A spectral multiplicity refinement of the Casazza DFT near-counterexample

Definitions and mathematical statement

The original bullet asks whether "Pete's example" can be improved, saying it affects the equivalences and is almost a counterexample. The source does not define Pete, the example, target inequality, or meaning of better. Its verification says neighboring bullets discuss matching quantitative lower bounds, an order-$\varepsilon^{-2}$ real-paving bound, and a constructive bound of "Pete C." No unique original mathematical problem is recoverable.

The report explicitly adopts one interpretation using the Casazza--Fickus--Mixon--Tremain two-block DFT projection. For that particular construction, it proposes that every first-block principal compression indexed by $S$, $|S|=s\geq n$, has \[ P[S]=\frac1{n+1}I+K_S,\qquad K_S\geq0,\qquad\operatorname{rank}K_S=n-1. \] Consequently its eigenvalue $1/(n+1)$ has exact multiplicity $s-n+1$. The defining DFT matrices, projection normalization, complete index range, and admissible $n$ are not supplied in the summary, so this is not a statement about arbitrary projections.

For every bipartition, the report asserts that the two complementary-projection compression spectra have together at least two eigenvalues at least $n/(n+1)$. In the natural paving interpretation these are the two block compressions of $I-P$; the full index convention is not printed. It further says one block of the associated zero-diagonal reflection has norm at least \[ \frac{n-1}{n+1}. \] A reflection is represented by $2P-I$, up to overall sign, in the constant-half-diagonal projection convention. These normalization qualifications are essential to the claimed spectral consequence.

The report additionally claims a Weyl-stability corollary and an exact even/odd replication no-amplification lemma, but gives neither their hypotheses nor formulas. It separately identifies a conference-matrix reading of the ambiguous workshop remark and derives its known order-$\varepsilon^{-2}$ paving lower bound. This alternative reading is not identified as the original example. The results are candidate refinements of specified surrogate constructions, not a recovery or resolution of the undefined "make it better" request.

Short English statement

The workshop example is unidentified. Under an explicit DFT-projection interpretation, the report proposes an exact small-eigenvalue multiplicity and consequent near-unpaved modes, while retaining a separate conference-matrix interpretation and missing construction details.

Sources

• AIM workshop "Beyond Kadison--Singer: paving and consequences," December 1--5, 2014, item b10: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit. The supplied verification places the bullet under a list headed by Petter Brändén; the source document was not accessed here.
• Local canonical registry aim-analysis-notes.json, notes[82], tag question; permanent UnsolvedMath ID $20000551$, source code catalogue entry. The original only refers to Pete's example. The verification gives neighboring "Pete C." and quantitative-paving context, but explicitly says no unique statement is recoverable. Original remarks and literature field are empty; the clean statement is absent.
• Supplied research attempt 1 adopts the Casazza--Fickus--Mixon--Tremain DFT construction and proposes spectral refinements. No authoritative full references, matrices, or exact replication/stability statements are supplied. The report is machine generated, not peer reviewed, and requires independent verification before citation.