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CS-674-A1 · revision r1 · on CS-674

gpt 6 astra ultra attempt on CS-674

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. Differential-Privacy Workshop Framing
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Record scope and qualifications.
  5. Statement recorded in the supplied source.
  6. Source status.
  7. Short English statement
  8. Sources
  9. Preparation and review.
  10. Research attempt
  11. Literature, source context and scope
  12. Randomization is necessary even before computational questions
  13. Verification and remaining gaps
  14. References and source attribution
  15. Integration review.
  16. Completion Estimate
  17. Final status and attribution

Differential-Privacy Workshop Framing

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.

Fix a neighboring-dataset relation, such as changing one row. A randomized mechanism $M$ is $(\epsilon,\delta)$-differentially private if $\Pr[M(D)\in E]\le e^\epsilon\Pr[M(D^{\prime})\in E]+\delta$ for every neighboring $D,D^{\prime}$ and measurable output event $E$, where $\epsilon\ge0$ and $0\le\delta<1$. A statistical query $q:\mathcal X\to[0,1]$ has empirical answer $q(D)=n^{-1}\sum_{i=1}^n q(x_i)$. An $\alpha$-accurate release approximates the requested answers within $\alpha$; the query class and probability of success must also be specified. Complexity depends on dataset size, dimension, query access, and these privacy and accuracy parameters.

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 registry flags this extraction as unrecoverable. It is displayed as a literal source fragment; the definitions provide context rather than a replacement formal statement.

Source-record limitation: Source flags the statement as unrecoverable: The official report uses this broad complexity prompt as workshop framing and supplies no determinate claim distinct from the concrete high-dimensional query problem retained later in the same report. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this record as unrecoverable. Its extract and terminology are documented, but a complete original problem has not been reconstructed.

Statement recorded in the supplied source.
What is the computational complexity of problems arising in differential privacy, including privately releasing statistical queries with useful accuracy? [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

The supplied fragment concerns differential-Privacy Workshop Framing. Its complete mathematical target is not recovered.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30002234 (OWR-12177-001). 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 Complexity Theory $2013$, Oberwolfach Reports, DOI 10.4171/owr/2012/54, pp. 3267--3304 https://doi.org/10.4171/owr/2012/54.
  • 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 30002234 and code OWR-12177-001 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.

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.

Randomization is necessary even before computational questions

The source record is broad workshop framing and is retained as such. We test a concrete proposed simplification: use a deterministic release to avoid the complexity of randomization. Assume finite databases of a fixed size over a finite nonempty universe, with one-row replacement as adjacency and pure differential privacy at some finite $\varepsilon\ge0$.
Proposition: Every deterministic pure-private mechanism on this database space is constant. For one binary record, a randomized binary estimate has worst-case probability of being correct at most $e^\varepsilon/(1+e^\varepsilon)$, and randomized response attains this value.
Proof: If neighboring databases $D,D'$ had different deterministic outputs, the event consisting of the output on $D$ would have probability one on $D$ and zero on $D'$, contradicting the privacy inequality. Any two fixed-size databases are joined by changing their rows one at a time, so the output is constant throughout the space. For a binary estimate let $p$ be its worst-case correct probability. Then $\Pr(M(1)=1)\ge p$ and $\Pr(M(0)=1)\le1-p$. Privacy for the event $\{1\}$ gives $p\le e^\varepsilon(1-p)$, hence the upper bound. Randomized response outputs the true input with probability $a=e^\varepsilon/(1+e^\varepsilon)$ and its complement with probability $1-a$. For either singleton output, the likelihood ratio between the two inputs is either $a/(1-a)=e^\varepsilon$ or its reciprocal. Summing over outputs verifies privacy for every event. Both inputs have success probability $a$, proving attainment. ∎

Verification and remaining gaps

At $\varepsilon=\log2$, the exact transition matrix has correct probabilities $2/3$ and incorrect probabilities $1/3$; all output events were enumerated. The failed deterministic route cannot release a nonconstant statistic under these assumptions. This is a standard one-record tradeoff, not a complexity classification for useful statistical analysis. Approximate privacy, larger data, query classes, utility measures and computational encoding must still be specified for the original broad question.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2012/54. Relevant passages read: 2012/54 introductory framing and pp. 3278--3279, differential privacy and statistical-query motivation.

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. 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 batch019/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.