= Four-tile mixed-boundary transplantation between a disk and a nonorientable surface
{title2=$\chi(D)=1,\quad\chi(B)=-1,\quad CM_s=N_sC$}
Take four congruent octagonal <regular polygons> with four alternate sides labelled $a,b,c,d$. The intervening sides have <Neumann boundary conditions>. Pair equal face parameters in the positive reference boundary direction. The complete gluing data are
$$
\begin{array}{c|c|c|c|c}
\text{face}&\text{pairs in }D&\text{Dirichlet tiles in }D&\text{pairs in }B&\text{Dirichlet tiles in }B\\ \hline
a&23&\varnothing&13&\varnothing\\
b&12&\varnothing&12&\varnothing\\
c&01&\{2,3\}&02&\{1,3\}\\
d&\varnothing&\{1,3\}&01,\ 23&\varnothing
\end{array}
$$
All unpaired labelled faces not in a Dirichlet column also have <Neumann boundary conditions>. Encode these rules by <signed permutation matrices>, using a positive swap for a paired interface and diagonal $-1$ or $+1$ for an unpaired <Dirichlet boundary condition> or <Neumann boundary condition>. The four-by-four <Hadamard matrix> $C$ above satisfies $CM_s=N_sC$ for every label. Explicitly, if its columns are $h_0,h_1,h_2,h_3$, applying $N_a,N_b,N_c,N_d$ gives respectively $(h_0,h_1,h_3,h_2)$, $(h_0,h_2,h_1,h_3)$, $(h_1,h_0,-h_2,-h_3)$, and $(h_0,-h_1,h_2,-h_3)$, which are exactly the columns of the corresponding $CM_s$. On every intervening face the face matrix is the identity.
Since $C^TC=4I$, the <orthogonal matrix> $C/2$ preserves the $L^2$ <inner product> and the sum of Dirichlet energies. Its intertwining relations map matching <Sobolev traces> and zero Dirichlet traces bijectively to those of the other assembly. It therefore preserves the <quadratic form> domains and the forms themselves. The associated <self-adjoint operators> are unitarily equivalent, proving equality of all <eigenvalues> with their <multiplicities> for the specified <mixed boundary conditions>.
The <topology of disk tiles glued along disjoint boundary arcs> shows that $D$, whose gluing <graph> is $0,1,2,3$ in a path, is a <closed disc> and is <simply connected> and <orientable>. The gluing <graph> of $B$ has edges $13,12,02,01,23$; it is connected with two independent cycles, so $\chi(B)=-1$ and $\pi_1(B)\cong F_2$. Its odd cycle $1,2,3,1$ makes it a <nonorientable surface>. Each assembled surface has nonempty polygonal boundary. This explicitly realizes isospectrality between a simply-connected orientable bordered surface and a nonorientable, non-simply-connected one. The boundary corners and <mixed boundary conditions> are essential qualifications; it does not contradict the <Euler characteristic from the heat trace of a bordered surface> for smooth boundaries and pure boundary conditions.
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