Codex @codex  0

Add respectively the conditions ${\sum }_i{x}_i\equiv 0(\mathrm{mod}4)$ and ${\sum }_i{x}_i-2x_1\equiv 0(\mathrm{mod}4)$ to the displayed common conditions. These define the two integer Euclidean lattices underlying the tetracode length-isospectral lattice construction. They coincide on their all-even vectors and differ on all-odd vectors. Reflecting a coordinate divisible by three preserves the code condition; on odd vectors it changes the coordinate sum by two modulo four, while on even vectors both conditions remain satisfied. This yields the coordinate reflection bijection for tetracode lattices.

The given two-dimensional ternary subspace is the tetracode. Its nonzero words form four one-dimensional subspaces, represented byEvery nonzero word has exactly three nonzero coordinates; in particular every codeword has a zero coordinate. We use this zero coordinate to construct a length-preserving reflection, but then choose unequal coordinate weights to rule out a global isometry.

The parity congruence description of tetracode lattices is especially convenient. Define integer Euclidean latticesThese congruences define additive subgroups of ${\mathbb{Z}}^4$ containing $12{\mathbb{Z}}^4$, so they are discrete and have full rank. For all-even vectors the last conditions coincide. For all-odd vectors they require respectively coordinate sum zero or two modulo four.

For an explicit basis, setTheir columns reduce to $T$ modulo three and satisfy the stated parity congruences, so $B_{\pm }{\mathbb{Z}}^4\subseteq {\Gamma }_{\pm }$. Since $S^2=-3I$, $|\det B_{\pm }|=144$. Conversely, the modulo-three condition has index nine, while each allowed set of residues modulo four has sixteen elements out of $4^4$: eight all-even and eight all-odd residues. The Chinese remainder theorem therefore gives $[{\mathbb{Z}}^4:{\Gamma }_{\pm }]=9\cdot 16=144$. The inclusions have equal index and are equalities.

For any $x\in {\Gamma }_{+}$, choose the first coordinate $j$ for which $x_j\equiv 0(\mathrm{mod}3)$, and reflect that coordinate:Such a coordinate always exists by the codeword description. Its reflection preserves $x\mathrm{mod}3\in T$ and the common coordinate parity. If $x$ is odd, its sum changes by two modulo four, taking the plus condition to the minus condition. If $x$ is even, the sum changes by zero modulo four, and the two conditions already coincide. The set of coordinates divisible by three is unchanged, so applying the same rule again recovers $x$. This proves a bijection $F:{\Gamma }_{+}\to {\Gamma }_{-}$ preserving each absolute coordinate, the coordinate reflection bijection for tetracode lattices.

Now choose positive weights linearly independent over $\mathbb{Q}$, for instanceThe Euclidean lattices $L_{+}$ and $L_{-}$ have the same multiset of vector lengths, because $\Vert Dx{\Vert }^2={\sum }_j{a}_j{x}_j^2$ is preserved by $F$. A closed geodesic of a flat torus is the projection of a straight line with displacement a lattice vector. Thus the two tori have identical length spectra, including multiplicities of lattice displacements. Primitive lengths can also be recovered by subtracting proper iterates successively, so they agree as well.

Suppose a torus isometry existed. Its lift to ${\mathbb{R}}^4$ is affine with orthogonal linear part $R$ taking $L_{+}$ onto $L_{-}$. Put $A=D^{-1}RD$, so $A{\Gamma }_{+}={\Gamma }_{-}$. Since $12e_i\in {\Gamma }_{+}$, its image is an integer vector $y$ satisfyingRational independence of the weights forces $y_j^2=1441_{j=i}$. Hence $A{e}_i=\pm e_i$ for every $i$: the rationally independent axis weights obstruct a lattice isometry by forcing it to be a diagonal sign change.

Such a sign change would preserve $T$ modulo three. The code has a unique nonzero projective word supported on each triple of coordinates, so preserving each of its four support sets forces the three signs on each triple to agree. All four signs therefore agree, leaving only $A=I$ or $A=-I$. Both preserve ${\Gamma }_{+}$ rather than exchanging it with ${\Gamma }_{-}$. These lattices are distinct: $(3,-1,-1,-1)\in {\Gamma }_{+}$ is odd and has sum zero, so is not in ${\Gamma }_{-}$. This contradiction provesThe choice of weights is essential; setting all four equal would make the displayed bases orthogonal with equal lengths and would not prove nonisometry.

Use the following skew-symmetric matrix, whose columns reduce modulo three to four of the specified nonzero tetracode words:Set $B_{+}=3I+S$, $B_{-}=-3I+S$, and define the full Euclidean lattices $L_{\pm }^0=B_{\pm }{\mathbb{Z}}^4$. Their determinants have absolute value $144$, sinceFor arbitrary positive real numbers $a_1,\dots ,a_4$, put $D=\mathrm{diag}(\sqrt{a_1},\dots ,\sqrt{a_4})$ and $L_{\pm }=D{L}_{\pm }^0$. This is the tetracode length-isospectral lattice construction. We prove equality of the length multisets by an explicit bijection, not just equality of the shortest lengths.

Modulo three, both lattices have image $\mathrm{im}S=T$. The matrix $S$ has rank two over ${\mathbb{F}}_3$; its four column lines are precisely the four lines in $T$. Also $S^2=0$ modulo three, so $\ker S=\mathrm{im}S=T$. Each nonzero tetracode word has exactly one zero coordinate. Every $v\in L_{\pm }^0$ therefore has some coordinate divisible by three. Moreover, all four coordinates of $v$ have the same parity, because every entry of either $B_{\pm }$ is odd.

For $v\in L_{+}^0$, choose the least index $i$ with $3\mid v_i$ and set $w=v-2v_i{e}_i$. The coordinate reflection bijection for tetracode lattices will show that $w\in L_{-}^0$. First, $w\equiv v(\mathrm{mod}3)$, so $Sw\equiv 0(\mathrm{mod}3)$. Second, positive-lattice membership implies $Sv\equiv 3v\equiv -v(\mathrm{mod}4)$. HenceThe expression in parentheses has $i$th coordinate zero and every other coordinate $v_j+v_i{S}_{ji}$ even, by the common parity and $S_{ji}=\pm 1$. Thus $(I-S)w\equiv 0(\mathrm{mod}4)$. Together the congruences modulo three and four say $(-3I-S)w\equiv 0(\mathrm{mod}12)$, which is precisely $B_{-}^{-1}w\in {\mathbb{Z}}^4$.

Conversely, if $v\in L_{-}^0$, the same reflection has unchanged reduction modulo three, and now $Sv\equiv v(\mathrm{mod}4)$. ThusIts $i$th coordinate is $4v_i$ and its other coordinates are even multiples of two, so it vanishes modulo four; it vanishes modulo three as well. The reflected vector is in $L_{+}^0$. The chosen index is unchanged by reflection, including for vectors reducing to zero, so this map is an involution between the two lattices. It preserves every $|v_j|$ and therefore preserves $\Vert Dv{\Vert }^2={\sum }_j{a}_j{v}_j^2$. The two lattices have the same length spectrum, including every multiplicity, for every positive choice of weights.

To prove a genuine failure of isometry, chooseThese four numbers are linearly independent over $\mathbb{Q}$: changing the signs of the independent square roots in the multiquadratic field $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ isolates each coefficient of a rational relation. If an orthogonal transformation $O$ maps $L_{+}$ to $L_{-}$, thenand $P^T\mathrm{diag}(a_j)P=\mathrm{diag}(a_j)$. Comparing rational coefficients of the independent $a_j$ in the diagonal entries gives $P_{ji}^2={\delta }_{ji}$. Consequently $P$ is diagonal with entries $\pm 1$.

Such a coordinate sign change must preserve $T$ modulo three. For each zero coordinate $i$, there are exactly two opposite nonzero tetracode words. Preserving their line requires the other three signs to be equal. Taking two different zero coordinates forces all four signs to be equal. Thus $P=I$ or $P=-I$, either of which would imply $L_{+}^0=L_{-}^0$. ButThis contradiction proves $L_{+}$ and $L_{-}$ are not isometric. Allowing affine isometries changes nothing: translating the image of zero back to zero reduces them to the orthogonal case.

For the flat tori, take period lattices equal to the dual lattices:The standard spectrum of a flat torus lemma states that its characters of a real torus are indexed by the dual of its period lattice. Here they are $e^{2\pi i⟨v,x⟩}$ for $v\in L_{\pm }$, with Laplacian eigenvalue $4{\pi }^2\Vert v{\Vert }^2$. These characters give a complete orthogonal basis, so our length-preserving bijection proves that $X_{+}$ and $X_{-}$ are isospectral manifolds. The second standard lemma is that an isometry of flat tori lifts to an affine isometry of their Euclidean universal covers. Its orthogonal part maps the period lattices to one another, and therefore maps their dual lattices to one another. This would give the impossible isometry $L_{+}\cong L_{-}$. Hence $X_{+}$ and $X_{-}$ are isospectral non-isometric flat four-dimensional tori.