Exact lower witnesses in the complete-edge Weaver family

Definitions and mathematical statement

Original workshop question. The note asks for constructive lower witnesses or sharpness information in the random MSS/Weaver setting, says Weaver has a construction requiring $\sqrt\varepsilon$ dependence, and contrasts a claimed absence of an MSS constructive lower bound with the statement that "Pete C." has one. It does not identify that person, the exact conjecture, normalization or requested optimal constant; the competing informal assertions remain attributed workshop context.

Attached unreviewed complete-edge family. For $n\ge2$, let $H_n=\{x\in{\mathbb{R}}^n:\sum_i x_i=0\}$ and take the edge vectors \[v_{ij}=\frac{e_i-e_j}{\sqrt n}\in H_n\quad(1\le i<j\le n).\] They have squared norm $\alpha_n=2/n$ and form a Parseval frame: $\sum_{i<j}v_{ij}v_{ij}^{\mathsf T}=I_{H_n}$. For signs $\sigma_{ij}\in\{\pm1\}$ write \[D_\sigma=\sum_{i<j}\sigma_{ij}v_{ij}v_{ij}^{\mathsf T},\qquad W(V_n)=\min_\sigma\|D_\sigma\|_{H_n\to H_n}.\] The aid asserts $\|D_\sigma\|\ge1/\sqrt n$ for every signing, with the stronger $1/\sqrt{n-1}$ bound for even $n$.

Let $A_\sigma$ be the symmetric $n$-by-$n$ matrix with zero diagonal and off-diagonal entries $\sigma_{ij}$. The claimed equality criterion for the first lower bound is that it be the core of a normalized symmetric conference matrix: \[C_\sigma=\begin{pmatrix}0&\mathbf1^{\mathsf T}\\\mathbf1&A_\sigma\end{pmatrix},\qquad C_\sigma^2=nI_{n+1}.\] A conference matrix here has zero diagonal and off-diagonal signs; normalization fixes its first row and column to $+1$ off the diagonal.

For every prime power $q\equiv1\pmod4$, the aid gives a Paley signing, indexing coordinates by ${\mathbb{F}}_q$ and using the quadratic character $\chi$ through $\sigma_{xy}=\chi(x-y)$ for $x\ne y$. Its reported exact values are \[W(V_q)=\frac1{\sqrt q}=\sqrt{\alpha_q/2},\qquad\alpha_q=2/q.\] For a two-partition into edge sets $E_+,E_-$, the block operators sum to $I_{H_q}$, and the asserted optimal larger block norm is \[\min_{E_+\sqcup E_-=E(K_q)}\max\left\{\left\|\sum_{e\in E_+}v_ev_e^{\mathsf T}\right\|,\left\|\sum_{e\in E_-}v_ev_e^{\mathsf T}\right\|\right\}=\frac12+\frac1{2\sqrt q}.\] This is an infinite explicit square-root lower-witness family in the specified Parseval/sign-discrepancy normalization. It does not identify the best universal Weaver constant, resolve the note's unspecified random-MSS comparison, or verify the "Pete C." attribution. All claims require independent review.

Short English statement

The informal source asks for constructive sharpness witnesses in MSS/Weaver problems. The aid supplies a normalized complete-edge frame, universal signing lower bounds and an exact conference-matrix equality condition. Paley signings give an explicit infinite family attaining square-root discrepancy, with an exact two-partition block norm.

Sources

• Local UnsolvedMath registry, permanent ID 20000550, source code catalogue entry; aim-analysis-notes.json, notes[81], tag question.
• AIM workshop Beyond Kadison-Singer: paving and consequences, source item b9, supplied Google document: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit. The note names Weaver, MSS and "Pete C." without full references. No document was visited and the abbreviated attribution is not expanded.
• Attached machine-generated research attempt 1 invokes Paley signings without a bibliographic locator. Result partial_result, status partially_solved, rigor label proved, novelty confidence low. It is not peer reviewed and requires independent verification before citation. Original literature field empty.