\section{Reconstructing the smaller six-distance cores}\label{sec:cores}
A \emph{distance pattern} on $r$ vertices is a symmetric matrix
$M\in\{0,1,\ldots,k\}^{r\times r}$ whose diagonal entries are zero and
whose off-diagonal entries are positive, with every color $1,\ldots,k$
used. A realization assigns pairwise distinct positive squared distances to the colors and
realizes the resulting matrix by distinct points of $\R^2$. Patterns are
identified under simultaneous permutations of rows and columns and under
arbitrary permutations of their positive colors. In particular, a color
label does not encode the numerical order of a distance.

The exhaustive generation in~\cite[Sections 3.1 and 4, Table 9]{SO}
leaves fifteen candidate patterns on twelve vertices and sixty-two on
eleven vertices, for \emph{at most} six colors. These are counts of
patterns passing necessary algebraic conditions, not counts of already
classified real configurations. Every planar realization must occur
among the candidates. Consequently, exhibiting that many inequivalent
realizable patterns reconstructs the entire candidate list. It remains
necessary to determine every metric realization of each pattern.

\subsection{The twelve-point patterns}
Use triangular coordinates
\[
 (a,b)\longmapsto \left(a+\frac b2,\frac{\sqrt3\,b}{2}\right),
 \qquad Q(a,b)=a^2+ab+b^2.
\]
Table~\ref{tab:lattice-cores} gives eleven twelve-point lattice sets
$L_0,\ldots,L_{10}$. In its row $i$, the entry in column $B_a$ means
that the points $(a,b)$ with $b\in B_a$ are included. A dash means that
$B_a$ is empty. The squared-distance palettes are
\[
 S_9=\{1,3,4,7,9\},\qquad S_{12}=S_9\cup\{12\},\qquad
 S_{13}=S_9\cup\{13\}.
\]
\begin{table}[ht]
\centering
\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{c c c c c c c}
\toprule
Set & $B_0$ & $B_1$ & $B_2$ & $B_3$ & $B_4$ & Palette\\
\midrule
$L_0$ & $0,1,2,3$ & $-1,0,1,2$ & $0,1$ & $-1,0$ & -- & $S_{13}$\\
$L_1$ & $0,1,2,3$ & $0,1,2$ & $-1,0,1,2$ & $0$ & -- & $S_{12}$\\
$L_2$ & $0,1,2,3$ & $0,1,2$ & $0,1$ & $-1,0,1$ & -- & $S_{13}$\\
$L_3$ & $0,1,2$ & $-2,-1,0,1$ & $-1,0,1$ & $-2,-1$ & -- & $S_{13}$\\
$L_4$ & $0,1,2$ & $-1,0,1,2$ & $-1,0,1$ & $-1,0$ & -- & $S_9$\\
$L_5$ & $0,1,2$ & $-1,0,1$ & $-2,-1,0,1$ & $-1,0$ & -- & $S_{12}$\\
$L_6$ & $0,1,2$ & $-1,0,1$ & $-1,0,1$ & $-2,-1,0$ & -- & $S_{13}$\\
$L_7$ & $0,2$ & $-2,-1,0,1$ & $-1,0,1$ & $-2,-1$ & $-1$ & $S_{13}$\\
$L_8$ & $0,2$ & $0,1$ & $-1,0,1$ & $-2,-1,0,1$ & $-1$ & $S_{13}$\\
$L_9$ & $0$ & $-2,-1,0,1$ & $-2,-1,0$ & $-3,-2,0$ & $-2$ & $S_{12}$\\
$L_{10}$ & $0$ & $-2,-1,0,1$ & $-2,0$ & $-3,-2,-1,0$ & $-2$ & $S_{12}$\\
\bottomrule
\end{tabular}
\caption{Eleven lattice realizations of twelve-point patterns. Entries
in each $B_a$ column list the permitted second coordinates.}
\label{tab:lattice-cores}
\end{table}

The other four patterns have the displayed realizations
\[
 R_{13}\setminus\{v\},\qquad R_{12},\qquad
 (R_{12}\setminus\{v\})\cup\{O\},\qquad R_{11}\cup\{O\},
\]
where $R_n$ is a regular $n$-gon and $O$ its center.
The pattern of each displayed set is obtained by assigning one color to
each distinct squared distance.

Here is an elementary certificate that the fifteen patterns are
inequivalent. For a color $c$, let $e_c$ be the number of edges of that
color. For each vertex $v$, let $d_c(v)$ be its number of incident edges
of color $c$, including zero values. Form the invariant
\[
 \mathcal I(M)=\left(
   \{\!\{e_c:c\}\!\},\quad
   \{\!\{\{\!\{(e_c,d_c(v)):c\}\!\}:v\}\!\}
 \right),
\]
where braces with doubled inner delimiters denote multisets.
This is unchanged by permutations of vertices or colors. Direct integer
calculation gives fifteen different invariants for the fifteen displayed
patterns. Together with Table 9 of~\cite{SO}, this exhausts the
candidate patterns on twelve vertices. Distinct invariant values prove
inequivalence; no converse assertion about this invariant is needed.

\subsection{The eleven-point patterns}
Order each $L_i$ lexicographically by $(a,b)$, with indices
$0,1,\ldots,11$; write $v_j$ for the point of index $j$. The machine-readable
name \texttt{Li minus j} means $L_i\setminus\{v_j\}$. Deleting the indices listed in
Table~\ref{tab:eleven-deletions}, one at a time, gives fifty-two lattice
patterns with distinct invariants $\mathcal I$. The remaining ten
circular patterns are given in Table~\ref{tab:circular-eleven}.
Throughout that table, identify the vertices of $R_n$ with
$1,\zeta_n,\ldots,\zeta_n^{n-1}$, where
$\zeta_n=\exp(2\pi i/n)$; a deleted index $a$ means the vertex
$\zeta_n^a$.

\begin{table}[ht]
\centering
\begin{tabular}{cl r}
\toprule
Source & Indices deleted, one at a time & Count\\
\midrule
$L_0$ & $0,1,2,3,4,5,6$ & 7\\
$L_1$ & $0,1,4,5,7,8,11$ & 7\\
$L_2$ & $0,1,4,5,7,10$ & 6\\
$L_3$ & $0,1,2,3,4,5,6,7,8,10,11$ & 11\\
$L_4$ & $0,4$ & 2\\
$L_5$ & $0,1,4,5,8,9$ & 6\\
$L_6$ & $4,5$ & 2\\
$L_7$ & $0,3,4$ & 3\\
$L_8$ & $2,4,5,8$ & 4\\
$L_9$ & $2,3,6,7$ & 4\\
\midrule
Total &&52\\
\bottomrule
\end{tabular}
\caption{Representatives of the fifty-two eleven-point lattice patterns.}
\label{tab:eleven-deletions}
\end{table}

\begin{table}[ht]
\centering
\begin{tabular}{c l c}
\toprule
Pattern & Displayed realization & Cyclotomic multipliers $j$\\
\midrule
$C_0$ & $R_{13}\setminus\{\zeta_{13}^{0},\zeta_{13}^{12}\}$ & $1,2,3,4,5,6$\\
$C_1$ & $R_{12}\setminus\{\zeta_{12}^{0}\}$ & $1,5$\\
$C_2$ & $(R_{12}\setminus\{\zeta_{12}^{0},\zeta_{12}^{11}\})\cup\{O\}$ & $1,5$\\
$C_3$ & $(R_{12}\setminus\{\zeta_{12}^{1},\zeta_{12}^{11}\})\cup\{O\}$ & $1,5$\\
$C_4$ & $(R_{12}\setminus\{\zeta_{12}^{2},\zeta_{12}^{11}\})\cup\{O\}$ & $1,5$\\
$C_5$ & $(R_{12}\setminus\{\zeta_{12}^{3},\zeta_{12}^{11}\})\cup\{O\}$ & $1,5$\\
$C_6$ & $(R_{12}\setminus\{\zeta_{12}^{5},\zeta_{12}^{11}\})\cup\{O\}$ & $1,5$\\
$C_7$ & $(R_{11}\setminus\{\zeta_{11}^{0}\})\cup\{O\}$ & $1,2,3,4,5$\\
$C_8$ & $R_{11}$ & $1,2,3,4,5$\\
$C_9$ & $R_{10}\cup\{O\}$ & $1,3$\\
\bottomrule
\end{tabular}
\caption{Ten circular eleven-point patterns. Multiplying every retained
vertex index by $j$ modulo $n$ gives the cyclotomic realizations to be
checked in the metric calculation. Some of these realizations have
different shapes despite having equivalent distance patterns.}
\label{tab:circular-eleven}
\end{table}

All sixty-two invariants are distinct. The lattice realizations are
checked using the integer form $Q$; the circular realizations follow from
the chord formulas. Hence the sixty-two patterns exhaust the published
candidate count at order eleven. Equivalently, one may generate all
one-vertex deletions from the fifteen twelve-point representatives,
retain one representative for each of the sixty-one distinct invariants
that occur, and adjoin $R_{10}\cup\{O\}$. This gives the same list.

\subsection{Exact metric realization checks}\label{subsec:core-metrics}
The pattern count alone does not classify actual point sets. To determine
metrics, take one of the reconstructed $r$-vertex patterns $M$ with $k$
colors. Normalize its first squared distance to $d_1=1$, set
$d_0=0$, and write $d_i=x_i$ for $2\le i\le k$. For indices
$1\le i,j\le r-1$, define
\[
 G_{ij}=d_{M_{0i}}+d_{M_{0j}}-d_{M_{ij}}.
\]
For a planar realization this is twice the Gram matrix of the vectors
based at vertex zero. Thus each principal three-by-three minor vanishes.
Only necessity is used: we do not assume that vanishing principal minors
suffices for arbitrary complex symmetric matrices to have rank two.

Let $I_M$ be the ideal over $\mathbb Q$ generated by these minors. Introduce
a variable $u$ and compute a lexicographic Gr\"obner basis of
\[
 I_M+\left\langle u\prod_{i=2}^{k}x_i-1\right\rangle
 \subset\mathbb Q[u,x_2,\ldots,x_k],
 \qquad u>x_2>\cdots>x_k.
\]
Every genuine realization satisfies this system, since none of its
pairwise distances is zero. Eliminating $u$ therefore gives necessary
polynomial equations for every genuine metric. No floating-point
calculation or approximate equality test is involved.

For a lattice pattern, the desired certificate is that the eliminated
basis consists precisely of the linear polynomials fixing all normalized
squared distances to those of its displayed coordinates. For a circular
pattern, the certificate is triangular: each $x_i$ except the last is a
polynomial in a final parameter $t$, with a nonzero constant coefficient
on $x_i$, followed by one univariate polynomial in $t$.

The circular solutions can be exhausted without numerical root finding.
For each multiplier $j$ in Table~\ref{tab:circular-eleven}, evaluate
\[
 d_a(j)=\frac{2-\zeta_n^{aj}-\zeta_n^{-aj}}
                  {2-\zeta_n^j-\zeta_n^{-j}},\qquad
 d_O(j)=\frac{1}{2-\zeta_n^j-\zeta_n^{-j}}
\]
for chord colors and the center color, respectively. All arithmetic is
in $\mathbb Q[z]/(\Phi_n(z))$. Check every eliminated equation, and check
that the resulting values of $t$ are pairwise distinct and that their
number equals the degree of its univariate equation. These checks prove
that no other complex solution for $t$ exists. Triangularity then fixes
all other squared distances. Since a complete squared-distance matrix
determines a Euclidean configuration up to isometry, this exhausts
all realizations up to similarity.

It is essential to distinguish pattern equivalence from similarity.
For example, the six multipliers for $C_0$ give the six possible separations
of two vertices deleted from $R_{13}$. They share a distance pattern
although their point sets need not be similar. For the extension argument,
all are covered by allowing the indicated regular-polygon deletion family.

\begin{lemma}[Computer-assisted classification]
\label{lem:eleven-classification}
Every eleven-point planar set with at most
six distances is similar either to one of the fifty-two lattice sets in
Table~\ref{tab:eleven-deletions}, or to one of the circular realizations
specified by Table~\ref{tab:circular-eleven} and its multipliers.
\end{lemma}
\begin{proof}
Every such set has a pattern in the published candidate list. The
sixty-two reconstructed inequivalent realizable patterns exhaust that
list. The exact metric checks described above exhaust the realizations of each
pattern. For the fifty-two lattice patterns, the eliminated equations are
linear and fix the displayed palettes. For the ten circular patterns, the
triangular equations have final degrees $6,2,2,2,2,2,2,5,5,2$ in the order
of Table~\ref{tab:circular-eleven}. The listed multipliers give that many
distinct solutions in every case. The verification computes the bases
over $\mathbb Q$ from the principal minors and verifies the cyclotomic
substitutions exactly.
\end{proof}