Two-Sided Paving Bounds

Definitions and mathematical statement

The supplied record asks about a reduction from partitioning to paving, but does not specify the objects being partitioned, the admissible matrices or operators, their normalization, or the desired spectral inequalities. Consequently, it does not determine a fully quantified mathematical conjecture.

To describe the missing data, a finite partition of an index set $I$ consists of nonempty, pairwise disjoint subsets $S_1,\ldots,S_r$ whose union is $I$. If a paving construction associates a block $B_j$ to each $S_j$, and each block has smallest and largest eigenvalues, simultaneous two-sided control would require inequalities of the form \[ L_j(\varepsilon)\leq \lambda_{\min}(B_j) \leq \lambda_{\max}(B_j)\leq U_j(\varepsilon), \qquad 1\leq j\leq r, \] for the same partition. Here $B_j$, $L_j(\varepsilon)$, and $U_j(\varepsilon)$ are placeholders for data absent from the record; these inequalities are explanatory notation, not a reconstruction of the intended theorem. In particular, no matrix class, dimension, normalization, or spectral center is being assumed.

Source question, with unresolved qualifications. Can the unspecified reduction from partitioning to paving be improved so that a single partition controls both the largest and smallest eigenvalues of its blocks, improving the block-count dependence on $\varepsilon$ from the supplied claimed bound $O(\varepsilon^{-4})$ toward the supplied claimed lower bound $\Omega(\varepsilon^{-2})$? The original wording asks whether the "cheating" used for one-sided bounds can be suspended to obtain both sides simultaneously; that procedure is not defined in the supplied record.

The asymptotic comparison concerns small positive $\varepsilon$, but its admissible range and the required uniformity of constants over the intended inputs are unspecified. The word "toward" does not prescribe a particular improved exponent or demand an $O(\varepsilon^{-2})$ upper bound. The two asymptotic bounds are reported source claims, not independently established facts in this draft.

Short English statement

The question asks whether one partition can simultaneously control both spectral extremes and improve the reported $O(\varepsilon^{-4})$ number of paving blocks toward the reported $\Omega(\varepsilon^{-2})$ lower bound. The supplied record omits the operator class, normalization, spectral targets, and reduction, so a precise quantified conjecture cannot yet be recovered.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002849$, title "Two-Sided Paving Bounds," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report citation: Arbeitsgemeinschaft: The Kadison-Singer Conjecture (2016), Oberwolfach Reports, pp. 927--969, DOI 10.4171/owr/2015/17. Supplied source URL and literature-source link, labeled "original OWR report": https://doi.org/10.4171/owr/2015/17. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature assessment, checked on 2026-08-21: the extracted phrase about suspending the "cheat" does not define a precise theorem or conjecture, and the curation identifies no verifiable resolution of the implied two-sided paving question. The supplied status is "open." This is supplied dated curation, not independently verified literature assessment; its time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and reports that the official passage was checked directly and its mathematical question, definitions, and notation restored without strengthening the claim. This is a supplied verification assertion; this draft has not visited or independently checked the report, and the complete record supplied here still lacks the necessary formulation data.