Explicit Deterministic Restricted Isometry Matrices

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.

A matrix $\Phi$ has restricted-isometry constant at most $\delta_s$ if $(1-\delta_s)\lVert x\rVert_2^2\leq\lVert\Phi x\rVert_2^2\leq(1+\delta_s)\lVert x\rVert_2^2$ for every vector with at most $s$ nonzero coordinates. Compressed sensing recovers structured vectors from measurements $\Phi x$.

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 acceptable isometry constant, measurement count, and sparsity tradeoff are not quantified in the supplied statement.

Source-record limitation: Small isometry constants and few measurements are not quantified, so the required construction regime cannot be recovered.

Statement recorded in the supplied source.

Can one construct deterministic explicit measurement matrices with small restricted isometry constants when the number of measurements is small, or equivalently when the sparsity level is large? [S2]

Source status.

The supplied catalogue labels this source record partially solved; 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

Explicitly build deterministic measurement matrices preserving the lengths of sparse vectors using few measurements.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30001526 (OWR-4414-001). 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 Wavelet and Multiscale Methods $2011$, Oberwolfach Reports, DOI 10.4171/owr/2010/33, pp. 1953--2025 https://doi.org/10.4171/owr/2010/33.
• S3 Bourgain--Dilworth--Ford--Konyagin--Kutzarova, Explicit matrices with RIP https://arxiv.org/abs/1403.3427.
• S4 Deterministic sensing matrices in compressive sensing, survey https://pmc.ncbi.nlm.nih.gov/articles/PMC3836388/.
• 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 30001526 and code OWR-4414-001 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

The primary report's compressed-sensing discussion asks for deterministic small restricted-isometry constants in compressed measurement regimes. The target does not fix a numerical optimal tradeoff. This attempt rederives an explicit finite-field graph construction and then audits the limitation of the coherence proof used to certify it. It does not claim that the classical square-root-scale construction solves the larger-sparsity agenda.

Research attempt: An explicit polynomial graph matrix and the coherence barrier

An explicit deterministic family

Let $q$ be prime and $1\leq r<q$. Index rows by $(x,y)\in\mathbb F_q^2$ and columns by the $q^{r+1}$ polynomials $p$ of degree at most $r$ over $\mathbb F_q$. Define \[ A_{(x,y),p}=\begin{cases}q^{-1/2}&y=p(x),\\0&\text{otherwise.}\end{cases} \] Each column has exactly $q$ nonzero entries and norm one. Distinct polynomials agree at at most $r$ field points, so their inner product is at most $r/q$. For completeness, the root bound follows by induction on degree: a root $a$ gives polynomial division by $X-a$, and every other distinct root is a root of the quotient. Thus $m=q^2$ measurements and $N=q^{r+1}$ columns have coherence $\mu\leq r/q$ and are explicitly listed by field evaluation.

For an $s$-sparse complex coefficient vector $z$, expansion of its squared image norm gives \[ \left|\lVert Az\rVert_2^2-\lVert z\rVert_2^2\right| \leq\mu\sum_{i\ne j}|z_i||z_j| =\mu\left(\left(\sum_i|z_i|\right)^2-\sum_i|z_i|^2\right) \leq\mu(s-1)\lVert z\rVert_2^2. \] The last inequality is Cauchy's inequality on at most $s$ nonzero entries. In particular the restricted isometry constant obeys $\delta_s\leq(s-1)r/q$. With $r=2$ and $q>2$ this has $N=q^3>m=q^2$ and certifies $\delta_s\leq\delta$ for $s\leq1+\delta q/2$. This is a genuine compressed family, but its certified sparsity grows only like $\sqrt m$ for fixed $\delta$.

Why this certificate cannot by itself pass the square-root scale

For any $m\times N$ matrix of unit columns with $N>m$, its Gram matrix $G=A^*A$ has trace $N$, at most $m$ nonzero nonnegative eigenvalues, and \[ \operatorname{tr}(G^2)\geq N^2/m \] by Cauchy's inequality on those eigenvalues. Its diagonal contributes $N$, and the $N(N-1)$ off-diagonal squared absolute entries are each at most $\mu^2$. Therefore \[ \mu^2\geq\frac{N-m}{m(N-1)}. \] Any proof using only the sufficient condition $(s-1)\mu\leq\delta$ must consequently have \[ s\leq1+\delta\sqrt{\frac{m(N-1)}{N-m}}. \] When $N/m$ is bounded away from one, this limits that proof method to a square-root order. It is a barrier to this certificate, not a lower bound on the actual restricted isometry of all deterministic matrices.

To see concretely that the coherence estimate can be loose, take unit columns $a_1=e_1$, $a_2=\mu e_1+\sqrt{1-\mu^2}e_2$, and $a_i=e_i$ for $i\geq3$, with $0<\mu<1$. Their Gram matrix has just one nonzero off-diagonal pair. Any support containing the first two columns has extremal eigenvalues $1\pm\mu$, and every other support is orthonormal. Thus $\delta_s=\mu$ for every $2\leq s\leq N$, whereas the generic certificate is $(s-1)\mu$. This example is square, not a proposed compressed solution; it disproves only the inference that the coherence bound must be sharp.

Verification and adversarial audit

The construction requires $r<q$, so distinct coefficient polynomials cannot coincide as functions on all field points. The complex-vector proof uses absolute values and is not restricted to nonnegative coefficients. The rank-based bound assumes $N>m$ and only constrains the coherence sufficient condition. The final example explicitly separates an analysis limitation from a construction meeting the full compressed target.

Audit corrections.

Retained the distinction between a certified restricted-isometry upper bound and the true constant; did not market a square-root-scale construction as an optimal deterministic solution.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The deterministic larger-sparsity or near-optimal measurement regime remains unresolved. This manuscript supplies a complete explicit known-style construction at a limited parameter range and proves why its elementary coherence analysis cannot establish the desired stronger tradeoff.

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/46289 (consulted 2026-10-11).

Model and contributor disclosure

The parent Codex GPT-6 model has reviewed the written mathematical arguments at their declared scopes. This is model-assisted review, not independent human verification. 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 and repository-format review; 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.