Fixed-diagonal geometry and an exact 4x4 two-paving profile

Definitions and mathematical statement

For an orthogonal projection $P=P^*=P^2\in M_n({\mathbb{C}})$ and a coordinate set $A$, let $Q_A$ be its diagonal coordinate projection. To fix the aid's convention, define for $1\le r\le n$ \[\pi_r(P)=\min_{\substack{\{1,\ldots,n\}=A_1\sqcup\cdots\sqcup A_r\\A_j\ne\varnothing}}\max_j\|Q_{A_j}PQ_{A_j}\|.\] Direct-compression $(r,c)$-pavability means $\pi_r(P)\le c$, while strict $r$-pavability means $\pi_r(P)<1$. This is not automatically the convention of paving the zero-diagonal matrix $P-\operatorname{diag}P$. The original does not specify a target constant or whether empty blocks are permitted.

Original request. Fix $r$ and find conditions on a projection's diagonal that guarantee $r$-paving, independently of the dimension, with interest in the geometry of the problem. A criterion for one specified projection, existence of some projection with a diagonal, and a guarantee for every projection with that diagonal are distinct possible targets; the source does not choose among them.

Attached unreviewed fixed-fiber theorem. In the complex rank-two fiber \[P\in M_4({\mathbb{C}}),\qquad\operatorname{diag}P=(1/2,1/2,1/2,1/2),\] the aid asserts \[\pi_2(P)=\frac12+\min\{|p_{12}|,|p_{13}|,|p_{14}|\}.\] Over this fiber it reports the exact range \[\left[\frac12,\frac12+\frac1{2\sqrt3}\right],\] with an explicit projection path realizing every value, although that path is not in the supplied summary. Every real projection in the same fiber is reported to have $\pi_2(P)=1/2$.

Consequently the existential optimal constant for this diagonal is $1/2$, whereas the universal worst-fiber constant is $1/2+1/(2\sqrt3)$. The aid says the diagonal therefore cannot decide whether an individual complex projection is $(2,3/4)$-pavable: the same diagonal admits both outcomes. This does not separate mere strict two-pavability, since every reported value is below one. The result is restricted to nonempty direct-compression blocks, $r=2$, $n=4$ and rank two; it is not a general dimension-independent classification.

The aid separately reports an exact rank-one characterization. For $P=vv^*$ with $\|v\|=1$ and diagonal entries $a_i=|v_i|^2$, compression on $A$ has norm $\sum_{i\in A}a_i$. Thus $(r,c)$-pavability is exactly the existence of $r$ nonempty bins partitioning the indices with each sum at most $c$, in the stated convention. This bin-packing condition is not claimed to reduce to a single largest-diagonal inequality or to give an efficient general algorithm. All assertions require independent review.

Short English statement

The source seeks dimension-independent diagonal paving conditions. The aid computes an exact complex $4$-by-$4$ two-paving fiber, separating individual quantitative pavability from existential and universal diagonal constants, while all examples remain strictly two-pavable. It also gives a diagonal bin-packing description in rank one.

Sources

• Local UnsolvedMath registry, permanent ID 20000566, source code catalogue entry; aim-analysis-notes.json, notes[97], tag question.
• AIM workshop Beyond Kadison-Singer: paving and consequences, source item b25, supplied Google document: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit. No proposer, definition or bibliography is supplied; no document was visited.
• Attached machine-generated research attempt 1: result partial_result, problem status at run open, rigor label proved, novelty confidence low. It is not peer reviewed and requires independent verification before citation. No authoritative reference accompanies the exact fiber or projection path.