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CS-367-A1 · revision r1 · on CS-367

gpt 6 astra ultra attempt on CS-367

Exploration · LLM-generated · gpt 6 astra ultra ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Square-Root Sparsity for Explicit RIP Matrices
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Statement recorded in the supplied source.
  5. Scope qualification.
  6. Source status.
  7. Short English statement
  8. Sources
  9. Preparation and review.
  10. Research attempt
  11. Literature, source context and scope
  12. The direct random-coherence route loses a logarithm
  13. A finite search is not the requested explicit construction
  14. Verification and remaining gaps
  15. References and source attribution
  16. Integration review.
  17. Completion Estimate
  18. Final status and attribution

Square-Root Sparsity for Explicit RIP Matrices

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.

A real $n\times p$ matrix $A$ has the restricted isometry property of order $s$ with distortion $\delta\in(0,1)$ if $(1-\delta)\|x\|_2^2\leq\|Ax\|_2^2\leq(1+\delta)\|x\|_2^2$ for every $x\in\mathbb R^p$ with at most $s$ nonzero coordinates. Here $n$ counts measurements and $p$ is the ambient dimension. The asymptotic question fixes a nontrivial distortion bound independently of $n,p$. An explicit construction produces the matrix deterministically with controlled construction time.

Statement recorded in the supplied source.
Does there exist an explicit construction of real-valued restricted-isometry matrices with sparsity $s\geq c\sqrt n$ in the polylogarithmic regime $n=(\log p)^{O(1)}$? [S2]
Scope qualification.
The source asks about the compressed regime $n<p$, with a fixed distortion below one. Sparsity must obey the feasibility restrictions of this regime; the supplied text does not state a range of exponents or a numerical distortion. Construction cost and representation precision must be stated in any answer.
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

Construct real restricted-isometry matrices reaching square-root sparsity using only polylogarithmically many measurements.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004126 (OWR-16931-010). 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 Combinatorics, Probability and Computing $2020$, Oberwolfach Reports, DOI 10.4171/owr/2019/19, pp. 1113--1189 https://doi.org/10.4171/owr/2019/19.
  • S3 Combinatorics, Probability and Computing, Oberwolfach Reports 19/2019, primary report PDF https://ems.press/content/serial-article-files/46798. 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 30004126 and source code OWR-16931-010 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.

The direct random-coherence route loses a logarithm

The primary report already discusses deterministic constructions close to the square-root scale and asks for a stronger polylogarithmic-regime result. We test a direct sign-matrix and exhaustive-search route. Let $n\ge1$, $p\ge2$ be integers and let $A$ have independent entries $\pm1/\sqrt n$ with equal probabilities. Its columns have norm one.
Proposition: With probability at least $1/2$, all distinct column inner products have magnitude at most \[ \mu=\sqrt{\frac{2\log(2p(p-1))}{n}}. \] Such a matrix has RIP distortion at most $(s-1)\mu$ at order $s$ whenever this expression is below one.
Proof: For a fixed pair of columns their inner product is $n^{-1}\sum_{j=1}^n X_j$, where the $X_j$ are independent uniform signs. The series comparison $(2k)!\ge2^k k!$ gives $\cosh t\le\exp(t^2/2)$. Thus Markov's inequality, independence and $t=\mu$ yield \[ \Pr\left\{n^{-1}\sum X_j>\mu\right\} \le\exp(-nt\mu)(\cosh t)^n\le\exp(-n\mu^2/2). \] The lower tail has the same bound. Summing over $p(p-1)/2$ pairs gives a failure probability at most $p(p-1)\exp(-n\mu^2/2)=1/2$. For a vector supported on at most $s$ columns, expansion of its squared output norm bounds the error by $\mu\sum_{i\ne j}|x_ix_j|\le\mu(s-1)\|x\|_2^2$, again by Cauchy--Schwarz. ∎
For fixed desired $0<\delta<1$, this proves only \[ s-1\le\delta\sqrt{\frac{n}{2\log(2p(p-1))}}. \] The proved guarantee is smaller than a fixed multiple of $\sqrt n$ by a factor growing as $\sqrt{\log p}$. This is a limitation of this elementary argument; it is not an upper bound on the achievable RIP order of random matrices.

A finite search is not the requested explicit construction

The probabilistic proof implies at least one acceptable sign matrix, so one may enumerate all $2^{np}$ sign patterns and compute every correlation. The verification can avoid irrational entries by using the integer unnormalized column dot products. This gives a finite deterministic search with an exponential worst-case bound, not a polynomial construction. The search space and the logarithmic loss remain even when $n=(\log p)^{O(1)}$. Thus substituting exhaustive search for randomness does not finish the target.

Verification and remaining gaps

Exact binomial probabilities for sums of 16 and 64 independent signs were compared with the two-tail bound, and the coherence inequality was checked on explicit sign matrices and sparse rational vectors. These checks are illustrative; the probability proof is the argument for arbitrary sizes. The source's more sophisticated explicit constructions are not superseded by this weaker baseline. No derandomization at the required sparsity or proof escaping the coherence loss is obtained.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2019/19. Relevant passages read: 2019/19 pp. 1132--1135, coherence constructions and the polylogarithmic square-root question.

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.