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CS-370 · computer science · mathematics · Computational algebra and number theory

Exact Higher-Order Matrix Decomposability

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

For $S\subseteq\{1,\ldots,n\}$ with $|S|=k$, let $E_S$ be the $n\times k$ matrix selecting the standard coordinate vectors indexed by $S$. An exact positive-semidefinite principal-block decomposition is an identity $Q=\sum_{|S|=k}E_S\Pi_SE_S^{\mathsf T}$ with real symmetric $\Pi_S\succeq0$. Exact means there is no remaining positive-semidefinite residual. The identity necessarily makes $Q$ real symmetric and positive semidefinite.

Statement recorded in the supplied source.
Given a matrix $Q$ and an integer $k>2$, find necessary and sufficient conditions for $Q$ to admit an exact $k\times k$ decomposition into positive-semidefinite principal-block contributions. [S2]
Source status.
The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Characterize which positive-semidefinite matrices can be assembled exactly from positive-semidefinite blocks supported on $k$ coordinates.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004154 (OWR-16938-002). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/26, pp. 1573--1637 https://doi.org/10.4171/owr/2019/26.
  • S3 Mixed-integer Nonlinear Optimization, Oberwolfach Reports 26/2019, primary report PDF https://ems.press/content/serial-article-files/46805. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30004154 and source code OWR-16938-002 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-370-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-370

Exact Higher-Order Matrix Decomposability

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

For $S\subseteq\{1,\ldots,n\}$ with $|S|=k$, let $E_S$ be the $n\times k$ matrix selecting the standard coordinate vectors indexed by $S$. An exact positive-semidefinite principal-block decomposition is an identity $Q=\sum_{|S|=k}E_S\Pi_SE_S^{\mathsf T}$ with real symmetric $\Pi_S\succeq0$. Exact means there is no remaining positive-semidefinite residual. The identity necessarily makes $Q$ real symmetric and positive semidefinite.

Statement recorded in the supplied source.
Given a matrix $Q$ and an integer $k>2$, find necessary and sufficient conditions for $Q$ to admit an exact $k\times k$ decomposition into positive-semidefinite principal-block contributions. [S2]
Source status.
The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Characterize which positive-semidefinite matrices can be assembled exactly from positive-semidefinite blocks supported on $k$ coordinates.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004154 (OWR-16938-002). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics $2020$, Oberwolfach Reports, DOI 10.4171/owr/2019/26, pp. 1573--1637 https://doi.org/10.4171/owr/2019/26.
  • S3 Mixed-integer Nonlinear Optimization, Oberwolfach Reports 26/2019, primary report PDF https://ems.press/content/serial-article-files/46805. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30004154 and source code OWR-16938-002 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems. Codex (GPT-6; exact serving revision not exposed) supplied the auxiliary proof route used here. OpenAI gpt-6-astra, reasoning ultra, independently worked through its mathematical steps, wrote this exposition, and performed the finite checks. This is model collaboration, not independent human verification.

An exact threshold on a symmetric matrix family

The source already formulates decomposition as an SDP and asks for structural conditions beyond its two-by-two characterization. The following calculation supplies a complete restricted family, not a general characterization. Let $2\le k<n$ be integers, $\lambda\ge0$, $J_n$ the all-ones matrix, and $Q=I_n+\lambda J_n$. A $k$-principal PSD decomposition means a sum of PSD matrices, each supported on a set of at most $k$ coordinates; smaller supports can be padded to size $k$.
Proposition: The matrix $Q$ has such a decomposition if and only if \[ \lambda\le\frac{k-1}{n-k}. \]
Proof: Write $a=\binom{n-1}{k-1}$ and $b=\binom{n-2}{k-2}$. For each $k$-subset $S$, embed in those coordinates the block $cI_k+dJ_k$, where \[ d=\lambda/b,\qquad c=(1+\lambda)/a-\lambda/b. \] Each coordinate occurs in $a$ subsets and each distinct pair in $b$, so the sum of these blocks has diagonal $a(c+d)=1+\lambda$ and off-diagonal $bd=\lambda$. It is exactly $Q$. Because $d\ge0$, a block is PSD if $c\ge0$. Using $a/b=(n-1)/(k-1)$, that condition is precisely $\lambda(n-k)\le k-1$. For necessity take $Z=I_n-J_n/k$. On a coordinate set $S$ of size at most $k$, Cauchy--Schwarz gives \[ x^TZ_{SS}x=\|x\|^2-k^{-1}(\mathbf1^Tx)^2\ge(1-|S|/k)\|x\|^2\ge0. \] Hence $\operatorname{tr}(ZR)\ge0$ for every PSD summand $R$ on such a support: diagonalize $R$ as a sum of nonnegative multiples of $vv^T$ to see each trace term is nonnegative. Any proposed decomposition therefore requires \[ 0\le\operatorname{tr}(ZQ)=\frac{n(k-1)}k-\frac{\lambda n(n-k)}k, \] which is the same threshold. ∎
For $k=n$, one block suffices for every PSD $Q$, and the displayed threshold with denominator $n-k$ is not used. For $k<n$, even the positive definiteness of $I+\lambda J$ does not guarantee a decomposition: every $\lambda$ above the threshold is a counterexample. The attempted route "positive definite, therefore split into small PSD blocks" thus fails on a fully explicit family.

Verification and remaining gaps

For $4\le n\le8$ and $2\le k<n$, rational blocks were reconstructed below and at the threshold, and the dual trace was checked to be negative above it. This exact family neither produces structural necessary-and-sufficient conditions for arbitrary $Q$ nor improves on the source's generic SDP reformulation. The credited collaborative proof is an auxiliary calculation and is not counted as resolution of the original question.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2019/26. Relevant passages read: 2019/26 pp. 1594--1596, exact two-by-two characterization, existing SDP formulation and extension to k>2.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, reviewed the written mathematical arguments, and contributed formatting/repository review. This additional model review is not independent human verification. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch022/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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