\documentclass[11pt,letterpaper,twoside]{article} % Page dimensions and typography follow the uploaded mathematical-paper model. \usepackage[left=1.38in,right=1.38in,top=0.43in,bottom=0.66in,headheight=11pt,headsep=11pt,footskip=25pt]{geometry} \usepackage[T1]{fontenc} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{enumitem} \usepackage{booktabs} \usepackage{xcolor} \usepackage{fancyhdr} \usepackage{titlesec} \usepackage{array} \usepackage{tabularx} \usepackage{url} \usepackage{hyperref} \usepackage[nameinlink,noabbrev]{cleveref} \definecolor{ResearchGreen}{RGB}{40,92,54} \definecolor{ResearchRed}{RGB}{145,48,48} \definecolor{ResearchYellow}{RGB}{126,92,0} \hypersetup{ colorlinks=true, linkcolor=black, citecolor=ResearchGreen, urlcolor=black, pdftitle={Finite Truncated Lattice-Length Data on Three-Dimensional Flat Tori}, pdfauthor={Pulin Shi}, pdfsubject={Finite observations, lattice geometry, flat tori, and quadratic forms}, pdfkeywords={lattices, flat tori, truncated length data, successive minima, theta series, quadratic forms} } \setlength{\parindent}{1.15em} \setlength{\parskip}{0pt} \setlist[itemize]{leftmargin=2em,itemsep=0pt,topsep=0.25em} \setlist[enumerate]{leftmargin=2em,itemsep=0pt,topsep=0.25em} \allowdisplaybreaks \emergencystretch=2em % Alternating running heads: author on even pages, short title on odd pages. \pagestyle{fancy} \fancyhf{} \fancyhead[LE]{\footnotesize\thepage} \fancyhead[CE]{\footnotesize\scshape Pulin Shi} \fancyhead[CO]{\footnotesize\scshape Finite Truncated Lattice-Length Data on Three-Dimensional Flat Tori} \fancyhead[RO]{\footnotesize\thepage} \renewcommand{\headrulewidth}{0pt} \renewcommand{\footrulewidth}{0pt} % First page: no running head, date at lower left, page number centred. \fancypagestyle{firstpage}{% \fancyhf{}% \fancyfoot[L]{\begin{minipage}[t]{2.25in}\rule{60pt}{0.4pt}\\[-2pt]\footnotesize\textit{Date:} October 26, 2026.\end{minipage}}% \fancyfoot[C]{\thepage}% \renewcommand{\headrulewidth}{0pt}% \renewcommand{\footrulewidth}{0pt}% } % Traditional mathematical-paper hierarchy: centred small-cap section heads, % bold run-in subsection heads, and no coloured or boxed headings. \titleformat{\section}[block] {\normalfont\scshape\centering} {\thesection.}{0.5em}{} \titlespacing*{\section}{0pt}{1.35\baselineskip}{1.05\baselineskip} \titleformat{\subsection}[runin] {\normalfont\bfseries} {\thesubsection.}{0.5em}{}[.] \titlespacing*{\subsection}{0pt}{0.8\baselineskip}{0.5em} \titleformat{\subsubsection}[runin] {\normalfont\itshape} {\thesubsubsection.}{0.5em}{}[.] \titlespacing*{\subsubsection}{0pt}{0.6\baselineskip}{0.5em} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\Z}{\mathbb Z} \newcommand{\T}{\mathbb T} \newcommand{\vol}{\operatorname{Vol}} \newcommand{\covol}{\operatorname{covol}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\detm}{\operatorname{det}} \newcommand{\LL}{\mathcal L} \newcommand{\VV}{\mathcal V} \newcommand{\rhoid}{\rho} \newcommand{\Ffun}{F} \newenvironment{paperabstract}{% \begin{center}\begin{minipage}{0.82\textwidth}\small\noindent\textsc{Abstract.}\ }% {\end{minipage}\end{center}} \begin{document} \thispagestyle{firstpage} \begin{center} \vspace*{0.48in} {\normalfont\bfseries\fontsize{11}{13}\selectfont FINITE TRUNCATED LATTICE-LENGTH DATA ON THREE-DIMENSIONAL FLAT TORI\\[0.35em] VOLUME NON-IDENTIFIABILITY, EXPLICIT CONSTRUCTIONS, AND IDENTIFICATION-RADIUS BOUNDS\par} \vspace{2.0em} {\normalfont\fontsize{10}{12}\selectfont PULIN SHI\par} \end{center} \vspace{2.9em} \begin{paperabstract} We study what finite truncations of the lattice-length multiset can determine about a three-dimensional flat torus. For a full-rank lattice $\Lambda\subset\R^3$, let $\lambda_1(\Lambda)\leq\lambda_2(\Lambda)\leq\lambda_3(\Lambda)$ be its successive minima, and let $\LL_T(\Lambda)$ record the lengths of all nonzero lattice vectors of norm at most $T$, counted with vector multiplicity. First, on the class defined by $\lambda_1=a$ and $\lambda_2>T$, we prove an exact volume classification: the set of possible covolumes is $(F(a,T),\infty)$, where $F(a,T)=\frac{aT}{2}\sqrt{3T^2-a^2}$. The lower bound uses Oppenheim's classical determinant-minimization theorem for Minkowski-reduced ternary forms; an explicit perturbed basis proves every larger volume is attained. Second, we give a pair of positive-definite Gram matrices with equal diagonal entries and matching two-coordinate data but different determinants. This yields a quantitative obstruction to uniform finite identification and gives the lower bound $\sqrt3$ for the worst-case identification-radius constant. Schiemann's finite theta-series determination theorem supplies the complementary upper bound $\sqrt{7/2}$ after conversion from squared lengths to radii. We also record a cubic sign-product identity for the eight signed basis vectors and explain its limitations. All contributions are separated from classical inputs, and the possible optimality of the constant $\sqrt3$ and the broader topological question are left open.\end{paperabstract} \begin{center} \begin{minipage}{0.82\textwidth}\small \noindent Keywords: lattice geometry; flat tori; truncated length data; successive minima; ternary quadratic forms; theta series; finite identifiability.\\[0.35em] \noindent AMS MSC2020: Primary 11H06; Secondary 11H55, 11E16, 53C22, 58J50. \end{minipage} \end{center} \vspace{1.5em} \section{Introduction}\label{sec:introduction} A flat three-dimensional torus is a quotient \[ M_\Lambda=\R^3/\Lambda, \] where $\Lambda$ is a full-rank lattice. Its metric geometry is encoded by the Euclidean lengths of lattice vectors. If only vectors up to a finite length threshold are observed, an inverse question arises: how much of the underlying lattice can the data determine? The present paper studies the finite, multiplicity-sensitive data \[ \LL_T(\Lambda)=\{\,\|v\|:v\in\Lambda\setminus\{0\},\ \|v\|\leq T\,\}, \] where repetitions are retained. This is a truncated lattice-length multiset, equivalently a truncation of the length data of closed geodesics on the flat torus. It is not, without passing to the dual lattice, the Laplace eigenvalue spectrum; we keep these notions distinct throughout. Our results have two complementary forms. In a regime where every vector up to the observation threshold lies on the shortest lattice line, we determine exactly the range of possible volumes compatible with the data. Separately, a Gram-matrix sign construction gives indistinguishable lattices whose first three successive minima agree, showing that a threshold proportional only to the shortest scale cannot identify all three-dimensional flat tori. \subsection{Summary of results} The principal statements are as follows. \begin{enumerate}[label=(\roman*)] \item If $\lambda_1(\Lambda)=a$ and $\lambda_2(\Lambda)>T\geq a$, then the truncated data are fixed by $a,T$: \[ \LL_T(\Lambda)=\{a,a,2a,2a,\ldots,Na,Na\},\qquad N=\lfloor T/a\rfloor. \] \item On this class, the exact set of possible covolumes is \[ \VV(a,T)=(F(a,T),\infty),\qquad F(a,T)=\frac{aT}{2}\sqrt{3T^2-a^2}. \] In particular, $F(a,T)$ is an infimum and is not attained. \item A sign-flip pair of Gram matrices has $\lambda_1=\lambda_2=\lambda_3=1$, identical truncated length data whenever $T<\sqrt{3-6\delta}$, and different covolumes. Consequently, the worst-case identification-radius constant is at least $\sqrt3$. \item Schiemann's finite determination result for ternary positive-definite quadratic forms gives the upper bound $\sqrt{7/2}$ for the same normalized worst-case constant, with the usual reduction and normalization hypotheses made explicit below. \item A third-order signed-basis identity isolates the product of the three off-diagonal Gram entries. It explains the sign mechanism in the construction, but it is not by itself an invariant of the unlabelled truncated multiset. \end{enumerate} \subsection{Relation to classical work} The volume lower bound uses Oppenheim's minimum-determinant result for reduced ternary forms; Barnes and Trenerry explicitly record this result in their discussion of determinant minima \cite{Oppenheim1946,Barnes1978,BarnesTrenerry1982}. Oppenheim's related work on one-sided inequalities for ternary forms gives further classical context \cite{Oppenheim1953}. The finite identification upper bound uses Schiemann's finite determination theorem for ternary positive-definite real quadratic forms \cite{Schiemann1997}. Background on lattice reduction, successive minima, and the geometry of numbers is available in Cassels, Mahler, van der Waerden, and Gruber--Lekkerkerker \cite{Cassels,Mahler1940,VanDerWaerden,GruberLekkerkerker}; Conway and Sloane and Martinet provide complementary treatments of lattice geometry \cite{ConwaySloane,Martinet}. The flat-torus and spectral remarks use standard references including Milnor, Charlap, and Wolf \cite{Milnor,Charlap,Wolf}. \subsection{Scope and non-claims}\label{subsec:scope} The results establish geometric non-identifiability from finite truncated lattice-length data. They do \emph{not} establish topological non-identifiability within the class of three-dimensional flat tori: every quotient $\R^3/\Lambda$ is homeomorphic (indeed diffeomorphic) to $\T^3$. No novelty claim is made for the classical determinant formula or Schiemann's theorem. We make no priority claim for the sign-flip construction or the signed-basis identity; their relationship to prior work remains to be established by a systematic literature search. \par\smallskip\noindent{\color{ResearchRed}\textbf{Observation model.}} The observation studied here is the \emph{primal-lattice vector-length multiset}. It is not the Laplace spectrum of the flat torus, which is described using the dual lattice. The results establish metric/geometric non-identifiability only; topology is unchanged throughout this model class.\par\smallskip \section{Preliminaries and observation model}\label{sec:notation} \subsection{Lattices and successive minima} A full-rank lattice in $\R^3$ is a subgroup $\Lambda=B\Z^3$ for some invertible real $3\times3$ matrix $B$. Its covolume is \[ \covol(\Lambda)=|\det B|. \] The associated flat torus has Riemannian volume $\vol(M_\Lambda)=\covol(\Lambda)$. For $j=1,2,3$, the $j$th successive minimum is \[ \lambda_j(\Lambda)=\inf\{r>0:\dim\operatorname{span}_{\R}(\Lambda\cap \overline B(0,r))\geq j\}. \] In particular, $\lambda_1$ is the shortest nonzero vector length. The quantity $\lambda_2$ is the least radius containing two linearly independent vectors, and $\lambda_3$ is the least radius containing three linearly independent vectors. We use Euclidean lengths, not squared lengths, for the $\lambda_j$. \subsection{Finite truncated data} \begin{definition} For $T\geq0$, the truncated lattice-length multiset is \[ \LL_T(\Lambda)=\{\,\|v\|:v\in\Lambda\setminus\{0\},\ \|v\|\leq T\,\}, \] with each vector counted separately. Thus $v$ and $-v$ contribute two copies of the same positive length. \end{definition} Two lattices have the same $T$-truncated data if these multisets agree, including every multiplicity. We write $\Lambda\cong\Gamma$ when the corresponding flat tori are isometric; for Euclidean lattices this is equivalent to the existence of an orthogonal transformation carrying one lattice to the other. In particular, unequal covolumes imply non-isometry. \subsection{Identification radius} \begin{definition} For a full-rank lattice $\Lambda$, define its identification radius by \[ \rho_{\mathrm{id}}(\Lambda)=\inf\Bigl\{T\geq0:\ \forall\Gamma\subset\R^3,\ \LL_T(\Gamma)=\LL_T(\Lambda)\Longrightarrow\Gamma\cong\Lambda\Bigr\}. \] The infimum is taken over thresholds for which the displayed uniqueness property holds. The property is monotone in $T$: once it holds at one radius, it holds at every larger radius. \end{definition} This definition uses the full truncated multiset of lengths, not merely the set of distinct length values. A finite determination theorem for theta series can therefore be applied after converting its cutoff in squared-length variables to a radius cutoff. \section{The one-direction observation regime}\label{sec:one-direction} \begin{theorem}[Structure of the truncated multiset]\label{thm:one-line} Let $a>0$ and $T\geq a$. If $\Lambda$ satisfies \[ \lambda_1(\Lambda)=a,\qquad \lambda_2(\Lambda)>T, \] then, with $N=\lfloor T/a\rfloor$, \[ \LL_T(\Lambda)=\{a,a,2a,2a,\ldots,Na,Na\}. \] \end{theorem} \begin{proof} Choose a shortest vector $u\in\Lambda$ of length $a$. Any lattice vector $v$ with $\|v\|\leq T$ must be linearly dependent over $\R$ with $u$, since otherwise the lattice would contain two linearly independent vectors of length at most $T$, contrary to $\lambda_2(\Lambda)>T$. As $u$ is primitive (a nontrivial integer multiple of a shorter lattice vector would contradict minimality), the lattice points on its line are exactly $\Z u$. Therefore the observed nonzero vectors are precisely $ku$ with $1\leq |k|\leq\lfloor T/a\rfloor$. Each positive integer $k$ gives two vectors $\pm ku$, both of length $ka$. \end{proof} \begin{remark} The hypothesis $\lambda_2>T$ is strict. At $T=\lambda_2$, a second independent direction may enter the data, and the stated list need not be complete. \end{remark} \section{Exact classification of compatible covolumes}\label{sec:volume} For fixed $a>0$ and $T\geq a$, define \[ \mathcal C(a,T)=\{\Lambda:\lambda_1(\Lambda)=a,\ \lambda_2(\Lambda)>T\} \] and \[ \VV(a,T)=\{\covol(\Lambda):\Lambda\in\mathcal C(a,T)\}. \] Put \[ F(a,T)=\frac{aT}{2}\sqrt{3T^2-a^2}. \] \subsection{The classical determinant input} For a Minkowski-reduced ternary positive-definite quadratic form with ordered diagonal coefficients $A\leq B\leq C$, Oppenheim's determinant-minimization result gives \begin{equation}\label{eq:oppenheim} \det G\geq ABC-\frac{AB^2}{4}-\frac{A^2C}{4}. \end{equation} The formula for the minimum, including its dependence on the ordered diagonal entries, is recorded in the literature on Minkowski reduction; see Oppenheim and the account by Barnes--Trenerry \cite{Oppenheim1946,BarnesTrenerry1982}. In applying it, the diagonal entries are the squares of the successive-minimum lengths of a suitable reduced basis in dimension three. This is a classical input, not a new result of this paper. \begin{lemma}[Covolume lower bound in terms of the first two minima]\label{lem:det-bound} For every full-rank three-dimensional lattice $\Lambda$, with $a=\lambda_1(\Lambda)$ and $b=\lambda_2(\Lambda)$, \begin{equation}\label{eq:det-minima} \covol(\Lambda)\geq \frac{ab}{2}\sqrt{3b^2-a^2}. \end{equation} \end{lemma} \begin{proof} Choose a reduced basis with Gram diagonal $A=a^2$, $B=b^2$, $C=\lambda_3(\Lambda)^2$, so $A\leq B\leq C$. From \eqref{eq:oppenheim}, \[ \det G\geq ABC-\frac{AB^2}{4}-\frac{A^2C}{4} =AC\Bigl(B-\frac A4\Bigr)-\frac{AB^2}{4}. \] Since $B-A/4>0$ and $C\geq B$, \[ \det G\geq AB\Bigl(B-\frac A4\Bigr)-\frac{AB^2}{4} =\frac{AB}{4}(3B-A). \] The square of the lattice covolume is $\det G$. Taking square roots and substituting $A=a^2$, $B=b^2$ gives \eqref{eq:det-minima}. \end{proof} \subsection{The strict lower bound} \begin{proposition}\label{prop:volume-lower} For every $\Lambda\in\mathcal C(a,T)$, \[ \covol(\Lambda)>F(a,T). \] \end{proposition} \begin{proof} Let $b=\lambda_2(\Lambda)>T$. By \cref{lem:det-bound}, \[ \covol(\Lambda)\geq H_a(b),\qquad H_a(x)=\frac{ax}{2}\sqrt{3x^2-a^2}. \] For $x\geq a>0$, the function $H_a$ is strictly increasing. Thus \[ \covol(\Lambda)\geq H_a(b)>H_a(T)=F(a,T). \] \end{proof} \subsection{An explicit construction attaining every larger volume} \begin{theorem}[Exact volume classification]\label{thm:volume-classification} For every $a>0$ and $T\geq a$, \[ \VV(a,T)=(F(a,T),\infty). \] Consequently, $F(a,T)$ is the infimum of compatible covolumes, but no lattice in $\mathcal C(a,T)$ attains it. \end{theorem} \begin{proof} The strict lower inclusion follows from \cref{prop:volume-lower}. For the reverse inclusion, fix any $V>F(a,T)$. Choose $\varepsilon>0$ sufficiently small and define \begin{align} s^2&=T^2-\frac{a^2}{4}+\varepsilon,\label{eq:s-def}\\ c&=1-\frac{T^2+\varepsilon}{2s^2},\label{eq:c-def}\\ h_0&=s\sqrt{1-c^2}.\label{eq:h0-def} \end{align} Since $T\geq a>0$, we have $s^2\geq 3T^2/4+\varepsilon$ and $0T$ whenever $h\geq h_0$. If $k=0,n=0$, the vector is a multiple of $u$ and every nonzero such vector has length at least $a$. If $k=0$ and $|n|=1$, the first term in \eqref{eq:norm-expansion} is at least $a^2/4$ and the second equals $s^2$, so \[ \|mu+nv\|^2\geq a^2/4+s^2=T^2+\varepsilon. \] If $k=0$ and $|n|\geq2$, then $|n|s\geq2s>T$. If $|k|\geq2$, then $\|mu+nv+kw\|\geq |k|h\geq2h_0>T$, because $c<1/2$ implies $h_0^2>s^2(3/4)>9T^2/16$. It remains to consider $|k|=1$. By replacing a vector by its negative, take $k=1$. Since $h^2\geq s^2(1-c^2)$, \[ \|mu+nv+w\|^2\geq a^2\left(m+\frac{n+1}{2}\right)^2+s^2(n+c)^2+s^2(1-c^2). \] For $n=0$, the first term is at least $a^2/4$ and the last two terms sum to at least $s^2$, giving $T^2+\varepsilon$. For $n=-1$, the first term is nonnegative and the last two terms are at least $2s^2(1-c)=T^2+\varepsilon$. For $n\geq1$, the last two terms are at least $2s^2(1+c)>T^2+\varepsilon$. For $n\leq-2$, they are at least $s^2((2-c)^2+1-c^2)>2s^2(1-c)=T^2+\varepsilon$. Thus every vector not parallel to $u$ has length strictly greater than $T$. Hence $\lambda_1=a$ and $\lambda_2>T$. The covolume at $h=h_0$ equals \[ V_\varepsilon=as h_0=\frac a2\sqrt{(T^2+\varepsilon)(3T^2-a^2+3\varepsilon)}. \] As $\varepsilon\downarrow0$, $V_\varepsilon\to F(a,T)$. Choose $\varepsilon$ small enough that $V_\varepsilonh_0$, the same successive-minimum conditions hold, and \[ \covol(\Lambda_h)=ash=V. \] Every $V>F(a,T)$ is therefore realized. Combined with \cref{prop:volume-lower}, this proves the equality. \end{proof} \begin{remark} The condition $\lambda_2>T$ is essential. The construction approaches the boundary $\lambda_2=T$ only in the limit $\varepsilon\downarrow0$; this explains why the lower endpoint is an infimum rather than an attained minimum. \end{remark} \par\smallskip\noindent{\color{ResearchYellow}\textbf{Literature status.}} The volume classification combines a classical determinant input with the explicit basis construction above. Its priority relative to the existing literature has not been established by a systematic search.\par\smallskip \section{A sign-flip construction beyond the first coordinate scale}\label{sec:signflip} The preceding theorem uses a regime in which only one independent direction is visible. We now build a different pair: both lattices have all three successive minima equal to $1$, yet their truncated data agree beyond the first-minimum scale. \subsection{The Gram matrices} For $0<\delta\leq1/6$, let \begin{equation}\label{eq:gram-pair} G_+=\begin{pmatrix}1&\delta&\delta\\\delta&1&\delta\\\delta&\delta&1\end{pmatrix},\qquad G_-=\begin{pmatrix}1&\delta&\delta\\\delta&1&-\delta\\\delta&-\delta&1\end{pmatrix}. \end{equation} Both matrices are positive definite. The eigenvalues of $G_+$ are $1+2\delta,1-\delta,1-\delta$, and those of $G_-$ are $1-2\delta,1+\delta,1+\delta$. \begin{lemma}[Successive minima]\label{lem:gram-minima} The lattices represented by $G_+$ and $G_-$ both satisfy \[ \lambda_1=\lambda_2=\lambda_3=1. \] \end{lemma} \begin{proof} The coordinate vectors have squared norm $1$, so $\lambda_3\leq1$. Any nonzero integer vector with one nonzero coordinate has squared norm at least $1$. If at least two coordinates are nonzero, then the Euclidean integer norm squared is at least $2$. The smallest eigenvalue of either Gram matrix is at least $1-2\delta\geq2/3$, and hence \[ Q_\pm(x)=x^TG_\pm x\geq (1-2\delta)\|x\|_2^2\geq4/3>1. \] Thus the only vectors of length $1$ are the six signed coordinate vectors. They span three dimensions, proving all three successive minima equal $1$. \end{proof} \subsection{Matching all vectors supported on at most two coordinates} For $x=(x_1,x_2,x_3)\in\Z^3$, write $Q_\pm(x)=x^TG_\pm x$. A vector is called supported on a coordinate subset $S\subset\{1,2,3\}$ if its coordinates outside $S$ are zero. \begin{lemma}[Two-coordinate matching]\label{lem:two-coordinate} For each fixed coordinate subset $S$ with $|S|\leq2$, the multiset of values $Q_+(x)$ over integer vectors with support exactly $S$ agrees with the corresponding multiset for $Q_-$. \end{lemma} \begin{proof} On the $\{1,2\}$- and $\{1,3\}$-coordinate planes, the two quadratic forms have identical restrictions. On the $\{2,3\}$-plane, the restrictions are $x_2^2+x_3^2+2\delta x_2x_3$ and $x_2^2+x_3^2-2\delta x_2x_3$; the substitution $x_3\mapsto-x_3$ is a multiplicity-preserving bijection between the corresponding integer vectors and preserves exact support. The coordinate axes agree trivially. \end{proof} \subsection{Equal truncated data and the first discrepancy} \begin{theorem}[Finite non-identifiability with $\lambda_3=1$]\label{thm:signflip} For $0<\delta\leq1/6$ and every $T<\sqrt{3-6\delta}$, \[ \LL_T(G_+)=\LL_T(G_-), \] where the notation identifies a positive-definite Gram matrix with its associated lattice. Nevertheless, the two lattices have different covolumes and are not isometric. \end{theorem} \begin{proof} By \cref{lem:two-coordinate}, all vectors with at most two nonzero coordinates can be matched, preserving length and multiplicity. Consider a vector with all three coordinates nonzero. Then $\|x\|_2^2\geq3$. The minimum eigenvalue bounds give \[ Q_-(x)\geq (1-2\delta)\|x\|_2^2\geq3-6\delta, \] and \[ Q_+(x)\geq (1-\delta)\|x\|_2^2\geq3-3\delta>3-6\delta. \] If one coordinate has absolute value at least $2$, then $\|x\|_2^2\geq6$, so $Q_\pm(x)\geq6(1-2\delta)\geq4>3-6\delta$. Therefore no vector with three nonzero coordinates enters either truncated multiset at radius $T<\sqrt{3-6\delta}$. All included vectors are covered by the matching lemma. Finally, direct calculation gives \[ \det G_+=1-3\delta^2+2\delta^3,\qquad \det G_-=1-3\delta^2-2\delta^3. \] Their difference is $4\delta^3>0$. The covolumes, which are $\sqrt{\det G_\pm}$, differ, so the lattices cannot be isometric. \end{proof} \par\smallskip\noindent{\color{ResearchGreen}\textbf{Construction.}} For each $0<\delta\leq 1/6$, this pair has $\lambda_3=1$, identical multiplicity-sensitive truncated data for every $T<\sqrt{3-6\delta}$, and unequal covolumes. It is therefore an explicit obstruction to any universal identification radius smaller than $\sqrt3\,\lambda_3$.\par\smallskip \begin{proposition}[The first distinguishing length for this pair]\label{prop:first-gap} For the pair \eqref{eq:gram-pair}, the truncated data differ as soon as $T\geq\sqrt{3-6\delta}$. In particular, the first distinguishing radius is exactly $\sqrt{3-6\delta}$. \end{proposition} \begin{proof} For $G_-$, the two integer vectors $(1,-1,-1)$ and $(-1,1,1)$ satisfy $Q_-(x)=3-6\delta$. By the proof of \cref{thm:signflip}, no vector with all coordinates nonzero in the $G_+$ lattice has squared length at most $3-6\delta$. The vectors with support of size at most two have identical length multiplicities under the matching in \cref{lem:two-coordinate}; the two displayed full-support vectors therefore create an additional multiplicity of at least two on the $G_-$ side at the threshold, which cannot be matched by full-support vectors of $G_+$. Thus the multisets differ at this radius. \end{proof} \begin{example}[A concrete pair]\label{ex:numeric} For $\delta=0.1$, both lattices have $\lambda_3=1$, and their truncated data agree for every $T<\sqrt{2.4}\approx1.54919$. In particular, they agree at $T=1.5$. Their covolumes are $\sqrt{0.972}$ and $\sqrt{0.968}$, respectively, so they are not isometric. \end{example} \section{Identification-radius bounds}\label{sec:radius} Let \[ C_3^*=\sup_{\Lambda}\frac{\rho_{\mathrm{id}}(\Lambda)}{\lambda_3(\Lambda)}, \] where the supremum ranges over all full-rank lattices in $\R^3$. Since both numerator and denominator scale linearly under a uniform dilation, the ratio is scale-invariant. \begin{theorem}[Current two-sided bound]\label{thm:radius-bounds} The known results discussed here imply \[ \sqrt3\leq C_3^*\leq\sqrt{\frac72}. \] \end{theorem} \begin{proof} For the lower bound, use \cref{thm:signflip}. Given any $T<\sqrt3$, choose \[ \delta(T)=\min\left\{\frac16,\frac{3-T^2}{12}\right\}>0. \] If the minimum is the second term, then $3-6\delta(T)=(3+T^2)/2>T^2$. If the minimum is $1/6$, then $T^2\leq1$ and $3-6\delta(T)=2>T^2$. Thus for every $T<\sqrt3$ there is a pair with equal truncated data through $T$, identical $\lambda_3=1$, and unequal covolumes. Hence no universal identification radius below $\sqrt3\lambda_3$ is possible, yielding $C_3^*\geq\sqrt3$. For the upper bound, Schiemann's theorem states that a ternary positive-definite real quadratic form, in a fixed reduced representative, is determined by the initial theta-series data through a bound that may be taken to be $\frac72 s_3$, where $s_3$ is the third successive minimum in the quadratic-form (squared-length) normalization \cite{Schiemann1997}. Converting squared length $s_3=\lambda_3^2$ to a radius gives $T=\sqrt{7/2}\,\lambda_3$. Therefore $\rho_{\mathrm{id}}(\Lambda)\leq\sqrt{7/2}\,\lambda_3(\Lambda)$, subject to the reduced-form normalization in Schiemann's theorem, and taking the supremum gives the upper bound. \end{proof} \par\smallskip\noindent{\color{ResearchYellow}\textbf{Literature-dependent bound.}} The lower bound is supplied by the explicit sign-flip family. The upper bound depends on applying Schiemann’s finite-determination theorem with the normalization stated above; the optimal constant is not determined here.\par\smallskip \begin{remark}[Status of the constant] The interval in \cref{thm:radius-bounds} is a rigorous target range, not a determination of the optimum. The assertion $C_3^*=\sqrt3$ is a natural conjectural target suggested by the lower-bound construction, but it is not proved here, and no claim is made that the question is absent from the literature. \end{remark} \section{A cubic identity for the three-way sign interaction}\label{sec:cubic} The matrices in \cref{eq:gram-pair} differ by the sign of one off-diagonal coefficient. A useful algebraic identity isolates the cyclic product of all three off-diagonal entries. \begin{proposition}[Signed-basis cubic identity]\label{prop:cubic} Let \[ G=\begin{pmatrix}d_1&g_{12}&g_{13}\\g_{12}&d_2&g_{23}\\g_{13}&g_{23}&d_3\end{pmatrix} \] be a real symmetric $3\times3$ matrix, and put $D=d_1+d_2+d_3$. For each $\epsilon\in\{\pm1\}^3$, let $q_\epsilon=\epsilon^TG\epsilon$. Then \begin{equation}\label{eq:cubic-identity} \sum_{\epsilon\in\{\pm1\}^3}(q_\epsilon-D)^3=384\,g_{12}g_{13}g_{23}. \end{equation} \end{proposition} \begin{proof} Expand \[ q_\epsilon-D=2\bigl(g_{12}\epsilon_1\epsilon_2+g_{13}\epsilon_1\epsilon_3+g_{23}\epsilon_2\epsilon_3\bigr). \] Upon cubing and summing over the eight sign triples, every monomial containing an odd power of some $\epsilon_i$ cancels. The only surviving mixed term uses each of the three pair products exactly once. It occurs in six permutations, has the factor $2^3$, and has the same value for all eight sign choices. Its total contribution is $8\cdot 8\cdot6\,g_{12}g_{13}g_{23}=384\,g_{12}g_{13}g_{23}$. \end{proof} The determinant itself is \begin{equation}\label{eq:det-expansion} \det G=d_1d_2d_3-d_1g_{23}^2-d_2g_{13}^2-d_3g_{12}^2+2g_{12}g_{13}g_{23}. \end{equation} Thus, when the diagonal entries and the absolute values of the three off-diagonal entries are fixed, the determinant is sensitive to the sign of the same cyclic product detected by \eqref{eq:cubic-identity}. \begin{remark}[What the identity does not say] The eight signed basis vectors are labelled by a chosen basis. An unlabelled truncated length multiset does not reveal which observed lengths came from which signed basis vectors. Accordingly, \eqref{eq:cubic-identity} is a basis-level algebraic diagnostic; it is not, by itself, an invariant computed from arbitrary truncated data, nor does it prove the universal upper bound $C_3^*\leq\sqrt3$. \end{remark} \section{Geometry, topology, and the meaning of the observation data}\label{sec:topology} \subsection{What is proved geometrically} If two lattices have different covolumes, the corresponding flat tori have different total Riemannian volumes and therefore cannot be isometric. The volume classification and the sign-flip construction consequently establish metric non-identifiability from the stated finite length observations. \subsection{What is not proved topologically} Every full-rank lattice $\Lambda=B\Z^3$ gives a quotient homeomorphic to $\R^3/\Z^3=\T^3$: the invertible linear map induced by $B$ identifies these quotient manifolds. This is part of the standard framework for compact flat manifolds; see Charlap and Wolf \cite{Charlap,Wolf}. The reduction-theoretic statements used earlier are also treated in Mahler, van der Waerden, and Barnes \cite{Mahler1940,VanDerWaerden,Barnes1978}. Our constructions change the metric lattice, not the underlying topological manifold. Thus a claim that the present examples prove non-identifiability of topology would be false. To pursue topology, the comparison class must be enlarged beyond flat three-tori and a finite observation map defined consistently across that larger class. \subsection{Length data versus Laplace spectrum} The length multiset used here comes from the primal lattice. For a flat torus, the Laplace eigenvalues are proportional to squared lengths in the dual lattice (with a convention-dependent factor such as $4\pi^2$). Classical work on isospectral manifolds, including Milnor\'s example, illustrates why the precise spectral data and dimension matter \cite{Milnor}. Therefore a theorem about the Laplace spectrum should not be transferred to our observation model without explicitly passing to the dual lattice. The relevant role of Schiemann\'s theta-series result is that a positive-definite ternary quadratic form has a finite determination bound from its own representation data; it is not a claim that the present truncated primal-length data are literally the Laplace spectrum. \section{Open problems and research programme}\label{sec:open} The results suggest several concrete next tasks. \begin{enumerate}[label=\textbf{P\arabic*.}] \item \textbf{Optimize the universal identification constant.} Determine whether $C_3^*=\sqrt3$, or find a construction with a strictly larger lower bound. This requires either a sharper general identification theorem or a better counterexample family. \item \textbf{Improve the sign-flip construction.} The present pair has first distinguishing radius $\sqrt{3-6\delta}\,\lambda_3$, tending to $\sqrt3\,\lambda_3$ from below. Unequal diagonal entries or a different integer-vector re-pairing may change the first distinguishing radius; no such improvement is established here. \item \textbf{Identify finite data more efficiently.} Schiemann's bound is sufficient but may not be optimal for every lattice. One can ask for lattice-dependent bounds in terms of gaps among $\lambda_1,\lambda_2,\lambda_3$ or the number of short vectors. \item \textbf{Study other dimensions.} The sign-product mechanism is specific to a three-edge cycle in dimension three. Higher dimensions may admit analogous cycle or graph invariants, but a dimension-$d$ theorem would require a separate proof. \item \textbf{Formulate a genuine topological question.} A topological non-identifiability theorem would need spaces of different topological types and an observation process defined on all of them; varying the lattice alone cannot do this. \item \textbf{Complete a priority search.} Search the literature for the exact volume-classification statement, the sign-flip truncated-multiset construction, the threshold constant problem, and the cubic identity before making a novelty claim. \end{enumerate} \section{Conclusion}\label{sec:conclusion} This paper gives an exact description of the covolumes compatible with a fixed finite length observation in the one-direction regime, using a classical ternary determinant bound and an explicit perturbation construction. It also gives a pair of three-dimensional lattices with all successive minima equal to one and identical truncated data up to a threshold approaching $\sqrt3$, despite different covolumes. This yields the interval \[ \sqrt3\leq C_3^*\leq\sqrt{7/2} \] for the worst-case identification-radius constant, with the upper endpoint supplied by Schiemann's finite theta-series determination theorem. The cubic identity records the algebraic sign mechanism that distinguishes the example, while its limitations are stated explicitly. The optimal constant and any extension to topological non-identifiability remain open within this project. \appendix \section{Parameter checks for the volume construction}\label{app:parameters} For completeness, the parameter inequalities in the proof of \cref{thm:volume-classification} are checked here. Since $T\geq a>0$, \[ s^2=T^2-\frac{a^2}{4}+\varepsilon\geq\frac{3T^2}{4}+\varepsilon>0. \] Moreover, \[ 1-c=\frac{T^2+\varepsilon}{2s^2}>0, \] so $c<1$. The condition $c>0$ is equivalent to \[ 2s^2>T^2+\varepsilon\iff T^2-\frac{a^2}{2}+\varepsilon>0, \] which holds because $T^2\geq a^2$. Finally, $c<1/2$ is equivalent to $T^2+\varepsilon>s^2$, or $a^2/4>0$. Thus $0\frac34s^2>\frac{9T^2}{16}, \] so $2h_0>3T/2>T$. They also give $T^2+\varepsilon=2s^2(1-c)$, which is the identity used in the cases $n=-1$ and $n\geq1$ in the main proof. \section{Multiplicity accounting for the sign-flip example}\label{app:multiplicity} The truncated data are a multiset over all integer vectors, not just primitive vectors or distinct length values. The matching lemma uses bijections on each exact support set: the identity map on supports $\{1,2\}$ and $\{1,3\}$, the sign change $x_3\mapsto-x_3$ on support $\{2,3\}$, and the identity on each coordinate axis. Since these support sets partition $\Z^3\setminus\{0\}$ according to the zero pattern, and no vector with support size three is observed below the stated threshold, the union of these bijections is a length-preserving, multiplicity-preserving matching of the entire truncated multisets. At the threshold $T_*^2=3-6\delta$, the two vectors $(1,-1,-1)$ and $(-1,1,1)$ enter for $G_-$, while the lower eigenvalue estimate for $G_+$ excludes any full-support vector at that length or below. Any two-support vectors at that numerical length have already matching multiplicities on both sides and cannot remove the additional count contributed by the full-support vectors of $G_-$. \begin{thebibliography}{99} \bibitem{Barnes1978} E. S. Barnes, ``On Minkowski's fundamental inequality for reduced positive quadratic forms (I),'' \emph{Journal of the Australian Mathematical Society, Series A}, 26 (1978), 46--52. \bibitem{BarnesTrenerry1982} E. S. Barnes and D. W. 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