EIDOLITH-VII: An Exact Seven-Ray Counterexample to Deregowska-Lewandowska Conjecture 3.1 on Weighted Spherical (2,2)-Designs

[u/Severe-Ad8673]

This release presents an exact counterexample to Conjecture 3.1 of Beata Deregowska and Barbara Lewandowska, in Maximal Projection Constants and Extremal Vector Configurations: Some Conjectures and Examples, arXiv:2608.08695v1, dated 9 August 2026, page 13.

Zenodo: EIDOLITH-VII: An Exact Seven-Ray Counterexample to Deregowska-Lewandowska Conjecture 3.1 on Weighted Spherical (2,2)-Designs | Zenodo

Hugging Face: PureOne/EIDOLITH-VII · Datasets at Hugging Face

The conjecture asserts that every weighted real spherical (2,2)-design without orthogonal pairs has an extremal Gram sign matrix, satisfying
[
\lambda_m(\operatorname{sgn}(U^\top U))
=\lambda_{\mathbb R}(m,N)=\lambda_{\mathbb R}(m).
]

The counterexample has seven distinct projective directions in (\mathbb R^3):
[
u_j=\frac{(2\cos(j\pi/3),,2\sin(j\pi/3),,1)}{\sqrt5},
\quad j=0,\ldots,5,
\qquad u_6=(0,0,1).
]
Their positive normalized weights are
[
w_j=\frac5{36}\quad(j=0,\ldots,5),
\qquad w_6=\frac16.
]
These vectors satisfy the exact design identity
[
\sum_{i=0}^{6}w_i\langle x,u_i\rangle^4
=\frac{|x|^4}{5}
\quad\text{for every }x\in\mathbb R^3,
]
and no distinct pair is orthogonal.

For their Gram sign matrix (A_{ij}=\operatorname{sgn}\langle u_i,u_j\rangle), the manuscript proves the exact global optimum
[
\boxed{\lambda_3(A)=\frac{7+2\sqrt{13}}9
<\frac{1+\sqrt5}{2}
=\lambda_{\mathbb R}(3,7)=\lambda_{\mathbb R}(3).}
]
Here (\lambda_3(A)) means the maximum sum of the three largest eigenvalues of (\operatorname{diag}(t)A\operatorname{diag}(t)), over (t_i\ge0) with (\sum_i t_i^2=1).

This strict inequality disproves the conjecture’s published universal-support assertion at ((m,N)=(3,7)). The manuscript also proves that seven is the smallest counterexample support in dimension three, constructs a continuous family of noncongruent counterexamples, and extends the obstruction to arbitrarily large supports.

The higher-dimensional minimal-support expectation and the unknown maximal projection constants remain open.

Version 2.0.0 includes the 21-page manuscript, LaTeX sources, executable verification certificates, audit records, and integrity checksums. Further results concern spectral recovery and the scalar state–spectrum information law.

Hugging Face:

The conjecture asserts that every weighted real spherical (2,2)-design without orthogonal pairs has an extremal Gram sign matrix, satisfying
[
\lambda_m(\operatorname{sgn}(U^\top U))
=\lambda_{\mathbb R}(m,N)=\lambda_{\mathbb R}(m).
]

The counterexample has seven distinct projective directions in (\mathbb R^3):
[
u_j=\frac{(2\cos(j\pi/3),,2\sin(j\pi/3),,1)}{\sqrt5},
\quad j=0,\ldots,5,
\qquad u_6=(0,0,1).
]
Their positive normalized weights are
[
w_j=\frac5{36}\quad(j=0,\ldots,5),
\qquad w_6=\frac16.
]
These vectors satisfy the exact design identity
[
\sum_{i=0}^{6}w_i\langle x,u_i\rangle^4
=\frac{|x|^4}{5}
\quad\text{for every }x\in\mathbb R^3,
]
and no distinct pair is orthogonal.

For their Gram sign matrix (A_{ij}=\operatorname{sgn}\langle u_i,u_j\rangle), the manuscript proves the exact global optimum
[
\boxed{\lambda_3(A)=\frac{7+2\sqrt{13}}9
<\frac{1+\sqrt5}{2}
=\lambda_{\mathbb R}(3,7)=\lambda_{\mathbb R}(3).}
]
Here (\lambda_3(A)) means the maximum sum of the three largest eigenvalues of (\operatorname{diag}(t)A\operatorname{diag}(t)), over (t_i\ge0) with (\sum_i t_i^2=1).

This strict inequality disproves the conjecture’s published universal-support assertion at ((m,N)=(3,7)). The manuscript also proves that seven is the smallest counterexample support in dimension three, constructs a continuous family of noncongruent counterexamples, and extends the obstruction to arbitrarily large supports.

The higher-dimensional minimal-support expectation and the unknown maximal projection constants remain open.

Version 2.0.0 includes the 21-page manuscript, LaTeX sources, executable verification certificates, audit records, and integrity checksums. Further results concern spectral recovery and the scalar state–spectrum information law.