-assisted. \end{theorem} \begin{proof} \emph{Diagonal.} $\mathrm{dr}(E)$ contains $\alpha_c$ exactly for the coordinates with $E_c\ne0$, since $-E_c\beta_c=\alpha_c-E_c\sum_{d\ne c}F_0[c][d]E_d$, and $\beta_c$ does not involve $\alpha_c$ when $E_c=0$. So $\mathrm{dr}(F_0)$ does not increase when every $\alpha_c$ is lowered to $m_c$, which keeps (B2) and (MIN); we assume $\alpha=m$. \emph{Minor tax.} At most one row of $F_0$ has all its off-diagonal entries in $\{\pm\frac12\}$: if rows $a\ne b$ did, the three ratios $F_0[a][c]/F_0[b][c]$, $c\notin\{a,b\}$, would lie in $\{\pm1\}$, two would be equal, and the $2\times2$ minor on those two columns would vanish. So at most one $m_c$ is $\frac12$: at most one row is \emph{light}, with all off-diagonal entries $\pm\frac12$, and the others are \emph{heavy}, and $\sum_c\alpha_c\ge4\cdot\frac32+\frac12=\frac{13}2$. \emph{Balance on the full steps.} A \emph{full} step has all $E_c=\pm1$. For $E\in\{\pm1\}^5$ (binding, as $5$ is odd), $\mathrm{dr}(E)=-E^{\!\top}F_0E=\sum_c\alpha_c+\phi(E)$ with $\phi(E)=\sum_{c26$ and gives $\delta N^{1/5}>1$ only from $p=89$ on ($\mathrm{LB}(83)=0.9975\ldots$, $\mathrm{LB}(89)=1.0056\ldots$, computed), against $53$ for $F_0^*$ (Figure~\ref{fig:half5}). Dominance, which Proposition~\ref{prop:Hk}(2) and Theorem~\ref{thm:HL} use to obtain a bound for all designs at once, is what the accounting says to avoid: by the diagonal step of the proof of Theorem~\ref{thm:D} it costs $\sum_c(\alpha_c-m_c)$ in drift. \end{remark} \begin{figure}[tbp] \centering \IfFileExists{figs/col_half5.pdf}{\includegraphics[width=\linewidth,height=0.45\textheight,keepaspectratio]{figs/col_half5.pdf}}{\fbox{\parbox{0.9\textwidth}{\centering figure \texttt{col\_half5} pending}}} \caption{\colHalfCaption\interactive{beyond.html}} \label{fig:half5} \end{figure} \emph{Searched designs.} Smaller constants than $B_s$ come from searched split-frame designs: random integer points with $u$ even and $v$ odd, then, column by column and level by level, short reachable columns (Proposition~\ref{prop:Gsplit}) in the coset $\frac12\mathbf h+\Z^{s-1}$ for a vector $\mathbf h$ of odd integers, found by lattice reduction; the diagonal is then chosen by evaluating the exact relaxation $\mathrm{LB}$ of Proposition~\ref{prop:Hk}(1), and it need not be dominant. The bound of Proposition~\ref{prop:HL5} is the exact value of the relaxation of Proposition~\ref{prop:Hk}(1); its design $H_5^*$ is also the one drawn in Figures~\ref{fig:arclarge}(c) and~\ref{fig:arcsplit}. \begin{proposition}[Dimension five; computer-assisted]\label{prop:HL5} Let $\mathcal H^{5*}_{k,p}=\mathcal P^{u,v,F}_{k,p}$ for the split frame $u=(-2,0,2,4,8)$, $v=(5,7,-1,3,1)$ and the design $F=H_5^*$ of Appendix~\ref{app:designs}. For every prime $p\notin\{2,3,5,7,97,101\}$ and every $k\ge1$ it is a digital $(0,5k,5)$-net over $\F_p$ with one point in every cell ($p=97,101$ enter only for $k\ge4$), and for every prime $p\ge1511$ and every $k\ge1$, \[ \delta N^{1/5}>1.0265 . \] \end{proposition} \begin{proof}[Proof (computer-assisted)] The design $H_5^*$ satisfies (Half); its geometric layer $F_0$ has the row scalars $\lambda'=\frac12(-9,-3,-3,3,9)$ and the column scalars $\mu=(5,5,5,-5,1)$. The primes of the point differences, of the units $\lambda'_c\mu_d$ and of the leading coefficients of the column polynomials $g_{d,j}$ (exact interpolation) are $2,3,5,7$ and $97,101$, the last two only in leading coefficients at carry depth $j=3$; Proposition~\ref{prop:Gsplit} gives the net statement. For the separation, $\max|\sigma_c|=253$, so Proposition~\ref{prop:Hk}(1) applies for $p>506$; the monotone bound $\mathrm{LB}^*$, minimized over the number of design levels $L=\min(k,4)\in\{1,2,3,4\}$ (that is, over $k$), is $1.026529\ldots$ at $p=1511$, the same value as $\mathrm{LB}$ (attained at $L=4$, i.e. $k\ge4$). As $\mathrm{LB}^*$ is nondecreasing in $p$ for each $L$, $\delta N^{1/5}\ge\mathrm{LB}^*(p)\ge\mathrm{LB}^*(1511)=1.026529\ldots>1.0265$ for every prime $p\ge1511$ and every $k$. \end{proof} \emph{Dimension three.} $D_3^+$ has minimum $\sqrt3/2<1$, so the mechanism above gives nothing at $s=3$; whether other layers with entries of order $p$ and a rational limit $A=\lim\tilde M/p$ can be realized for every $k$ in dimension three is open (Remark~\ref{rem:open}(7)). \section{Verification status}\label{sec:formal} Every result is proved in the text, by hand or, where stated, with a computer-assisted finite certificate. The numbers in the paper fall into three classes. \begin{itemize} \item \emph{Computer-assisted proofs} rest on exact finite certificates: Propositions~\ref{prop:korobov4}, \ref{prop:4D2}, \ref{prop:HL5} and~\ref{prop:half5}; part (4) of Proposition~\ref{prop:wnet} (exact minimum distances at the primes $17\le p\le89$); the part of Theorem~\ref{thm:D} for layers that are not skew; the rows $s\le6$ of Table~\ref{tab:Kdesigns} and $s\ge4$ of Table~\ref{tab:practical} (with Proposition~\ref{prop:k1designs}); the values of Table~\ref{tab:S1E}; the per-prime certificates of Remark~\ref{rem:perprime} (Appendix~\ref{app:perprime}); and Theorem~\ref{thm:base23} (Table~\ref{tab:base23}). The certificates of Propositions~\ref{prop:korobov4}, \ref{prop:4D2}, \ref{prop:HL5} and~\ref{prop:half5}, of part (4) of Proposition~\ref{prop:wnet}, of Theorem~\ref{thm:D}, of Table~\ref{tab:S1E}, of Appendix~\ref{app:perprime}, of parts (2) and (3) of Theorem~\ref{thm:base23} and the per-prime completions of Proposition~\ref{prop:k1designs} are also checked in the Lean kernel. Part (1) of Theorem~\ref{thm:base23}, the entries $.4699$ and $.2586$ of Table~\ref{tab:base23}, and the thresholds and scans of Tables~\ref{tab:Kdesigns} and~\ref{tab:practical} are checked by scripts only (Appendix~\ref{app:formal}). The certificates of Theorems~\ref{thm:base23} and~\ref{thm:D}, Propositions~\ref{prop:4D2}, \ref{prop:half5}, \ref{prop:wnet} and~\ref{prop:k1designs} and Tables~\ref{tab:Kdesigns} and~\ref{tab:practical} were moreover reproduced by two independent implementations, that is, by two separately written programs (for Theorem~\ref{thm:base23}, two C++ checkers written from the mathematics, without shared code). The scripts of the accompanying software deposit (Appendix~\ref{app:verify}) rerun the certificates of Theorem~\ref{thm:base23}, Propositions~\ref{prop:half5} and~\ref{prop:wnet} and Theorem~\ref{thm:D}. \item \emph{Computed values} are labelled where they occur and are used in no proof: the decay of the net of \cite{GK09} (Section~\ref{sec:intro}); the bracket of Remark~\ref{rem:cubecost}; the exact minimum distances of Table~\ref{tab:c}; the values $a^*_s$ of Table~\ref{tab:astar}; the rows $s=7,8$ of Table~\ref{tab:Kdesigns}; the effect of Remark~\ref{rem:projlines}; the per-prime nets listed after Corollary~\ref{cor:frameid} other than the one of Proposition~\ref{prop:korobov4}; the computed values after Proposition~\ref{prop:twocoset}, including the comparison with the column nets; those after Proposition~\ref{prop:wnet}, including the layer at $m=3$ in which two coordinates share their geometric digit (value $\sqrt{233}/15$ in the limit model of Section~\ref{sec:wnet}); the shifts of Remark~\ref{rem:wshift} that break the lock; the comparison of $\mathcal W^*_p$ with $\mathcal W_p$ in Remark~\ref{rem:wstar}; the comparison nets of Figure~\ref{fig:drift}(c), whose second digit is the same in all coordinates (Section~\ref{sec:drift}); the exact minimum distances of $\mathcal Q^*_p$ after Proposition~\ref{prop:half5}; the orbits after Theorem~\ref{thm:D}; and the values for $M_5$ in Remark~\ref{rem:M5}. Appendix~\ref{app:verify} lists the scripts that reproduce them. \item \emph{Heuristic statements} are marked as such where they occur (for instance in Remarks~\ref{rem:glue}, \ref{rem:diag0}, \ref{rem:skelr0}, \ref{rem:skelconst} and~\ref{rem:S1Elim}, and the limit model of Section~\ref{sec:wnet}). \end{itemize} \emph{Lean.} Every theorem, proposition, lemma and corollary, every definition except Definition~\ref{def:geom} (which only introduces names), and the proved statements of the remarks listed in Table~\ref{tab:lean} are formalized in Lean~4 \cite{dMU21} with Mathlib \cite{Mathlib20}, apart from finite computations that are checked by scripts instead: part (1) of Theorem~\ref{thm:base23}, the tables of designs and the computed values above (the list is in Appendix~\ref{app:formal}). One formalized result, Proposition~\ref{prop:cube}, uses classical theorems that are not in Mathlib as explicit hypotheses: Haj\'os's theorem, Keller's conjecture for $s\le7$ and the optimal packing bounds in dimensions two and three; so does the growth statement for the design $U_s$ of Remark~\ref{rem:Us} (the prime number theorem). Consequences of other cited theorems, such as the asymptotics after Proposition~\ref{prop:E} and Theorem~\ref{thm:Lstar}, are cited and not formalized. Appendix~\ref{app:formal} describes the method and maps the sections of the paper to the statement files and proof modules of the formalization (Table~\ref{tab:lean}). The formal proofs complement, and do not replace, the arguments in the text. \section{Open problems}\label{sec:remarks} \fussy We list what remains open, by the regimes of the lattice picture (Section~\ref{sec:prelim}): (I) geometric layers $\tilde M$ with entries bounded independently of $q$, at $m=sk$; (II) lattice skeletons, $m\not\equiv0\pmod s$; (III) layers with entries of size $\Theta(q)$. Results stated earlier are not repeated. Recall that the (first-order) deficit of a family of nets is the $K$ in $\delta N^{1/s}=1-K/q+O(q^{-2})$, and the certified deficit $K(F)$ of a design is the value that the one-digit relaxation certifies (Definition~\ref{def:K}). \begin{remark}[Open problems]\label{rem:open} \emph{(I) Bounded layers.} (1) \emph{Growth of the deficit.} Let $K(s)$ be the best first-order deficit $K$ in $\delta N^{1/s}=1-K/q+O(q^{-2})$, $q\to\infty$, over families of digital $(0,sk,s)$-nets over $\Z/q\Z$ whose geometric layers have entries bounded independently of $q$ and no zero diagonal entry. Lower bounds: $K(s)\ge\sqrt{(s-3)/8}$ (Theorem~\ref{thm:L}). For designs on the places $\infty,0,\dots,s-2$: $K(F)\ge s^{1-o(1)}$ (Theorem~\ref{thm:Lstar}); if moreover the row scalars $\lambda'_c$ (Proposition~\ref{prop:F}) take one value on the finite places, $K(F)\ge\operatorname{lcm}(1,\dots,s-3)$ (Proposition~\ref{prop:E}); and the two-axis bound $K_{\mathrm{pair}}$ is at least $2$ for $s\ge5$ (Theorem~\ref{thm:O5}). Upper bounds: the constructions here achieve $1$ for $s=3$, $2$ for $s=4$ (Proposition~\ref{prop:4D2}), $5$ and $10$ for $s=5,6$ (Table~\ref{tab:Kdesigns}), and a certified deficit at most $a^*_s\le e^{s\log s+O(s)}$ for every $s\ge4$ (Theorem~\ref{thm:A}). Does $K(s)$ grow polynomially or exponentially? On these places a certified deficit polynomial in $s$ would need row scalars that differ between the finite places (Proposition~\ref{prop:E}). Also open: $K=1$ for $s=4$; whether layers with zero diagonal do better (Remark~\ref{rem:diag0}); and the optimal second-order constant $C$ in $\delta N^{1/3}=1-1/p+C/p^2+o(p^{-2})$ in dimension three. For nets with the layer of $\mathcal P_{k,p}$ it is at most $4$ for $k\ge4$ (after Corollary~\ref{cor:ceil3}); for other bounded layers Proposition~\ref{prop:ceil}(i) bounds it by $\frac12\bigl((|\tilde M_{ba}|+1)^2+(|\tilde M_{b'a}|+1)^2\bigr)$ for a coordinate $a$ with $\tilde M_{aa}=-1$, and for zero diagonals (Remark~\ref{rem:diag0}) we have no bound. (2) \emph{Small bases.} Theorem~\ref{thm:A} needs $q\ge5a^*_s+4$, and the designs of Table~\ref{tab:Kdesigns} certify smaller bases only by finite scans; for $s=4$ the bases $q\le15$ have only weaker proved constants, and for $s\ge4$ Theorem~\ref{thm:A} excludes even $q$ (it needs $\operatorname{lpf}(q)\ge s-1$; for $q=2$ no $(0,m,s)$-net with $s\ge4$ exists, as $s\le q+1$). In dimension three the bases $2$ and $3$ are covered for every $m$ (Theorem~\ref{thm:base23}, computer-assisted); at $m=3$ the column nets cover every base coprime to $6$ (Theorem~\ref{thm:col}), column nets do not exist in even bases (Theorem~\ref{thm:O}), and the other bases $q\ge4$ divisible by $2$ or $3$ are open at $m=3$. \emph{(II) Lattice skeletons.} (3) For $s\ge5$ the residues $1\le r\le s-3$ ($\ell\ge3$ skeleton coordinates) are open for all $p$; per-prime nets exist for $s=5$, $r=2$ at seven primes (computed, after Corollary~\ref{cor:frameid}), and a family for all large $p$ is open. Lattices with the optimal minimum order $p^{\ell/s}$ exist by a counting argument (after Lemma~\ref{lem:skellat}); explicit ones are known here only for $\ell=1$, $\ell=2$ and $r=1$, and for $3\le\ell\le s-2$ an explicit family is open (a mixed radix in the ring of integers of a degree-$\ell$ field is a natural candidate). The bottleneck is to realize such a skeleton with bounded faults: inside the frame orbit Lemma~\ref{lem:frameid} excludes the Korobov skeleton digits of $\mathcal B^1$ with small faults (Corollary~\ref{cor:frameid}); other Korobov vectors, faults growing slowly with $p$, and completions outside the frame orbit are not excluded, and per prime frame-orbit skeletons with faults $\pm1$ work (Proposition~\ref{prop:korobov4}). For one skeleton coordinate it is open whether the certified Euclidean constants tend to $\sqrt{\gamma_s}$ (Remark~\ref{rem:S1Elim}) and which explicit $\kappa$ attain it; for two skeleton coordinates, Euclidean (rather than maximum-norm) bounds are open. (4) In dimension four: $m\equiv1\pmod4$ for primes $p>83$ (no closed form is known; per-prime nets are known at every prime $7\le p\le83$, after Corollary~\ref{cor:frameid} (computed), Proposition~\ref{prop:korobov4} works out $p=13$, and we know of no obstruction for larger $p$), $m=4$, and small primes ($m\equiv1$ at $p=5$, $m\equiv2$ at $p\le19$, $m\equiv3$ at $p\le13$). Also open: an explicit $p_0(s)$ in Theorem~\ref{thm:S2} for $s\ge9$, and a covering radius uniform in $p$ for $\ell\ge2$. (5) In dimension three: a mesh ratio uniform in $p$ for $m\equiv1$; $m\in\{4,7\}$ for $p\le31$ and $m=5$ for $p\le7$; and an explicit family for all large primes with limit $>1$ at the resolutions $m\ge4$ (at $m=3$ the nets $\mathcal T_p$ and $\mathcal W_p$ of Propositions~\ref{prop:twocoset} and~\ref{prop:wnet} are such families). At $m=3$, among constants below $1$: whether digital $(0,3,3)$-nets that are not column point sets have smaller deficits than $\frac13$ and $\frac23$ (Theorem~\ref{thm:O} settles this among column point sets; outside them the drift-and-jitter bookkeeping of Section~\ref{sec:drift} is only a heuristic). \emph{(III) Large-entry layers.} (6) (a) Smaller bases for $s\ge5$: Proposition~\ref{prop:HL5} needs $p\ge1511$, and at one resolution it is open whether some $(0,5,5)$-net over $\F_p$ with $p\le31$ has $\delta N^{1/5}>1$ (Proposition~\ref{prop:half5}). (b) The drift-and-jitter bookkeeping for half-integer layers at $k\ge2$, where the split-frame layers are Cauchy matrices, and for $s\ge6$. (c) Split-frame designs with explicit constants of moderate size for $s=6,7,8$. (d) Half-integer layers at the resolutions $m\not\equiv0\pmod s$. (e) Other split-frame layers beyond $D_s^+$. (7) In dimension three: whether a geometric layer $\tilde M\approx qA$ with a fixed rational matrix $A$ can be realized for every $k$ with $\delta N^{1/3}$ bounded above $1$. At $k=1$, layers whose geometric digits wrap together in two or three coordinates (locked wraps, Section~\ref{sec:wnet}) give the limits $\sqrt{66}/8$ (Proposition~\ref{prop:twocoset}), $\sqrt{281}/16$ (Proposition~\ref{prop:wnet}) and $\mu^*=1.0480\ldots$ (Remark~\ref{rem:wstar}), the optimum of the layers of rank one with image $\F_p(2,2,1)$ (Proposition~\ref{prop:wfamily}); the optimum over the other images of rank one, over layers of rank two and over non-digital point sets is open. At $k\ge2$ the geometric digits of any two coordinates are jointly uniformly distributed (Lemma~\ref{lem:shared}, as $2(k+1)\le3k$), so no relation between the geometric digits of two coordinates survives, and neither the two-coordinate lock of $\mathcal T_p$ nor the three-coordinate lock of $\mathcal W_p$ extends to $k\ge2$. For digital nets the only linear relation among the three geometric digits of one index that this count leaves is one of full support, $h_xr_x+h_yr_y+h_zr_z\equiv0$ with all $h_c\ne0$; whether such a relation, or a correlation that involves digits of other indices, can raise the constant above $1$ is open. \emph{The case $k=1$ and digital shifts.} (8) At $k=1$: whether $K_1=1$ is possible for $s=5$, or $K_1=2$ for $s=6$ (Proposition~\ref{prop:k1designs}); frameless layers ($F_0$ chosen directly, not from a frame, Proposition~\ref{prop:k1designs}(2)) for $s\ge7$; and the choice of the deeper rows, which are free at $k=1$ apart from $t=0$ and change the exact separation at small $q$ (Proposition~\ref{prop:k1} and the remark after it). (9) In the bases two and three the constants of Theorem~\ref{thm:base23}(3) are certified lower bounds, not exact values; for $b=3$ and $m\equiv2\pmod3$ the infimum over $m$ and $\tau$ for these nets lies between $0.0770$ and $0.0901$, the latter the exact value at $m=20$ (computed). Whether other seeds or lift parameters give larger every-shift constants in base three is open. \emph{Sequences.} (10) All nets here depend on $m$; they are not prefixes of one sequence, and the question of \cite{DGS2} for sequences remains open in dimension $s\ge3$. \emph{Beyond digital nets.} (11) Can nets that are not digital, for instance permutation nets, exceed the constants of the digital nets here, for instance by avoiding the slip planes that linear offsets force (Corollary~\ref{cor:linoff})? \end{remark} \section{Relation to prior work}\label{sec:prior} Where we say below that we found no prior work, this reflects a literature search (web searches of October 2026) that was not exhaustive; it is not a claim of priority. \emph{The frame and unit minors.} The unit-minor criterion over any finite commutative ring is Niederreiter's \cite[Thm.~6.1]{Nie87}; the M\"obius frame $(T^2,I,T)$ of Section~\ref{sec:frames}, which by Proposition~\ref{prop:faure} is the net with the generator matrices $(J,P,I)$ up to scrambles, is his construction \cite[Thm.~6.2]{Nie87} with the elements $\beta=(0,1)$ of $\F_q$, whose box determinants are $\pm\prod(\beta_k-\beta_h)^{d_hd_k}=\pm1$, and \cite[Cor.~6.3]{Nie87} gives $(0,m,3)$-nets in every base (Theorem~6.2 is stated over a product of finite fields; since the determinant identity is polynomial, the same integer matrices work over $\Z/q$ by Theorem~6.1). For integer systems the reduction to prime-power and prime moduli is \cite[Lemmas~8--9]{LNS96}, and the admissible dimensions over a ring, in particular $s\le\operatorname{lpf}(q)+1$ over $\Z/q\Z$, are \cite[Thm.~2, Cor.~2]{LNS96}; Lemmas~\ref{lem:hermite} and~\ref{lem:Ts} are therefore classical in substance. The same matrices reappear as universally decodable matrices in coding theory \cite{VG06}. \emph{Scrambles and $t$-preserving moves.} That lower-triangular scrambles preserve $t$ is standard \cite{Mat98,Tez95,DP10} (as do Owen's nested scrambles \cite{Owe95}, a nonlinear family that contains the lower-triangular ones); Proposition~\ref{prop:faure} places $\mathcal P_{k,q}$ in Tezuka's generalized Faure class (for each $m$). Lemma~\ref{lem:box} (the gauge moves, which preserve $t$) is elementary; the block scalings are not per-coordinate lower-triangular scrambles (whether they leave the orbit of the nested scrambles we have not checked), and the nets $\mathcal B^1$ use them. The occupied fine cells of $\mathcal B^1$ form, per coarse block, cosets of a Korobov lattice \cite{Kor59,SJ94}. \emph{Separation.} In two dimensions, well-separated nets are known: the Larcher--Pillichshammer net \cite{LP01,DGS2} and, in coding language, a digitally shifted Hammersley set with a product-distance bound \cite{TV06}. In every dimension Sobol' and Shukhman \cite{SS07} bound the minimum distance of Sobol' points from below, by a bound that decays faster than $N^{-1/s}$. Without the digital structure, quasi-uniform low-discrepancy point sets are known in every dimension: Frolov point sets \cite[Thm.~3.1]{DGLPS1}, Kronecker sequences \cite[Thm.~3.9 and the text after Cor.~3.13]{DGLPS1} (with discrepancy $N^{-1+\varepsilon}$), and (by an existence argument) rank-$1$ lattices \cite[Thm.~3.7]{DGLPS1}. These are not digital nets in a fixed base (Section~\ref{sec:intro}), and we do not claim the broad statement. In two dimensions the even-$m$ permutation nets of \cite[Sec.~3]{GK09} follow the pattern of this paper: a lattice of lines, van der Corput slides along these lines carrying $t=0$, and a torus tiling adapted to that lattice (Remark~\ref{rem:gk2d}; their net property is proved in \cite{Rud26}). \emph{Large-entry layers.} The packings $D_s^+$ and $E_8$ are classical \cite{CS99}, and so are the bounds of Cohn and Elkies \cite{CE03} and the optimality of $E_8$ \cite{Via17}. We found no digital nets with a $D_s^+$ or $E_8$ local structure, and no construction that reads the cell and the offset of a coordinate at different points, as the split frames of Section~\ref{sec:large} do. Their box-minor lemma (Lemma~\ref{lem:Tsplit}) is classical in substance: it is the confluent Vandermonde determinant of Hermite interpolation, as in Niederreiter's frames \cite[Thm.~6.2]{Nie87}. Suzuki \cite{Suz26} studies the separation of scrambled digital nets: under full Owen scrambling fixed-$t$ nets are not quasi-uniform (the mesh ratio is $\Omega(N^{1/(2s)})$ in probability), and matrix (linear) scrambling of binary fixed-$t$ digital nets gives mesh ratio $O(\log N)$ in probability in every dimension $s\ge2$. Both results are in probability, for randomized nets, and short of quasi-uniformity; they do not overlap with the explicit constructions here. They concern random scrambles; a particular nested scramble can be well separated: the two-dimensional nets of Remark~\ref{rem:gk2d} are a nested scrambling \cite{Owe95} of the Hammersley net (Remark~\ref{rem:gk2d}(iv)), and they are not digital \cite{GK09,Rud26}. The closest work we found is: \cite{DGS2}, whose quasi-uniform digital example is two-dimensional; He \cite{He17,He21}, who builds rotated sphere packing designs and designs from rotated lattices (rotated $\Z^s$, densest packings including $E_8$, thinnest coverings, interleaved lattices) with quasi-optimal separation distance on all projections, which are not digital nets; He \cite{He19}, who builds maximin designs from interleaved lattices (unions of shifted copies of a lattice), which are not nets; and the search for $(t,m,s)$-nets with maximized minimum distance \cite{GHSK08,GK09}. $D_s$ is Construction~A of the even-weight code, and for even $s$, $2D_s^+$ is Construction~D applied to the repetition code contained in the even-weight code \cite{BS83}; whether such one-point-per-cube descriptions have been used for nets before we have not checked systematically. \emph{One point per cube.} The theorems behind Proposition~\ref{prop:cube} are classical: Haj\'os's theorem on lattice cube tilings \cite{Haj42}, Keller's conjecture, proved for $s\le6$ by Perron \cite{Per40} and for $s=7$ by Brakensiek, Heule, Mackey and Narv\'aez \cite{BHMN20}, and false for $s\ge8$ \cite{LS92,Mac02}, and the optimality of the face-centred cubic packing \cite{Hal05}, of $E_8$ \cite{Via17} and of the Leech lattice \cite{CKMRV17}. We found no prior use of the cube-tiling theorems for the separation of nets. Maximin Latin hypercube designs, in which every coordinate takes each of $N$ grid values exactly once, have separation bounds under this marginal one-per-row constraint \cite{vDHHM07,vDRH09}; we found no prior separation analysis under the constraint of one point per cube. \emph{Codes.} Each ingredient of Remark~\ref{rem:codes} is classical: Construction~A and Construction~D \cite{CS99,BS83}, the characterization of MDS codes by nonsingular submatrices \cite{MS77} and of generalized Reed--Solomon codes by Cauchy redundancy matrices \cite{RS85}, the spectral test \cite{CM67,Knu97}, and the equivalence of $t=0$ with the MDS property in the Rosenbloom--Tsfasman metric, with Reed--Solomon codes of that metric \cite{RT97,Skr01,Nie87}; over a field, the zero count after Lemma~\ref{lem:hermite} is Skriganov's MDS proof for these codes. We found no prior work that relates the Euclidean separation of digital nets to Construction~A, Lee-metric codes or the spectral test, and no treatment of digital-net skeletons as generalized Reed--Solomon Construction~A lattices or of geometric layers $\tilde M\approx qA$ with $A$ rational of denominator $e$ as codes over $\Z/e^2$. The carry criterion of Corollary~\ref{cor:linoff}(b) is the analogue, for carries, of the criterion of Kositwattanarerk and Oggier \cite{KO14}: the code formula $C_0+2C_1+\dots+2^{a-1}C_{a-1}+2^a\Z^n$ of nested binary linear codes is a lattice if and only if the codes are closed under the Schur product, whereas Construction~D always gives a lattice. \emph{What is new.} We did not find the following in the literature. \begin{itemize} \item Part~I: the scramble $L$; the residue description of the geometric digits (Proposition~\ref{prop:cell} and Lemma~\ref{lem:Rs}); a single choice of scrambles that realizes a design for every number $k$ of cell digits (Proposition~\ref{prop:Gs}); a separation bound of the optimal order $N^{-1/s}$ for a family of digital nets in a fixed base ($N=q^m$, $m\to\infty$) in any dimension $s\ge3$ (decaying lower bounds are known \cite{SS07}); the lattice of admissible design columns with its adelic Chebyshev identity (Lemmas~\ref{lem:sat}, \ref{lem:adelic}); the deep-shift lift $m\to m+3$ (Definition~\ref{def:lift}), which carries a separation certificate (Lemma~\ref{lem:liftsep}); and lattice skeletons of digital nets for the resolutions $m\not\equiv0\pmod s$ (Section~\ref{sec:skel}; the lattices themselves are classical congruence and Korobov lattices). \item Column nets: column nets themselves and an exact minimum-distance formula for point sets on lines parallel to the body diagonal (Theorems~\ref{thm:colmin} and~\ref{thm:col}); their two-dimensional analogue is the union of translated diagonals with van der Corput slides of \cite[Sec.~3]{GK09}, and exact formulas are known for particular two-dimensional nets: the $\ell^\infty$ separation of the Sobol' points \cite{Suz25} and the dispersion of the Hammersley net \cite{Kri21}. Also the optimality of the column nets among column point sets and the column ceiling (Theorem~\ref{thm:O}(3), Corollary~\ref{cor:colceil}; part (2) of Theorem~\ref{thm:O} is classical in substance \cite{Pai51,HP55}). \item Part~II: ceilings of the type of Proposition~\ref{prop:ceil} and Theorem~\ref{thm:L}; lower bounds for the certified deficit of frame designs by counting cross-ratios (Theorem~\ref{thm:Lstar}); the bookkeeping of the first-order deficit by drift and jitter with its holonomy obstruction (Lemma~\ref{lem:drift}); the description of the local configurations of digital nets by linear offsets, with the one-point-per-cube ceilings of Section~\ref{sec:whylat}; split frames with their non-symmetric Cauchy layers, and digital nets whose separation exceeds the cubic value $1$ at every resolution $m=sk$ (Theorem~\ref{thm:HL}); digital $(0,5,5)$-nets with $\delta N^{1/5}>1$, and the least drift $\mathrm{dr}=\frac{17}2$ over the half-integer layers of Theorem~\ref{thm:D} (Proposition~\ref{prop:half5}); explicit digital $(0,3,3)$-nets whose local configuration is a union of two lattice cosets, with separation above the cubic value (Proposition~\ref{prop:twocoset}); and a digital $(0,3,3)$-net with a layer of rank one and $\delta N^{1/3}\to\sqrt{281}/16$, with the optimum $\mu^*$ of its family (Propositions~\ref{prop:wnet} and~\ref{prop:wfamily}). \end{itemize} More generally, we found no digital $(0,3,3)$-net (polynomial lattice point sets included, as they are digital nets) with separation above the cubic value in the literature; the closest constructions we found are lattices, not nets: lattice designs \cite{He17,He21}, some of them built from packings denser than the cubic lattice, and the quasi-uniform rank-$1$ and Korobov lattice designs of Sakai and Goda \cite{SG26}. \subsection*{Use of AI systems} The constructions, proofs and Lean formalization were produced by AI systems (Anthropic Claude Opus~5.5; some early exploratory results used Claude Fable~5.1) under the author's direction. The author selected the problems, steered the research and reviewed the formal statements, but has not verified every proof by hand. AI systems are not listed as authors; responsibility for the deposit rests with the author. \appendix \section{Designs}\label{app:designs} A design $(F_0,\dots,F_{L-1})$ prescribes, for each pair of coordinates $(c,d)$, how the geometric digit of $c$ depends on the last $L$ cell digits of $d$ (Section~\ref{sec:prelim}). With Definition~\ref{def:Ns} each design below defines the nets $\mathcal P^F_{k,q}$; the deficits and base thresholds they certify are in Table~\ref{tab:Kdesigns}, and the values at $k=1$ in Table~\ref{tab:practical}. Conventions: entry $(c,d)$ is in row $c$ and column $d$, with the coordinates in the order of the places listed; $F_n$ is the target at carry depth $n$ (the level $k-1-n$) of row $0$ of the block $(c,d)$, diagonal included. \subsection{The designs of Table~\texorpdfstring{\ref{tab:Kdesigns}}{tab:Kdesigns}} \emph{$s=4$}, places $(\infty,0,1,2)$, row scalars $\lambda'\equiv1$; this is the deficit-two design of Proposition~\ref{prop:4D2}: \[ F_0=\begin{psmallmatrix}-2&-2&-1&-2\\1&-2&1&1\\1&-2&-2&2\\1&-1&-1&-1\end{psmallmatrix},\ F_1=\begin{psmallmatrix}0&3&1&1\\0&1&-2&-1\\-1&1&0&-3\\-2&1&0&0\end{psmallmatrix},\ F_2=\begin{psmallmatrix}0&-2&-1&0\\0&0&1&-1\\0&0&0&0\\1&0&1&0\end{psmallmatrix}. \] \emph{$s=5$}, places $(\infty,0,1,2,3)$, row scalars $\lambda'=(\frac12,1,1,1,1)$ (Proposition~\ref{prop:F}; the $\frac12$ needs $q$ odd): \[ F_0=\begin{psmallmatrix}-4&-3&1&-1&3\\-1&-5&-2&1&-2\\-1&-6&-5&2&-3\\-1&-3&2&-5&-6\\-1&-2&1&-2&-5\end{psmallmatrix},\ F_1=\begin{psmallmatrix}-1&5&-1&1&-8\\1&0&-2&-2&4\\0&-2&0&-6&5\\-1&2&-6&1&4\\-2&2&-2&-2&1\end{psmallmatrix},\] \[ F_2=\begin{psmallmatrix}1&0&1&1&4\\-1&1&1&0&-3\\1&5&1&3&-4\\2&2&1&0&1\\2&1&1&3&1\end{psmallmatrix},\ F_3=\begin{psmallmatrix}0&-2&-1&-1&1\\1&0&1&1&0\\-1&3&0&1&1\\-1&-3&3&0&0\\0&-1&0&-1&-1\end{psmallmatrix}. \] \emph{$s=6$}, places $(\infty,0,1,2,3,4)$, $\lambda'=(\frac12,1,1,1,1,1)$: \[ F_0=\begin{psmallmatrix}-8&-6&3&1&-3&6\\2&-9&-6&-1&2&-3\\2&-12&-8&-2&3&-4\\2&-6&6&-7&6&-6\\2&-4&3&2&-9&-12\\2&-3&2&1&-6&-9\end{psmallmatrix},\ F_1=\begin{psmallmatrix}-1&7&-7&-1&7&-13\\-1&-1&2&0&-6&5\\-2&-10&0&-2&-10&6\\-3&1&-26&-1&-26&7\\-4&2&-10&-6&2&2\\-5&2&-6&-2&2&1\end{psmallmatrix},\ F_2=\begin{psmallmatrix}0&-4&0&1&-9&10\\1&4&2&0&7&-1\\0&20&1&4&11&0\\0&9&26&-1&23&3\\1&4&5&0&-2&8\\3&2&2&1&-1&1\end{psmallmatrix},\] \[ F_3=\begin{psmallmatrix}1&8&7&-1&2&-4\\-2&-3&-1&0&-4&0\\1&3&0&-1&-7&0\\1&-3&-6&3&-3&1\\0&-1&-1&3&3&0\\0&0&1&-1&2&0\end{psmallmatrix},\ F_4=\begin{psmallmatrix}-1&-3&-3&-3&3&1\\0&4&0&1&1&-1\\-1&-1&1&1&3&-2\\0&-1&0&-2&0&-5\\3&-1&3&1&-1&1\\0&-1&1&1&0&3\end{psmallmatrix}. \] Each design is reachable: the interpolating polynomials $g_{d,j}$ of Proposition~\ref{prop:F} have degree $j$ and leading coefficients that are units of the ring $R'$ of Definition~\ref{def:Ns} (checked by exact interpolation). The designs for $s=7,8$ are in the software deposit (Appendix~\ref{app:verify}). \subsection{The layers of Table~\texorpdfstring{\ref{tab:practical}}{tab:practical}} At $k=1$ a design is a single geometric layer $F_0$, of two kinds (Proposition~\ref{prop:k1designs}). The \emph{frame layers} are, off the diagonal, the Cauchy layers $\lambda'_c\mu_dw_{cd}$ that every scrambled frame on the places $\infty,0,1,\dots,s-2$ has (Proposition~\ref{prop:Ns}), with a diagonal chosen for $k=1$: for $s=4$ the off-diagonal part of $U^*_4$, for $s=5$ that of the $s=5$ design of Table~\ref{tab:Kdesigns}, and for $s=6$ the searched scalars $\lambda'=(1,\frac23,-2,-\frac23,\frac23,-2)$, $\mu=(\frac32,18,3,3,9,6)$: \[ \begin{psmallmatrix}-1&2&1&2\\1&-1&-1&-1\\1&2&-2&-2\\1&1&1&-2\end{psmallmatrix},\qquad \begin{psmallmatrix}-3&-3&1&-1&3\\-1&-3&-2&1&-2\\-1&-6&-3&2&-3\\-1&-3&2&-3&-6\\-1&-2&1&-2&-3\end{psmallmatrix},\qquad \begin{psmallmatrix}-6&18&3&3&9&6\\1&-4&-2&-1&-2&-1\\-3&-36&-6&6&9&4\\-1&-6&-2&-6&6&2\\1&4&1&2&-4&-4\\-3&-9&-2&-3&-18&-3\end{psmallmatrix}. \] The \emph{frameless layers}, integer matrices chosen directly, for which a $t=0$ completion exists for each admissible prime ($s=5$, $K_1=2$; $s=6$, $K_1=3$; $s=6$, $K_1=5$), are \[ \begin{psmallmatrix}-2&1&-1&-2&-1\\-1&-2&-2&2&-1\\-1&1&-1&-1&2\\-2&1&-2&-2&1\\-1&-2&1&1&-2\end{psmallmatrix},\qquad \begin{psmallmatrix}-3&-3&2&-1&2&1\\1&-3&2&1&3&-2\\1&-2&-3&3&-1&-1\\2&3&1&-2&-1&1\\-3&-2&1&1&-3&2\\-1&-1&3&2&2&-3\end{psmallmatrix},\qquad \begin{psmallmatrix}-4&1&3&-1&2&4\\-2&-4&1&-1&1&3\\-4&-2&-4&4&1&-2\\-2&4&-2&-4&-4&-2\\1&-3&3&1&-5&2\\-1&4&4&-2&1&-5\end{psmallmatrix}; \] their deeper rows are chosen per prime factor of $q$ (Proposition~\ref{prop:k1designs}). For the finitely many admissible primes below the Schwartz--Zippel bound $\mathrm{SZ}(s)$ of that proof, where the generic argument does not apply, explicit completions with $t=0$ were constructed and checked; they are in the deposit. \subsection{The five-dimensional split-frame design \texorpdfstring{$H_5^*$}{H5*}} This design gives the nets $\mathcal H^{5*}_{k,p}$ of Proposition~\ref{prop:HL5}, with $\delta N^{1/5}>1.0265$ for every prime $p\ge1511$. Its split frame (Definition~\ref{def:split}) has the offset points $u=(-2,0,2,4,8)$ and the cell points $v=(5,7,-1,3,1)$; the conventions are those stated at the start of this appendix: \[ 2F_0=\begin{psmallmatrix}-147&5&45&-9&3\\3&-61&-15&-5&3\\5&3&-53&-15&-3\\-15&-5&3&-61&1\\15&45&5&-9&-45\end{psmallmatrix},\quad F_1=\begin{psmallmatrix}0&23&21&12&-1\\-2&0&38&5&-2\\-3&15&0&-10&-1\\4&-28&-4&0&0\\-14&-18&-6&21&0\end{psmallmatrix}, \] \[ F_2=\begin{psmallmatrix}0&-19&-54&-7&-2\\-1&0&-39&-2&1\\-2&-13&0&19&2\\2&26&0&0&-1\\5&-6&-1&-18&0\end{psmallmatrix},\quad F_3=\begin{psmallmatrix}0&-7&10&-1&1\\1&0&8&-1&-1\\2&-4&0&-2&0\\1&4&2&0&0\\1&1&4&1&0\end{psmallmatrix}. \] Off the diagonal, the geometric layer is $M^{(0)}_{cd}=\lambda'_c\mu_d/(u_c-v_d)$ with $\lambda'=\frac12(-9,-3,-3,3,9)$ and $\mu=(5,5,5,-5,1)$; all partial sums $M^{(j)}$ lie in $\frac12+\Z$ (off-diagonal entries of absolute value at most $\frac{87}2$), and $M^{(3)}=-\frac12$ off the diagonal. The interpolating polynomials $g_{d,j}$ of Proposition~\ref{prop:Gsplit} have degree $j$, and their leading coefficients have only the primes $2,3,5,7$, except $-97$ ($d=1$, $j=3$) and $\frac{101}{12}$ ($d=2$, $j=3$). This is why Proposition~\ref{prop:HL5} excludes $p\in\{2,3,5,7,97,101\}$, and why $97$ and $101$ matter only from $k\ge4$ on, when carry depth $3$ is used. \section{Deferred proofs and side results}\label{app:proofs} This appendix holds one deferred proof and two heuristic remarks. \subsection{Proof of Proposition~\ref{prop:P2}} \begin{proof} Let $e=\e_a+\varepsilon\e_b$ be the step of the proposition, from the cell $n$ to the cell $n+e$. Write $\rho'_b$ and $\rho''_b$ for the second offset digit of $b$ at $n$ and at $n+e$; along $e$ it changes by a constant $\sigma'_b$ modulo $q$. Count the cells $n$ with the following properties: \begin{enumerate} \item[(a)] the last digits of $n_a$ and $n_b$ cause no carry along $e$; \item[(b)] $r_a+\sigma_a\ge q$, so that $a$ wraps ($\sigma_a$ values of $r_a$); \item[(c)] $r_b$ does not wrap ($q-|\sigma_b|$ values); \item[(d)] $\rho'_b$ lies in a nonempty set $\mathcal V_b$ of digit values chosen so that $\varepsilon(\rho''_b-\rho'_b)\le0$; \item[(e)] $r_u+\sigma_u\in[0,q)$ for every coordinate $u\notin\{a,b\}$. \end{enumerate} For one $u$ at a time, the digits involved form a box with depths $k+1$ on $a$, $k+2$ on $b$, $k+1$ on $u$ and $0$ elsewhere, of total $3k+4\le sk$, so they are jointly uniform; the conditions (a)--(d) have positive density and the fraction of cells violating (e) at $u$ is $|\sigma_u|/q$, so some cell satisfies all of them, because $\sum_u|\sigma_u|0$) uses, in the same way, the boxes with depths $k+1$ on $a$, $b$ and $u$ and $0$ elsewhere (total $3k+3\le sk$), with $r_a$ and $r_b$ both wrapping, so that $|\Delta_a|<(\sigma_a+1)/q$ and $|\Delta_b|<(|\sigma_b|+1)/q$. The pair lies inside the cube, so the bound holds for the toroidal and the Euclidean distance. \end{proof} \subsection{A conjectural limit for one skeleton coordinate (heuristic)} \begin{remark}[The limit $\sqrt{\gamma_s}$]\label{rem:S1Elim} Table~\ref{tab:S1E} suggests that $\max_\kappa\lambda_{\mathrm{jit}}(\kappa)p^{-1/s}\to\sqrt{\gamma_s}$, the square root of the Hermite constant ($1.122$, $1.189$, $1.231$, $1.290$, $1.345$, $\sqrt2$ for $s=3,\dots,8$); here $\lambda_{\mathrm{jit}}(\kappa)$ is the Euclidean minimum of the skeleton lattice $\Lambda_\kappa$ after one fine cell of jitter is subtracted in each coordinate (Proposition~\ref{prop:S1E}). The upper bound holds by the definition of $\gamma_s$, since $\lambda_{\mathrm{jit}}\le\lambda_2(\Lambda_\kappa)\le\sqrt{\gamma_s}\,p^{1/s}$. For the lower bound it would suffice that the index-$p$ sublattices of $\Z^s$, scaled to covolume $1$, equidistribute in $\mathrm{SL}_s(\Z)\backslash\mathrm{SL}_s(\R)$ as $p\to\infty$. Indeed, the set of unimodular lattices with $\lambda_1>\sqrt{\gamma_s}-\varepsilon$ is open and nonempty, so for large $p$ it would contain a scaled $\Lambda_\kappa$ ($\lambda_1$ is continuous, and the sublattices containing a unit vector have normalized minimum at most $p^{-1/s}$), and then $\lambda_{\mathrm{jit}}\ge\lambda_2-\sqrt s$. Such equidistribution is a theorem of Clozel, Oh and Ullmo on Hecke points \cite{COU01}, but as usually stated it is for simply connected $G$ and Hecke operators $T_a$ with $a\in G(\mathbb Q)$, which does not literally contain the index-$p$ correspondence of $\mathrm{diag}(p,1,\dots,1)\notin\mathrm{SL}_s(\mathbb Q)$. The version of Eskin and Oh \cite[Thm.~1.2]{EO06} allows $a$ in the commensurator of the arithmetic group; we have not carried out the reduction in detail, so we do not state the limit as a theorem. The limit would hold at the residue $m\equiv-1\pmod s$, as $p\to\infty$ uniformly in $k$, and would exceed the limit $\min(\sqrt s/2,\sqrt2)$ of Theorem~\ref{thm:HL} (at the residue $m\equiv0$) for $4\le s\le7$ and equal it at $s=8$. \end{remark} \subsection{Size of the constants in Theorem~\ref{thm:S2} (heuristic)} \begin{remark}[Constants]\label{rem:skelconst} $B(p)$ and $\rho(p)$ are nondecreasing, so the bounds are positive for every $p\ge p_0$. For $s\ge5$ a rough count gives $B\approx(p/\sqrt2)^{2/s}$, i.e. $\delta N^{1/s}\to2^{-1/s}$ (heuristic). For $s\ge9$ the route of Lemma~\ref{lem:G2N1}(c) needs $B$ of order $(2s/\pi)^2$, which suggests $p_0(s)=e^{O(s\log s)}$ (heuristic). Nothing is optimized: the faults cost $g^*$ cells at small $p$, and for even $r$ a \emph{block radix}, in which each of the two rows of $A$ is a radix chain on its own group of the full coordinates, has a better constant. \end{remark} \section{Per-prime designs in dimension three}\label{app:perprime} \begingroup\small With Theorem~\ref{thm:base23} ($p=2,3$) and Corollary~\ref{cor:allm} ($p\ge37$), these designs give well-separated families of digital $(0,m,3)$-nets in every prime base (Remark~\ref{rem:dgsfinite}). Proposition~\ref{prop:ppsep} reduces the separation bound of Table~\ref{tab:perprime}, for every $k$ and every graph form with the prescribed rows, to one finite computation per design; Propositions~\ref{prop:pp2} and~\ref{prop:pp1} show by hand that a net with the prescribed rows exists for every $k$ if the design satisfies finitely many polynomial conditions modulo $p$, which hold for the $37$ designs. All of this is formally verified (Proposition~\ref{prop:pp1} for odd $p$; every tabulated design has $p\ge5$). The appendix sets up the notation, defines the invariants of the moves, proves existence for every $k$, proves the separation bound for every $k$, and lists the finite checks. \emph{Setting.} We use the notation of Section~\ref{sec:allm}: $p$ is prime, $m=3k+r$, $r\in\{1,2\}$, the depths are $\mathbf k=(k+1,k,k)$ resp. $(k+1,k+1,k)$, and $\Phi^F$ is the integer graph form of the frame with depths $\mathbf k$; all its box minors are $\pm1$. An offset row is named by its coordinate and its position: $X0$ is the first offset row of $x$, $Y1$ the second offset row of $y$, and so on. The \emph{prescribed rows} are the first offset row of a full coordinate and the first two offset rows of a skeleton coordinate: $X0,Y0,Z0=S_z,Z1$ for $r=2$, and $X0,Y0=S_y,Y1,Z0=S_z,Z1$ for $r=1$. The first offset row of a skeleton coordinate is its skeleton digit (Section~\ref{sec:skel}). The other prescribed row of each coordinate gives its \emph{inner digit}: the first digit after the fine cell of the point (Section~\ref{sec:fine}), which places the point inside its fine cell up to $1/p$ of the fine unit. For a full coordinate the inner digit is the first offset digit, the analogue of the geometric digit (Definition~\ref{def:geom}); Lemma~\ref{lem:skelred} counts it among the deeper digits, whereas the designs below prescribe it. $R|d_l$ is the coefficient of an offset row $R$ on level $l$ of the cell of $d$, and for rows $R,R'$ and columns $c,c'$ (levels, possibly of different coordinates) $[R,R'\,|\allowbreak\,c,c']=R|c\;R'|c'-R|c'\;R'|c$. The \emph{window} of the cell of $d$ is its last $L_d$ levels, $L_x=4$ if $r=1$ and $L_d=3$ otherwise. \begin{definition}\label{def:ppdesign} A \emph{per-prime design} for $(p,r)$ assigns to each prescribed row $R$ and each coordinate $d$ integers $R|d^{(i)}$, $0\le i0}^3$ a search radius. A \emph{configuration} is a pair $(E,j)$ of a cell difference $E\in\Z^3\setminus\{0\}$, $|E_c|\le B_c$, and carry depths $j_c\in\{0,\dots,L_c-1\}$, with $j_c=0$ if $E_c=0$. Put $\Delta_{d,0}=E_d$, $\Delta_{d,i}=\sgn(E_d)\,[i\le j_d]$ for $1\le iB_c$ for a full $c$, or $|E_c|\ge2$ for a skeleton $c$, the claim follows from $T\le p^2B_c^2$ resp. $T\le p^4$. Otherwise $|E_c|\le B_c{\raggedright\arraybackslash}p{2.0cm}>{\raggedright\arraybackslash}p{8.1cm}>{\raggedright\arraybackslash}p{4.5cm}@{}} \toprule part of the paper & results & statement file; proof modules\\ \midrule Sec.~\ref{sec:prelim} & Lems.~\ref{lem:nie}, \ref{lem:box}, \ref{lem:Ss}; Rem.~\ref{rem:K}; Def.~\ref{def:dshift} & \texttt{Prelim} (19); \texttt{F*}, \texttt{DShift}\\ Sec.~\ref{sec:frames} & Lems.~\ref{lem:hermite}, \ref{lem:Ts}, \ref{lem:Rs}, \ref{lem:CH}; Props.~\ref{prop:faure}, \ref{prop:Ns}, \ref{prop:Gs}, \ref{prop:F}; Def.~\ref{def:Ns}, Cor.~\ref{cor:tsk}; Rem.~\ref{rem:projlines} & \texttt{Frames} (50); \texttt{F*}, \texttt{Proj*}, \texttt{Obs*}\\ Sec.~\ref{sec:dims} & Thm.~A, Prop.~\ref{prop:D}, Lems.~\ref{lem:sat}, \ref{lem:adelic}, \ref{lem:babai}, Def.~\ref{def:Ustar}; the design $U_s$ of Rem.~\ref{rem:Us} (assuming PNT) & \texttt{ThmA} (66); \texttt{ThmA*}, \texttt{SU*}; Theorem~A also \texttt{Headline}\\ Sec.~\ref{sec:skel} & Lems.~\ref{lem:skelred}, \ref{lem:skelnec}, \ref{lem:skellat}, \ref{lem:G2N1}, \ref{lem:frameid}; Prop.~\ref{prop:skelcomp}, Cors.~\ref{cor:skelreach}, \ref{cor:frameid}; Defs.~\ref{def:S1}, \ref{def:G2rule}, \ref{def:S2}; Thms.~\ref{thm:S1}, \ref{thm:S2}; Prop.~\ref{prop:korobov4}; mathematical claims of Rem.~\ref{rem:codes} & \texttt{Skeleton} (67); \texttt{Sk*}, \texttt{S1*}, \texttt{S2*}, \texttt{T3*}\\ Sec.~\ref{sec:dim3} & Def.~\ref{def:net}, Thms.~\ref{thm:t0}, \ref{thm:main}, \ref{thm:sep}, Prop.~\ref{prop:cell}, Rem.~\ref{rem:lattice3}; Def.~\ref{def:col}, Thms.~\ref{thm:colmin}, \ref{thm:col}, Lem.~\ref{lem:shared}, Props.~\ref{prop:colfree}, \ref{prop:colshift}; Def.~\ref{def:B}, Prop.~\ref{prop:Bnet}, Thm.~\ref{thm:allm}, Rem.~\ref{rem:B2}; Cor.~\ref{cor:allm}, Rem.~\ref{rem:perprime} & \texttt{Dim3} (67; Def.~\ref{def:net} in \texttt{Frames}); \texttt{Main*}, \texttt{Net*}, \texttt{Sep*}, \texttt{Col*}, \texttt{D3*}, \texttt{KN*}\\ Sec.~\ref{sec:base23} & Def.~\ref{def:lift}, Lems.~\ref{lem:lift}, \ref{lem:liftsep}, \ref{lem:shiftcert}; Thm.~\ref{thm:base23}(2), (3), and (1) beyond $L$ given its certificate; Rem.~\ref{rem:shiftparity}; the proved statements of Rem.~\ref{rem:base2} & \texttt{Base23} (21); \texttt{B23*}\\ Sec.~\ref{sec:dgs} & Lem.~\ref{lem:shift}, Props.~\ref{prop:shift}, \ref{prop:linf}; Cors.~\ref{cor:allmshift}, \ref{cor:dgs}; Rems.~\ref{rem:shiftshared}, \ref{rem:dgsfinite} & \texttt{DGS} (64); \texttt{*Shift*}, \texttt{DGS*}\\ Secs.~\ref{sec:deficit}--\ref{sec:drift} & Def.~\ref{def:K}, Props.~\ref{prop:Dprime}, \ref{prop:ceil}, Lem.~\ref{lem:Kone}, Cors.~\ref{cor:ceil3}, \ref{cor:ceils}, Props.~\ref{prop:4D2}, \ref{prop:k1}, \ref{prop:k1designs}; Rem.~\ref{rem:pm1}; the proved statements of Rem.~\ref{rem:sign}; Prop.~\ref{prop:P2}, Def.~\ref{def:Ktrue}, Thms.~\ref{thm:L}, \ref{thm:O5}; Lems.~\ref{lem:O}, \ref{lem:C}, Prop.~\ref{prop:E}, Thm.~\ref{thm:Lstar}; Lem.~\ref{lem:drift}, Cor.~\ref{cor:colceil}, Thm.~\ref{thm:O} & \texttt{Deficit} (75); \texttt{Deficit*}, \texttt{Obs*}, \texttt{Ceil}, \texttt{D4D2*}, \texttt{K1*}, \texttt{Lstar*}, \texttt{Col*}\\ Sec.~\ref{sec:whylat} & Lems.~\ref{lem:hom}, \ref{lem:phase}, Cor.~\ref{cor:linoff}; Prop.~\ref{prop:cube} (assuming Haj\'os, Keller for $s\le7$, packing bounds) & \texttt{WhyLat} (14); \texttt{WL*}\\ Secs.~\ref{sec:S1E}--\ref{sec:wnet} & Prop.~\ref{prop:S1E}, Table~\ref{tab:S1E}; Prop.~\ref{prop:twocoset}; Props.~\ref{prop:wnet}, \ref{prop:wfamily}, Lem.~\ref{lem:wwrap}, Rems.~\ref{rem:wstar}, \ref{rem:wshift} & \texttt{Large} (81); \texttt{S1E*}, \texttt{SkCert*}; \texttt{QUNetsM3/TC*}, \texttt{QUNetsM3/W*}\\ Secs.~\ref{sec:Pnohalf}--\ref{sec:halfms} & Prop.~\ref{prop:P}; Def.~\ref{def:split}, Lems.~\ref{lem:Tsplit}, \ref{lem:Rsplit}, Prop.~\ref{prop:Gsplit} and the remark after it; Lem.~\ref{lem:Hs}, Prop.~\ref{prop:Hk}, Thm.~\ref{thm:HL}, Prop.~\ref{prop:HL5}; Lem.~\ref{lem:R2}, Prop.~\ref{prop:half5}, Rem.~\ref{rem:accounting}, Thm.~\ref{thm:D} & \texttt{Split} (78); \texttt{Split*}, \texttt{HL*}, \texttt{Half*}, \texttt{Q5*}, \texttt{TD*}, \texttt{Acct*}\\ App.~\ref{app:perprime} & Defs.~\ref{def:ppdesign}, \ref{def:pprelax}, Prop.~\ref{prop:ppsep}, Props.~\ref{prop:pp1}, \ref{prop:pp2}, Lem.~\ref{lem:ppinv}, Rem.~\ref{rem:ppshift}; Table~\ref{tab:perprime} & \texttt{Appendix} (14); \texttt{PP*}, \texttt{KN*}\\ \bottomrule \end{tabular} \medskip \caption{Formal verification: for each part of the paper, the statement file of the Lean library that states its results (in \texttt{QUNets/Statements/}, with the number of its theorems; $616$ in all) and the principal proof modules (in \texttt{Z3Net/} unless stated; \texttt{*} stands for any part of a name). Several results are also used, or restated, in other files. ``Assuming~$X$'' means that the result uses the classical theorems~$X$, which are not in Mathlib, as explicit hypotheses (PNT, the prime number theorem; Appendix~\ref{app:formal}). \texttt{Headline} is the self-contained entry for Theorem~A.} \label{tab:lean} \end{table} \section{Computations}\label{app:verify} The computed values printed in the paper, none of which is used in a proof, the finite certificates of Theorem~\ref{thm:base23}, Theorem~\ref{thm:D} and Propositions~\ref{prop:korobov4} and~\ref{prop:half5}, and the figures are reproduced by the programs in the folder \texttt{scripts/} of the accompanying software deposit (Python~3, run with \texttt{uv}, and C++). A driver script builds and runs all of them in a few minutes on one laptop (the certificates of Theorem~\ref{thm:base23} use several cores if available); it fails if a value differs from the one printed here or a regenerated figure differs from the published one. The README of the folder lists, for each program, what it reproduces: \begin{itemize} \item the base-$2$ $(0,m,3)$-net of \cite[Sec.~2]{GK09}, built from its generator matrices and, independently, by a re-implementation of the C routine printed in \cite{GK09}, with the $(0,m,3)$ property by box counting and the exact toroidal minimum distance: $\delta N^{1/3}=.750$, $.649$, $.486$, $.433$, $.472$, $.466$, $.424$, $.447$, $.375$, $.265$, $.186$ for $m=3,\dots,13$ (Section~\ref{sec:intro}); \item the two-dimensional nets of Remark~\ref{rem:gk2d} for $m=4,6,8,10$: the $(0,m,2)$ property; for (iii), all slides along their lines that keep $t=0$ and their distances from lattice slides ($m=4,6,8$), and, for even $8\le m\le20$, the $(0,m,2)$ property and the exact minimum distances, in the tiling metric, of the re-scrambled nets and of the nets themselves; \item the enumeration of one-per-cube lattices in $\frac1e\Z^3$, $e\le11$, and the affinity check of Remark~\ref{rem:cubecost}; \item the exact minimum distances of Table~\ref{tab:c}, by a toroidal neighbour-cell scan of nets rebuilt from Definition~\ref{def:net}, with the other values of this table; \item the computed values after Proposition~\ref{prop:twocoset} (Figure~\ref{fig:arctwocoset}(c)); \item the evaluations of proved formulas in Table~\ref{tab:skel}; \item Table~\ref{tab:astar} ($a^*_s$ and the largest entries computed from Definition~\ref{def:Ustar}, $5a^*_s+4$, the bound of Theorem~A, $a_s$ of $U_s$) and the values derived from it in the Introduction and in Table~\ref{tab:Kdesigns}; \item the certified deficits, $\sigma_{\max}(F)$, $q_{\mathrm{tail}}$ and thresholds of Table~\ref{tab:Kdesigns} ($K$ and thresholds for $s\le6$; $\sigma_{\max}$ and $q_{\mathrm{tail}}$ for $s\le8$) and the finite parts of Proposition~\ref{prop:4D2}, by the exact relaxation of Lemma~\ref{lem:Ss} on the designs of Appendix~\ref{app:designs}; and the $K_1$, thresholds and sizes of Table~\ref{tab:practical} (the per-prime $t=0$ completions of Proposition~\ref{prop:k1designs} are not rerun by these scripts; they were reproduced by independent implementations and are checked in the Lean kernel, Section~\ref{sec:formal}); \item the certificates of Theorem~\ref{thm:base23} (Table~\ref{tab:base23}): the chains replayed from the seeds, $t=0$ at every resolution, the integers $v^*_L$ of Lemma~\ref{lem:liftsep} with the proviso $g_L\le b^{L/3}-Wb^{-2L/3}$, the exact minima between the seed and the certificate resolutions, and the small-$m$ nets, by the checker of the deposit (with OpenMP about half a minute on ten cores); for part~(3), the integers $v^\circ_L$, the exact minima over all digital shifts at the resolutions up to $L$ and of the small-$m$ nets (Lemma~\ref{lem:shiftcert}), and the freezing of the chains, by a second checker (a few seconds); \item the column nets of Section~\ref{sec:m3}: Theorem~\ref{thm:colmin} on random column point sets against an exact brute-force distance, the closed forms of Theorem~\ref{thm:col} against two exact kernels at every $q\le241$ coprime to $6$ (composites included), the box minors, all digital shifts at small $q$ against Proposition~\ref{prop:colshift}, the class maximum of the digital column nets against Theorem~\ref{thm:O}, the comparison with $\mathcal T_p$ after Proposition~\ref{prop:twocoset}, and the comparison nets with a common second digit of Figure~\ref{fig:drift}(c); \item the rank-one nets of Section~\ref{sec:wnet}, Proposition~\ref{prop:wnet}: the box minors of $\mathcal W_p$, Lemma~\ref{lem:wwrap} by enumeration of all cells at small $p$, the numerical steps of the proof of Proposition~\ref{prop:wnet} in exact arithmetic (the rules of part~(2) against the lemma, Table~\ref{tab:wcontacts} and the tangent-line constants, and, independently, the full quadratic of every option in each of the $24$ classes of $p$ modulo $48$), the relaxation at every prime $53\le p\le3000$, the pair of part~(3), the exact minimum distances of $\mathcal W_p$ at every prime $5\le p\le199$ and at $307$, $353$ (the finite part of (4)), the comparison with $\mathcal T_p$, the layer with a geometric digit shared by two coordinates at $p=307$ and its value in the limit model, and the digital shifts of Remark~\ref{rem:wshift}; \item for Proposition~\ref{prop:wfamily} and Remark~\ref{rem:wstar}: the constants and every numerical fact of the proof in exact arithmetic, the value $F(\psi^*)=\mu^{*2}$ from all options, the identity $F_p(u)=F(u/p)$ and $\max_uF_p(u)<\mu^{*2}$ over all $u$ at small primes, the box minors and the bound with $K$ for $\mathcal W^*_p$, the thresholds, and the exact minimum distances of $\mathcal W_p$ and $\mathcal W^*_p$ at the primes $p\le1009$ where they differ (recomputed by the scripts at $193$, $241$, $337$; at the other five primes the values come from two separate distance programs that agree, and are not rerun by the driver); \item the certificates of Proposition~\ref{prop:half5} (the $126$ box minors, the exact scans of Lemma~\ref{lem:R2} and of Proposition~\ref{prop:Hk}(1) with their monotone tails, the finite check of the limit), the constants of Remark~\ref{rem:accounting} and Remark~\ref{rem:M5}, the exact minimum distances of $\mathcal Q^*_p$ for $7\le p\le61$, and the enumerations of Theorem~\ref{thm:D} and of the text after it; \item the data printed in Proposition~\ref{prop:korobov4}, Remark~\ref{rem:cubecost} and Tables~\ref{tab:ppdesigns}, \ref{tab:S1Ekappa}, \ref{tab:b23seeds}, compared with their sources entry by entry, with the finite checks in the proof of Proposition~\ref{prop:korobov4}, the layer of Remark~\ref{rem:accounting} and the identity and estimates in the proof of Proposition~\ref{prop:twocoset}; \item every printed decimal bound, recomputed in exact arithmetic and checked to be truncated or rounded as stated in Section~\ref{sec:conv} (this check also reads the integers $T_p$ of Table~\ref{tab:perprime} from the formalization); \item the figures, including those of Part~I (Figures~\ref{fig:thmA}, \ref{fig:base23}, \ref{fig:projlines}, \ref{fig:col}) and of the accounting (Figures~\ref{fig:drift}, \ref{fig:half5}, \ref{fig:wnet}), each regenerated by its script, which also asserts the properties stated in its caption. \end{itemize} \begin{thebibliography}{99} \bibitem{BHMN20} J.~Brakensiek, M.~Heule, J.~Mackey, D.~Narv\'aez, \emph{The resolution of Keller's conjecture}, J. 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