gpt 6 astra ultra attempt on CS-706

Research-attempt provenance

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

For a facility set $X$, each site pays its minimum cost to a facility in $X$. The ordered-median objective sorts those site costs and weights their ranks by the given $\lambda_k$.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The supplied text defines an optimization problem but does not identify an unresolved theorem or a requested complexity/approximation result.

Source-record limitation: The supplied assessment flags this as an optimization-problem definition, not an open statement or complexity conjecture.

Statement recorded in the supplied source.

Given $n$ sites, a nonnegative cost matrix $C=(c_{j\ell})$, and $p<n$ facilities, choose $X\subseteq\{1,\ldots,n\}$ with $|X|=p$ to minimize the ordered weighted cost $\sum_{k=1}^n\lambda_k c_{\sigma_X(k)}(X)$, where $c_j(X)=\min_{\ell\in X}c_{j\ell}$ and $\sigma_X$ sorts these costs. [S2]

Source status.

The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Choose the stated number of facilities to minimize ordered site costs.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30003653 (OWR-15957-006). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Copositivity and Complete Positivity $2018$, Oberwolfach Reports, DOI 10.4171/owr/2017/52, pp. 3071--3120 https://doi.org/10.4171/owr/2017/52.
• Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.

Preparation and review.

The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30003653 and code OWR-15957-006 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Literature and context check

Puerto's primary OWR 52/2017 talk, pp. 3103--3105, was read. It defines nondecreasing ordering of costs, nonnegative rank weights, and free self-service $c_{jj}=0$, and presents an existing completely-positive reformulation. The imported record is a problem definition rather than a specified unresolved complexity theorem. The following direct algorithm and restricted linearization are known elementary routes rederived here, not a claimed resolution of a new open theorem.

Research attempt: Exact finite search and a conditional ordered-cost linearization

Finite exact algorithm with all order weights retained.

Assume $1\le p<n$ and rational nonnegative costs and weights. The minimum over an empty facility set is not defined by the source, so $p=0$ would need a separate convention. For each of the $\binom np$ subsets $X$, compute the $n$ minima $c_j(X)$ using $O(np)$ comparisons, sort them using $O(n\log n)$ comparisons, and evaluate their weighted sum. Return the least value and its set. This is exact because every admissible set is examined. Ties in sorted costs do not affect the sum: permuting equal values leaves every rank contribution unchanged. The total arithmetic/comparison count is \[ O\left(\binom np(np+n\log n)\right). \] It is polynomial for fixed $p$, in particular $O(n^2\log n)$ for $p=1$, but not a polynomial guarantee when $p$ is part of the input. Rational bit costs also depend on the supplied coefficient lengths.

Weights cannot be discarded. With $p=1$ and \[ C=\begin{pmatrix}0&3&9&9\\1&0&9&9\\1&3&0&9\\5&3&9&0\end{pmatrix}, \] the first two facility columns have sorted costs $(0,1,1,5)$ and $(0,3,3,3)$. Equal weights choose the first column, of total cost $7$ rather than $9$; weights $(0,0,0,1)$ choose the second, of maximum cost $3$ rather than $5$. The other columns have total cost $27$ and maximum $9$. Conversely, with free self-service and weights $(1,0,\ldots,0)$, every nonempty facility set has objective zero, since its smallest service cost is zero.

A convex sorting formula under an additional hypothesis.

For a vector $z$ let $z_{(1)}\le\cdots\le z_{(n)}$. If $0\le\lambda_1\le\cdots\le\lambda_n$, then \[ \sum_{i=1}^n\lambda_i z_{(i)} =\lambda_1\sum_jz_j+ \sum_{r=2}^n(\lambda_r-\lambda_{r-1})\sum_{i=r}^n z_{(i)}. \] This follows by collecting the coefficient of each sorted entry. The sum of the largest $L$ entries, for $1\le L\le n$, equals \[ \min_{t\in\mathbb R}\left(Lt+\sum_j(z_j-t)_+\right). \] For any $t$, the sum over the largest $L$ entries is at most the displayed expression, by bounding each $z_j$ in that set by $t+(z_j-t)_+$. Equality holds when $t$ lies between the largest excluded and smallest included values, with $t\le z_{(1)}$ when $L=n$. Hence epigraph variables $u_j\ge z_j-t$, $u_j\ge0$ give a linear formulation of each top-$L$ sum. Nonnegative weight differences allow the minima to be combined independently.

This can be combined with binary facility variables $x_\ell$, $\sum_\ell x_\ell=p$, and continuous assignment variables $a_{j\ell}\ge0$, $\sum_\ell a_{j\ell}=1$, $a_{j\ell}\le x_\ell$, setting $z_j=\sum_\ell c_{j\ell}a_{j\ell}$. For fixed open facilities every such $z_j$ is at least its nearest-facility cost. Every order statistic is coordinatewise nondecreasing: if $z\le z'$ coordinatewise, the number of entries below any threshold for $z'$ is at most the corresponding number for $z$, implying $z_{(i)}\le z'_{(i)}$. With nonnegative $\lambda$, replacing assignments by nearest facilities therefore cannot increase the objective. Thus this mixed-integer linear route is exact for the additional nondecreasing-weight case.

It does not make the original facility problem convex or eliminate its binary search. Moreover the source allows nonnegative weights without ordering them. For $\lambda=(1,0)$ the ordered objective is $\min(z_1,z_2)$, which is not convex: it is zero at $(0,2)$ and $(2,0)$ but one at their midpoint. Negative differences in the displayed decomposition invalidate the independent epigraph-minimization argument. This is the precise gap in extending that convenient formulation to arbitrary source weights.

Verification and adversarial audit

The finite search, ties, fixed-$p$ count, zero-objective self-service boundary and explicit four-site example were checked. The top-$L$ formula was proved with both inequality and attaining threshold. The assignment relaxation is justified by coordinatewise monotonicity, while the stronger nondecreasing-weight condition is imposed only for the convex sorting representation.

Audit corrections.

Recovered nonnegative weights and free self-service from the primary passage. Kept the nondecreasing-weight linearization separate from arbitrary nonnegative rank weights, and exposed the nonconvex two-cost counterexample.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The source record does not identify a requested unresolved theorem, approximation ratio or complexity target. Exact enumeration and a restricted standard formulation therefore do not amount to a new complete solution. No general efficient algorithm for variable $p$ or arbitrary ordered weights is produced, and the source's completely-positive formulation is not re-proved.

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% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://ems.press/content/serial-article-files/46715 (consulted 2026-10-11).

Model and contributor disclosure

OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration, repository-format review, and written-proof review of this attempt; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.