A mixed-discriminant reconstruction of the five-way spanning problem

Definitions and mathematical statement

The literal workshop note, associated with Mirko Visontai, asks about partitioning indexed vectors into five spanning sets and an identity-augmented positive-semidefinite matrix construction, but leaves the tested property as "xxxx." It does not define a well-posed algorithm or theorem. The following is the supplied aid's explicit reconstruction, not an assertion that the omitted property has been recovered from the original source.

Take $\mathbb F={\mathbb{R}}$ or ${\mathbb{C}}$, integers $m,n,r\geq1$, and indexed vectors $v_1,\ldots,v_m\in\mathbb F^n$. For $J\subset[m]=\{1,\ldots,m\}$, put $\operatorname{rank}(J)=\dim_{\mathbb F}\operatorname{span}\{v_j:j\in J\}$. The adjoint $v_i^*$ is transpose or conjugate transpose as appropriate. For $h$ positive-semidefinite $h$-by-$h$ matrices, fix the mixed-discriminant normalization \[ D(B_1,\ldots,B_h)=\frac1{h!} [t_1\cdots t_h]\det\!\left(\sum_{i=1}^h t_iB_i\right). \] The scalar normalization does not affect positivity.

Suppose $m\geq rn$, put $q=m-rn$, and define $m$-by-$m$ positive-semidefinite matrices \[ A_i=(I_r\otimes v_iv_i^*)\oplus I_q\qquad(1\leq i\leq m), \] where a zero-size identity block is omitted. The reconstructed theorem states the equivalence of:

1. $D(A_1,\ldots,A_m)>0$;
2. the indexed vector family contains $r$ pairwise disjoint bases of $\mathbb F^n$;
3. $[m]$ admits a partition into $r$ classes, each of whose vectors spans $\mathbb F^n$;
4. for every $J\subset[m]$, \[ |[m]\setminus J|\geq r\left(n-\operatorname{rank}(J)\right). \]

Taking $r=5$ models the five-way spanning question. Positivity here is a joint mixed-discriminant condition on the whole tuple, not an unspecified test on each individual matrix.

For the other side of the identity-padding dictionary, suppose $m\leq rn$ and set $B_i=I_r\otimes v_iv_i^*$, now of size $rn$. The supplied tuple-padding theorem is \[ D(B_1,\ldots,B_m,\underbrace{I_{rn},\ldots,I_{rn}}_{rn-m\ \mathrm{arguments}})>0 \] if and only if the indexed vectors partition into $r$ linearly independent classes, equivalently if and only if \[ |J|\leq r\operatorname{rank}(J)\qquad\text{for every }J\subset[m]. \] Empty classes are allowed in this independent-coloring statement. At $m=rn$, the two encodings coincide and the independent classes must be bases. For $m>rn$, the common identity block belongs inside every matrix; for $m<rn$, the identities are extra tuple arguments. These distinct operations must not be conflated. The aid also reports that literal uncolored repeated-vector constructions collapse to ordinary independence or spanning of the original family and cannot encode the intended multi-class condition; their exact uncolored matrix models are not included in the excerpt.

Short English statement

The source note omits the matrix property and construction needed to define its proposed algorithm. The supplied reconstruction replaces this gap with a precise mixed-discriminant criterion for five disjoint spanning classes, and a separate identity-padding criterion for independent classes. These are repaired formulations rather than a verification of the literal note.

Sources

• Local registry: permanent UnsolvedMath ID $20000562$, exact public code catalogue entry; aim-analysis-notes.json, notes[93].
• AIM workshop Beyond Kadison--Singer: paving and consequences, item b21, note ending "Mirko Visontai": https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit. The original note refers to Rado--Horn but provides no complete citation.
• Supplied research summary, "A mixed-discriminant reconstruction of the five-way spanning problem," attempt 1. No authoritative external citation for the mixed-discriminant equivalences or claimed reconstruction is supplied; the original remarks and literature fields are empty.