June 09, 2026
We prove a conjecture stated in [4], which asserts that no \([10,10,16]_\mathbb{Z}\) formula can arise as a restriction of any Hurwitz\(-\)Radon formula. Consequently, the unique \([10,10,16]_\mathbb{Z}\) formula provides the first known example of a composition formula that cannot be obtained from classical Hurwitz–Radon formulas by a process of restrictions and direct sums.
Given integers \(r, s\) and \(n\), a composition or a formula of size \([r,s,n]_\mathbb{R}\) is a square identity of type \[(x_1^2 + \cdots + x_r^2)(y_1^2 +\cdots + y_s^2) = z_1^2 + \cdots + z_n^2\] where \(x=(x_1, ..., x_r)\) and \(y=(y_1, ..., y_s)\) are systems of indeterminates and each \(z_k = z_k(x,y)\) is a bilinear form in \(x\) and \(y\) with coefficients in \(\mathbb{R}\). Given \(r, s\), \(r*s\) is the smallest number of n such that there exists an \([r,s,n]_\mathbb{R}\) formula. Such a formula is equivalent to a normed bilinear map \(f: R^r \times R^s \rightarrow R^n\) satisfying \[|f(x,y)|=|x| |y|, x \in \mathbb{R}^r, y \in \mathbb{R}^s\]
Two \([r,s,n]_\mathbb{R}\) formulas are said to be equivalent if their associated normed bilinear maps \(f,g\) differ by orthogonal changes of coordinates, namely the following diagram is commutative:
\[\xymatrix{ \mathbb{R}^r \times \mathbb{R}^s \ar[r]^{g} \ar[d]_{\alpha \times \beta} & \mathbb{R}^n \ar[d]^{\gamma } \\ \mathbb{R}^r \times \mathbb{R}^s \ar[r]^{f} & \mathbb{R}^n }\] where \(\alpha, \beta\) and \(\gamma\) are isometries.
Replace \(\mathbb{R}\) with \(\mathbb{Z}\) (integers), then the \([r,s,n]_\mathbb{Z}\) formula is over integers and each \(z_k(x,y)\) is a bilinear form in x and y with coefficients in \(\mathbb{Z}\). Given \(r,s\), \(r *_\mathbb{Z}s\) is the smallest number of n such that there exists an \([r,s,n]_{\mathbb{Z}}\) formula. Since an \([r,s,n]_\mathbb{Z}\) formula is also an \([r,s,n]_\mathbb{R}\) formula, we have \(r *_\mathbb{Z}s \ge r *_\mathbb{R}s\).
A matrix \(M\) of size \(r \times s\) is an intercalate matrix if:
All entries are nonnegative integers (called colors).
The colors along each row (resp. column) are distinct.
If \(M(i, j ) = M(i',j')\) then \(M(i, j') = M(i',j )\).(intercalacy)
An intercalate matrix M is consistently signed if there exist \(\epsilon_{ij} = \pm 1\) such that \(\epsilon_{ij} \epsilon_{ij'} \epsilon_{i'j} \epsilon_{i'j'} = -1\) whenever \(M(i, j ) = M(i',j')\) and \(i \ne i'\) and \(j \ne j'\).
It is well known that there exists an \([r,s,n]_\mathbb{Z}\) formula if and only if there is a consistently signed \(r\times s\) intercalate matrix with \(n\) colors.
Given a consistently signed intercalate matrix \(M\), the corresponding normed bilinear map \(f=(z_1,z_2,...,z_n)\) can be defined as follows: each \(z_k\) is determined by color \(k\), if color \(k\) appears in position \((i,j)\) with a sign \(c=\pm 1\), then \(z_k\) has a term \(cx_iy_j\). For example, if \(M\) is the following matrix: \[M = \left[ \begin{array}{rrrrrrrrrr} 1 & 2 & 3 & 4 \\ 2 & -1 & 4 & -3 \\ 3 & -4 & -1 & 2 \\ 4 & 3 & -2 & -1 \end{array} \right]\]
then the normed bilinear map is \[f(x, y) = \left( \begin{align} z_1 &= x_1y_1 - x_2y_2 - x_3y_3 - x_4y_4 \\ z_2 &= x_1y_2 + x_2y_1 + x_3y_4 - x_4y_3 \\ z_3 &= x_1y_3 - x_2y_4 + x_3y_1 + x_4y_2 \\ z_4 &= x_1y_4 + x_2y_3 - x_3y_2 + x_4y_1 \end{align} \right)\]
Lemma 1. If \(f=(z_1,z_2,...,z_n)\) is the normed bilinear map of an \([r, s, n]_\mathbb{Z}\) formula, then every term \(x_iy_j\) of the map \(f\) has coefficient \(\pm 1\) and only appears in one \(z_k\).
Pick two vectors \(x=(0,…,a_i,0,…,0)\in \mathbb{R}^r\), \(y=(0,…,b_j,0,…,0)\in \mathbb{R}^s\), where \(a_i=1\) at the ith position and \(b_j=1\) at the jth position. Then \(1=|x|^2|y|^2=|f(x,y)|^2=z_1(x,y)^2+...+z_n(x,y)^2\), since \(f\) has integer coefficients, if \(x_iy_j\) appears in \(z_k\), then \(z_k(x,y)^2= c^2\ge 1\), where \(c\) is the coefficient of \(x_iy_j\), so \(c=\pm 1\) and all other \(z_k(x,y)^2=0\), if follows that \(x_iy_j\) has coefficient \(\pm 1\) and only appears in one \(z_k\).
\(\square\)
A classical result of Hurwitz and Radon states that an \([r, n, n]_\mathbb{R}\) formula exists if and only if \(r \leq \rho(n)\), where \(\rho(n)\) is the Hurwitz\(-\)Radon function defined as follows: if \(n = 2^{4a+b}n_0\) where \(n_0\) is odd and \(0 \leq b \leq 3\), then \(\rho(n) = 8a + 2b\). A \([\rho(n), n, n]_\mathbb{R}\) formula is called a Hurwitz\(-\)Radon formula. From [2] and [4], we know that for every \([r, s, n]_\mathbb{R}\) with \(n - r \leq 5\), there is a formula built from the classical Hurwitz\(-\)Radon formulas by a process of restrictions and direct sums. A natural question is whether the statement remains true for \(n-r>5\), it is difficult to answer. Even for the smallest case \([10,10,16]\), we know there exists a \([10,10,16]_\mathbb{Z}\) formula that is unique and not a direct sum of any other formulas, however, we do not know if it is a restriction of some Hurwitz\(-\)Radon formulas, Shapiro conjectured in [4] that no \([10,10,16]_\mathbb{Z}\) formula can be a restriction of a Hurwitz\(-\)Radon formula in 2000. In this paper, we are going to prove this conjecture. Hence the \([10,10,16]_\mathbb{Z}\) formula is the first example that cannot be built from the classical Hurwitz\(-\)Radon formulas by a process of restrictions and direct sums.
Let \(F\) be the normed bilinear map of a Hurwitz\(-\)Radon formula \([h,m,m]_\mathbb{R}\), a restriction of the Hurwitz\(-\)Radon formula is an \([r,s,n]_\mathbb{R}\) formula with \(r \leq h, s \leq m, n \leq m\) such that there exist subspaces \(X\subseteq\mathbb{R}^h, Y\subseteq\mathbb{R}^m\), and \(Z\subseteq\mathbb{R}^m\) of dimensions \(r,s\), and \(n\), respectively, where \(Z\) contains the image of \(X \times Y\) under F, and the normed bilinear map of the \([r,s,n]_\mathbb{R}\) formula is the restriction of \(F\) on \(X \times Y\), namely, the normed bilinear map of the \([r,s,n]_\mathbb{R}\) formula \(f: X \times Y \rightarrow Z\) is defined as \(f(x,y)=F(x,y)\).
If a restriction \([r,s,n]_\mathbb{R}\) formula is equivalent to an integral formula \([r,s,n]_\mathbb{Z}\) with bilinear map \(g\), then there are isometries \(\alpha :\mathbb{R}^r \rightarrow X, \beta :\mathbb{R}^s \rightarrow Y, \gamma : \mathbb{R}^n \rightarrow Z\) such that the following diagram is commutative: \[\xymatrix{ \mathbb{R}^r \times \mathbb{R}^s \ar[r]^{g} \ar[d]_{\alpha \times \beta} & \mathbb{R}^n \ar[d]^{\gamma } \\ X \times Y \ar[r]^{f} & Z }\]
Images of standard orthonormal bases of \(\mathbb{R}^r\), \(\mathbb{R}^s\) and \(\mathbb{R}^n\) are orthonormal bases of \(X\),\(Y\) and \(Z\). Clearly, these bases can be extended to orthonormal bases for \(\mathbb{R}^h\), \(\mathbb{R}^m\) and \(\mathbb{R}^m\) so that the first r-dimensional subspace of \(\mathbb{R}^h\) is \(X\), the first s-dimensional subspace of \(\mathbb{R}^m\) is \(Y\) and the first n-dimensional subspace of \(\mathbb{R}^m\) is \(Z\). The restriction \(F\) on \(X \times Y\) is the normed bilinear map \(g\).
We use \(x=(x_1,…,x_r,…,x_h)\), \(y=(y_1,…,y_s, …,y_m)\) and \(z=(z_1,…,z_n,…,z_m)\) as vectors of \(\mathbb{R}^h\), \(\mathbb{R}^m\) and \(\mathbb{R}^m\), where \(x=(x_1,…,x_r)\), \(y=(y_1,…,y_s)\) and \(z=(z_1,…,z_n)\) as vectors of \(X\), \(Y\) and \(Z\) and every \(z_k=z_k(x,y)\) is a bilinear form and the normed bilinear map \(f\) can be expressed as \(f(x,y)=g(x,y)=(z_1,...,z_n)\) by removing all the terms of \(cx_iy_j\) with \(i>r\) or \(j>s\).
We say \(x_iy_j\) appears in \(z_k\) if \(z_k\) has a term \(cx_iy_j\) with \(c\ne 0\).
Lemma 2. Bilinear forms \(z_1\),…, \(z_n\) have the following properties:
(1) If \(x_iy_j\) with \(i \leq r\), \(j \leq s\) appears in some \(z_k\), then \(x_iy_j\) does not appear in any other \(z_{k'}\).
(2) If \(x_iy_j\) with \(i\leq r, j\leq s\) and \(x_{i’}y_{j’}\) with \(i’\leq r\) or \(j’\leq s\) appear in \(z_k\), then \(x_iy_{j’}\) and \(x_{i’}y_j\) must appear in a unique \(z_{k'}\). Further more, if coefficients of \(x_iy_j, x_{i’}y_{j’}, x_iy_{j’}, x_{i’}y_j\) are \(c_{ij},c_{i'j'},c_{ij'},c_{i'j}\) in \(z_k\) and \(z_{k'}\), then \(c_{ij}c_{i'j'}=-c_{ij'}c_{i'j}\).
(3) Every \(z_k\) has a term \(x_iy_j\) for every \(x_i\).
(1) Since \(i \leq r,j \leq s, x_iy_j\), by Lemma 1, it only appears in one \(z_k\) with \(k\leq n\) and has coefficient \(\pm 1\). Let \(a_i=b_j=1\), \(x=(0,…,a_i,0,…,0)\in \mathbb{R}^h\), \(y=(0,…,b_j,0,…,0)\in \mathbb{R}^m\), then \(1=|x|^2|y|^2=|F(x,y)|^2=z_1(x,y)^2+… +z_m(x,y)^2\), since \(z_k(x,y)^2=1\), no other \(z_{k'}(x,y)^2>0\), namely, \(x_iy_j\) does not appear in any other \(z_k\).
(2) \(z_k^2\) contains a term \(2c_{ij}c_{i'j'}x_iy_jx_{i’}y_{j’}\), by the formula identity, this term is cancelled from terms of other \(z_{k'}^2\). By (1), \(x_iy_j\) appears only in \(z_k\), hence, there must exist a \(z_{k'}\) that contains both \(x_iy_{j’}\) and \(x_{i’}y_j\). If \(i’\leq r\), since \(j\leq s\), \(x_{i’}y_j\) appears only in one \(z_{k'}\) by (1), no other \(z_{k''}\) contains these two terms. To cancel the term \(2c_{ij}c_{i'j'}x_iy_jx_{i’}y_{j’}\), coefficients of these terms must satisfy \(c_{ij}c_{i'j'} = -c_{ij'}c_{i'j}\). If \(j'<s\), we can prove the same result with a similar argument.
(3) Let \(x =(0,…,x_i,0,…,0)\) with \(x_i=1\) be a basis vector in \(\mathbb{R}^h\), since F is a normed bilinear map, \(F\) defines a map \(F_x: \mathbb{R}^m \rightarrow \mathbb{R}^m\): \(y \rightarrow F(x,y)\) that is isometric, it follows that the image of \(F_x\) has dimension \(m\). Hence there must be some vector \(y\), \(F_x(y)\) is not zero in \(z_k\). this means there exists some \(y_j\) such that \(x_iy_j\) appears in \(z_k\).
\(\square\)
For \([10,10,16]_\mathbb{Z}\), Yiu proved in [6] that every \([10,10,16]_\mathbb{Z}\) is equivalent to the formula with the following consistently signed intercalate matrix:
\[\left[ \begin{array}{rrrrrrrrrr} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 2 & -1 & 4 & -3 & 6 & -5 & -8 & 7 & 10 & -9 \\ 3 & -4 & -1 & 2 & 7 & 8 & -5 & -6 & 11 & 12 \\ 4 & 3 & -2 & -1 & 8 & -7 & 6 & -5 & 12 & -11 \\ 5 & -6 & -7 & -8 & -1 & 2 & 3 & 4 & 13 & 14 \\ 6 & 5 & -8 & 7 & -2 & -1 & -4 & 3 & 14 & -13 \\ 7 & 8 & 5 & -6 & -3 & 4 & -1 & -2 & 15 & -16 \\ 8 & -7 & 6 & 5 & -4 & -3 & 2 & -1 & 16 & 15 \\ 9 & -10& -11& -12& -13& -14& -15& -16& -1 & 2 \\ 10& 9 & -12& 11& -14& 13& 16& -15& -2 & -1 \end{array} \right]\]
Let \(x=(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10})\), \(y=(y_1,y_2,y_3,y_4,y_5,y_6,y_7,y_8,y_9,y_{10})\) and \(z=(z_1,z_2,z_3 ,z_4,z_5,z_6,z_7,z_8,z_9,z_{10},z_{11},z_{12},z_{13},z_{14},z_{15},z_{16})\) be points in \(R^{10}\), \(R^{10}\) and \(R^{16}\). Then the corresponding bilinear map \(f: \mathbb{R}^{10} \times \mathbb{R}^{10} \rightarrow \mathbb{R}^{16}\) defined as
\[f(x, y) = \left( \begin{align} z_1 &= x_1y_1 - x_2y_2 - x_3y_3 - x_4y_4 - x_5y_5 - x_6y_6 - x_7y_7 - x_8y_8 - x_9y_9 - x_{10}y_{10} \\ z_2 &= x_1y_2 + x_2y_1 + x_3y_4 - x_4y_3 + x_5y_6 - x_6y_5 - x_7y_8 + x_8y_7 + x_9y_{10} - x_{10}y_9 \\ z_3 &= x_1y_3 - x_2y_4 + x_3y_1 + x_4y_2 + x_5y_7 + x_6y_8 - x_7y_5 - x_8y_6 \\ z_4 &= x_1y_4 + x_2y_3 - x_3y_2 + x_4y_1 + x_5y_8 - x_6y_7 + x_7y_6 - x_8y_5 \\ z_5 &= x_1y_5 - x_2y_6 - x_3y_7 - x_4y_8 + x_5y_1 + x_6y_2 + x_7y_3 + x_8y_4 \\ z_6 &= x_1y_6 + x_2y_5 - x_3y_8 + x_4y_7 - x_5y_2 + x_6y_1 - x_7y_4 + x_8y_3 \\ z_7 &= x_1y_7 + x_2y_8 + x_3y_5 - x_4y_6 - x_5y_3 + x_6y_4 + x_7y_1 - x_8y_2 \\ z_8 &= x_1y_8 - x_2y_7 + x_3y_6 + x_4y_5 - x_5y_4 - x_6y_3 + x_7y_2 + x_8y_1 \\ z_9 &= x_1y_9 - x_2y_{10} + x_9y_1 + x_{10}y_2 \\ z_{10} &= x_1y_{10} + x_2y_9 - x_9y_2 + x_{10}y_1 \\ z_{11} &= x_3y_9 - x_4y_{10} - x_9y_3 + x_{10}y_4 \\ z_{12} &= x_3y_{10} + x_4y_9 - x_9y_4 - x_{10}y_3 \\ z_{13} &= x_5y_9 - x_6y_{10} - x_9y_5 + x_{10}y_6 \\ z_{14} &= x_5y_{10} + x_6y_9 - x_9y_6 - x_{10}y_5 \\ z_{15} &= x_7y_9 + x_8y_{10} - x_9y_7 - x_{10}y_8 \\ z_{16} &= -x_7y_{10} + x_8y_9 - x_9y_8 + x_{10}y_7 \end{align} \right)\]
Theorem 1. The normed bilinear map of a \([10,10,16]_\mathbb{Z}\) formula is not a restriction of any normed bilinear map of size \([h,m,m]\).
Let \(F=(z_1,z_2,…,z_{16},z_{17},…,z_m)\) be normed bilinear map of size \([h,m,m]\), if the normed bilinear map of a \([10,10,16]_\mathbb{Z}\) formula is a restriction of \(F\), as in the previous section, we can assume \(F\) restricts to \(\mathbb{R}^{10} \times \mathbb{R}^{10}\rightarrow \mathbb{R}^{16}\) is the normed bilinear map defined above, then every bilinear map \(z_i\) of \([h,m,m]\) formula with \(i \leq 16\) contains terms listed in the above table with respect to \(z_i\).
By Lemma 2 (3), every bilinear map \(z_i\) of \([h,m,m]\) formula has a term starting with \(x_9\), hence, for example, \(z_3\) must have a term \(cx_9y_j\) with \(j>10\) because every \(x_9y_j\) with \(j \leq 10\) must already appear in the above table. Without loss of generality, we can assume \(y_j=y_{11}\), then \(z_3\) of the restriction of \(F\) on \(\mathbb{R}^{10} \times \mathbb{R}^{11}\), \(z_3\) has a term \(cx_9y_{11}\). Since \(x_1y_3\) is in \(z_3\) and does not appear in any other \(z_k\), so \(x_1y_{11}\) and \(x_9y_3\) appear in a unique \(z_k\) by Lemma 2 (2). From the above table, \(x_9y_3\) appear in \(z_{11}\) and does not appear in any other \(z_i\) by Lemma 2 (1), it follows that \(x_1y_{11}\) must appear in \(z_{11}\) as well and the coefficient is also \(c\). Similarly, since \(-x_2y_3\) is in \(z_3, -x_2y_{11}\) must appear in \(z_{12}\), similarly for other terms in \(z_3\) as well, we can show that \(z_9,z_{10},…,z_{16}\) have terms: \(-cx_3y_{11}, cx_4y_{11}, cx_1y_{11}, -cx_2y_{11}, -cx_7y_{11}, -cx_8y_{11}, cx_5y_{11}, cx_6y_{11}\) respectively:
| \(z_9 = x_1y_9 -x_2y_{10} +x_9y_1 +x_{10}y_2 -cx_3y_{11}\) | |
| \(z_{10} = x_1y_{10} +x_2y_9 -x_9y_2 +x_{10}y_1 +cx_4y_{11}\) | |
| \(z_{11} = x_3y_9 -x_4y_{10} -x_9y_3 +x_{10}y_4 +cx_1y_{11}\) | |
| \(z_{12} = x_3y_{10} +x_4y_9 -x_9y_4 -x_{10}y_3 -cx_2y_{11}\) | |
| \(z_{13} = x_5y_9 -x_6y_{10} -x_9y_5 +x_{10}y_6 -cx_7y_{11}\) | |
| \(z_{14} = x_5y_{10} +x_6y_9 -x_9y_6 -x_{10}y_5 -cx_8y_{11}\) | |
| \(z_{15} = x_7y_9 +x_8y_{10} -x_9y_7 -x_{10}y_8 +cx_5y_{11}\) | |
| \(z_{16} = -x_7y_{10} +x_8y_9 -x_9y_8 +x_{10}y_7 +cx_6y_{11}\) |
For every \(z_k\) with \(k\geq 9\), there is a new term \(x_iy_{11}\), by the above argument, some new terms should appear in other \(z_k\). For example, for \(z_{11}\), we have the following:
| \(z_{3} = x_1y_3 -x_2y_4 +x_3y_1 +x_4y_2 +x_5y_7 +x_6y_8 -x_7y_5 -x_8y_6 +cx_9y_{11}\) | |
| \(z_{4} = x_1y_4 +x_2y_3 -x_3y_2 +x_4y_1 +x_5y_8 -x_6y_7 +x_7y_6 -x_8y_5 -cx_{10}y_{11}\) | |
| \(z_9 = x_1y_9 -x_2y_{10} +x_9y_1 +x_{10}y_2 -cx_3y_{11}\) | |
| \(z_{10} = x_1y_{10} +x_2y_9 -x_9y_2 +x_{10}y_1 +cx_4y_{11}\) |
Now, \(z_{4}\) has a term \(x_{10}y_{11}\), applying the above argument for \(z_3\), we get the following:
| \(z_{9} = x_1y_9 -x_2y_{10} +x_9y_1 +x_{10}y_2 +cx_3y_{11}\) | |
| \(z_{10} = x_1y_{10} +x_2y_9 -x_9y_2 +x_{10}y_1 -cx_4y_{11}\) | |
| \(z_{11} = x_3y_9 -x_4y_{10} -x_9y_3 +x_{10}y_4 -cx_1y_{11}\) | |
| \(z_{12} = x_3y_{10} +x_4y_9 -x_9y_4 -x_{10}y_3 +cx_2y_{11}\) | |
| \(z_{13} = x_5y_9 -x_6y_{10} -x_9y_5 +x_{10}y_6 -cx_7y_{11}\) | |
| \(z_{14} = x_5y_{10} +x_6y_9 -x_9y_6 -x_{10}y_5 -cx_8y_{11}\) | |
| \(z_{15} = x_7y_9 +x_8y_{10} -x_9y_7 -x_{10}y_8 +cx_5y_{11}\) | |
| \(z_{16} = -x_7y_{10} +x_8y_9 -x_9y_8 +x_{10}y_7 +cx_6y_{11}\) |
Notice that \(z_{11}\) that has a term \(-cx_1y_{11}\), but \(z_{11}\) has a term \(cx_1y_{11}\) in the previous table. This sign difference means \(c=0\) that is a contradiction. This completes the proof. \(\square\)
Since every Hurwitz\(-\)Radon formula \([h,m,m]_\mathbb{R}\) or \([h,m,m]_\mathbb{Z}\) has a normed bilinear map, Theorem 1 implies the following:
Corollary 1. No \([10,10,16]_\mathbb{Z}\) formula is a restriction of any Hurwitz\(-\)Radon formula.
We proved Shapiro’s conjecture, hence, not every \([r,s,n]_\mathbb{R}\) or \([r,s,n]_\mathbb{Z}\) formula can be built from Hurwitz\(-\)Radon formulas by a process of restrictions and direct sums. The \([10,10,16]_\mathbb{Z}\) formula has a very interesting structure, it is not just the smallest formula that is not a restriction of any Hurwitz\(-\)Radon formulas but also the largest case for \(r*r = r\circ r\), where \(r\circ r\) is the smallest integer \(n\) such that there exists an \(r\times r\) intercalate matrix with \(n\) colors.
For a consistently signed intercalate matrix, a submatrix is also a consistently signed intercalate matrix and its normed bilinear map is a restriction of the original normed bilinear map. Hence, by Theorem 1, a consistently signed intercalate matrix of a Hurwitz\(-\)Radon formula \([h,m,m]_\mathbb{Z}\) has no submatrices of type \([10,10,16]_\mathbb{Z}\).
It is also not difficult to create more \([r,s,n]_\mathbb{Z}\) formulas with \(n-s>6\) that is not a restriction of any Hurwitz\(-\)Radon formula by extending \([10,10,16]_\mathbb{Z}\) formula. For example, the consistently signed intercalate matrix of a \([12,12,26]_\mathbb{Z}\) formula in [5, (7.2)] has a submatrix of type \([10,10,16]_\mathbb{Z}\), so it cannot be a restriction of any Hurwitz\(-\)Radon formula.
The author would like to thank Professor K. Y. Lam for his valuable comments and encouragement.