Nonexistence of the complex (8,3) ETF and a triangle-balance certificate

Definitions and mathematical statement

The supplied record concatenates two workshop questions. Problem 20 asks whether a complex equiangular tight frame of $N=8$ vectors in $\mathbb C^M$, $M=3$, exists. Problem 21 asks for a small-parameter catalogue indexed by $(N,M)$, recording existence, nonexistence or open status and the methods supporting settled entries. Its original examples are $(7,3)$ by difference sets and $(8,3)$ then described as open. The trailing "3" is pagination, not an additional mathematical condition.

A unit-norm complex ETF is a collection $v_1,\ldots,v_N\in\mathbb C^M$ such that \[ |\langle v_i,v_j\rangle|=\alpha\ (i\ne j),\qquad \sum_{i=1}^N v_iv_i^*=\frac NM I_M. \] For $N>M$, tightness forces $\alpha^2=(N-M)/(M(N-1))$. Equiangular lines or a spanning equiangular frame without tightness are different objects; the merged catalogue wording must specify which is intended.

The aid reports a negative answer to Problem 20: Szollosi proved nonexistence in 2014 by an exact Gröbner-basis calculation. It also reports a 2026 Singer--Zauner gap theorem excluding complex $d$-by-$n$ ETFs when \[ d^2-d+1<n<d^2, \] which excludes $(n,d)=(8,3)$. These are supplied dated literature claims, not independently checked here. The aid identifies Fickus--Mixon small-parameter tables and says they need a 2026 update. No complete catalogue or numerical range for "small" is provided. It separately reports eight equiangular lines in $\mathbb C^3$ that are not tight, so the ETF nonexistence cannot be applied to that broader interpretation.

For the reported exact certificate, let $S$ be an order-eight Hermitian Seidel matrix: $S_{ii}=0$ and $|S_{ij}|=1$ for $i\ne j$. Set \[ \alpha=\sqrt{5/21},\qquad c=\frac2{3\alpha},\qquad G=I+\alpha S. \] The aid gives the switching-invariant identity \[ \left\|G^2-\frac83G\right\|_F^2 =2\alpha^4\sum_{i<j} \left|\sum_{k\notin\{i,j\}}S_{ij}S_{jk}S_{ki}-c\right|^2, \] where $\|\cdot\|_F$ is the Frobenius norm. Thus global tightness is measured by local triangle-balance defects. After diagonal-unitary dephasing, write $S=\left(\begin{smallmatrix}0&\mathbf1^*\\\mathbf1&C\end{smallmatrix}\right)$. Exact vanishing is reported equivalent to the order-seven core conditions \[ C=C^*,\quad C_{ii}=0,\quad |C_{ij}|=1\ (i\ne j),\qquad C\mathbf1=c\mathbf1,\qquad C^2=7I+cC-J, \] where $J=\mathbf1\mathbf1^*$. The forced core spectrum is \[ \left\{c\text{ once},\quad\frac{\sqrt{105}}3\text{ twice}, \quad-\frac{\sqrt{105}}5\text{ four times}\right\}. \] This is a reported feasibility characterization, not a construction of such a core. The nonexistence conclusion requires the supplied external results or an additional contradiction, not just the displayed spectral list.

Short English statement

The merged source asks about the complex $(8,3)$ ETF and a small-parameter catalogue. The aid reports that the ETF does not exist, with 2014 and 2026 literature explanations, and provides a triangle-balance/Seidel-core certificate. It distinguishes this from eight equiangular lines without tightness, which the aid says do exist.

Sources

• Local source record: catalogue entry, catalogue entry; canonical aim-geometry-notes.json, notes[205], canonical source item 20. Title: "Nonexistence of the complex (8,3) ETF and a triangle-balance certificate."
• AIM list Open Problems in Frame Theory/Phase Retrieval, Problems 20 and 21, following the July--August 2013 workshop "Frame theory intersects geometry," source reference [1]: https://aimath.org/pastworkshops/frametheorygeomproblems.pdf. The supplied verification identifies the OCR concatenation on page 3. This draft uses the supplied text without accessing a PDF.
• Supplied literature attributions: Szollosi's 2014 exact Gröbner-basis nonexistence result; the 2026 Singer--Zauner gap theorem; Fickus--Mixon small-parameter tables. No full paper titles, identifiers or theorem numbers are supplied. The original $(7,3)$ example cites the difference-set method without a full reference.
• Attempt 1 supplies the Seidel-core characterization, triangle-defect identity and non-tight eight-line example. Registry dated 14 August 2026: "solved," "SOLVED-IN-LITERATURE," statement status "exact"; research-run status refuted, attempt partial, rigor "proved," novelty confidence low. The machine-generated, non-peer-reviewed aid requires independent verification before citation.